Open Access
Add to BibSonomyAbstract
Intrinsic mode coupling in a graded-index plastic optical fiber (GI POF) is investigated using the developed coupled power theory for a GI POF with a microscopic heterogeneous core. The results showed that the intrinsic material properties can induce random power transitions between all the guided modes, whereas the structural deformation of microbending results in nearest-neighbor coupling. It was numerically demonstrated that efficient group-delay averaging due to intrinsic mode coupling brings the pronounced bandwidth enhancement in fibers with much shorter length than the case of glass multimode fibers.
©2013 Optical Society of America
1. Introduction
A flexible and easily handled graded-index plastic optical fiber (GI POF) is a promising candidate for transmission cable in short-reach communication networks. The bandwidths have expanded to achieve 40 Gbps transmission over a 100-m GI POF through the development of low-dispersive materials and GI profile control techniques [1]. It has also been reported that the transmission characteristics of GI POFs are significantly affected by much stronger mode coupling than glass multimode fibers (MMFs) [2–5] whereas the underlying physics are not well understood. However, this suggests that the GI POF has mode coupling origins other than microbending, which is a predominant factor for mode couplings in glass MMFs [6]. Therefore, the mode coupling in the GI POF cannot be analyzed based on conventional diffusion theory of coupled power equation [7].
Optical fiber materials have density fluctuations because of their amorphous nature. In glass MMFs, these fluctuations cause Rayleigh scattering because the refractive index varies at much smaller length scales than the guided light wavelengths. Rayleigh scattering induces random mode coupling, but the effect is too weak to influence the transmission bandwidth of glass MMFs with lengths of less than several hundred meters. However, polymers such as poly(methyl methacrylate) (PMMA) have microscopic heterogeneous structures with larger length-scale fluctuations [8,9]. Recently, we have developed a coupled power equation that considers microscopic heterogeneities, which proves stronger mode coupling due to greater forward scattering in GI POFs than in glass MMFs [10]. In this paper, we investigate the influence of intrinsic mode coupling on optical pulse transmissions through the GI POF based on analyses using the developed coupled power theory. It is clarified that intrinsic mode coupling can induce more efficient group delay averaging than glass MMFs.
2. Intrinsic mode coupling
2.1 Developed coupled power equation
Light scattering by microscopic heterogeneities depends on the spatial correlation properties of the dielectric constant fluctuation δε [11]. For Gaussian-correlated heterogeneities, the autocorrelation function is given by:
where the mean square fluctuation 〈δε2〉 and the correlation length D are measures of fluctuation amplitude and scale, respectively. The angular dependence of the scattered light intensity can be obtained by a Fourier transform of the correlation function under the Rayleigh−Debye approximation. Therefore, the scattering profile is predominantly determined by the fluctuation scale or the correlation length, whereas the fluctuation amplitude controls the scattering efficiency. Thus the larger-scale fluctuations in the GI POF materials result in more forward scattering than in glass MMF materials, which is one of main reasons for much higher attenuations in the GI POF. Because of the intrinsic forward scattering, we can observe the streak of the light trajectory in the GI preform as shown in Fig. 1. The microscopic heterogeneities in bulk polymers have been well studied, and as a result, their correlation lengths can be sufficiently reduced [8,9]. However, the microscopic heterogeneities in actual GI POF cores are not well understood, because the correlation characteristics can be influenced by the fabrication process.
Fig. 1 Visually observable light trajectory due to forward scattering by microscopic heterogeneous structures in PMMA-based GI preform.
Mode couplings due to random perturbations in MMFs can be statistically analyzed with the coupled power equation [12]:
where hij is the power coupling coefficient from mode j to mode i, τi is the group delay per unit length for mode i, αi is the attenuation coefficient for mode i, N is the total mode number, and Pi(j) is the ensemble-averaged mode power over slightly different MMFs with statistically similar perturbations. In the original derivation of the coupling coefficient, directional perturbations along the fiber axis (z axis) have been assumed because the core diameter variations and fiber axis fluctuations could be the dominant coupling origins in glass MMFs. However, recently we have shown that the microscopic heterogeneities in POF core materials are basically isotropic perturbations, and the corresponding coupling coefficient have the following form [10]:whereHere ω is the angular frequency of the guided light, Δβ is the propagation constant difference, and Ei(j) is the transverse electric field vector for guided mode i(j) of the ideal optical fiber without the fluctuation. As shown in Eq. (3), the coupling coefficients depend on the scattering characteristics of the materials and the field-intensity overlap, given by Cij and ∬|Ei*·Ej|2dxdy, respectively. For typical correlation lengths of the heterogeneities, the coupling coefficients barely depend on the propagation constant difference [10]. Therefore, the mode dependence of the coupling coefficients is predominantly determined by the field-intensity overlap, which depends on the structural parameters of an ideal optical fiber without any perturbations.2.2 Random power transition
Based on the developed coupled power theory, we evaluated random mode coupling due to microscopic heterogeneities in PMMA-based GI POF. We analyzed the parabolic refractive-index profile whose guided modes can be classified into a principal mode group with almost the same group velocity, which helps in understanding the optical pulse transmission of the GI POF. The index profile along the radial direction ρ in the core is expressed as n(ρ) = n1[1 – 2Δ(ρ/a)2]1/2 where a = 25 μm, and Δ = (n12 – n22)/2n12. For typical PMMA cores doped with diphenyl sulfide, n1 = 1.503, and n2 = 1.490 for a laser source wavelength of 650 nm [1]. To calculate the power coupling coefficients using Eqs. (3) and (4), the propagation constants, the mode fields, and the group delays of the GI POF were calculated using finite element analysis of the scalar wave equation for the LPm,l modes, where m is the azimuthal mode number, and l is the radial mode number. The GI POF has 147 LP modes, which can be classified into 24 principal mode groups. The highest-order mode group was ignored for the analyses of the impulse response described in the next section because the modes in the group can easily radiate from the fiber owing to their much lower power confinement in the core than that from other groups.
Figure 2(a) shows power coupling coefficients for the intrinsic mode coupling due to microscopic heterogeneities with D = 300 nm and 〈δεr2〉 = 2.0 × 10−8 [10], where δεr is relative dielectric constant fluctuation, as a function of the propagation constant difference of the coupled mode pair. It was confirmed that the Δβ values can be approximately classified into 24 discrete values corresponding to all the mode-group pairs because of the parabolic index profile. Therefore, the result indicates that microscopic heterogeneities can induce random power transitions between all the group pairs. As shown in Fig. 2(b), we also calculated the coupling coefficient for microbending-induced coupling under the assumption that the bending curvature is Gaussian-correlated [13]. The standard deviation and the correlation length of the curvature are 2 × 10−4 mm−1 and 1.0 mm, which are close to the reported values for glass MMFs. The microbending results in only the coupling between the nearest-neighbor groups, obeying the selection rules for the mode coupling [7]. Note that the average coupling coefficient of 4.1 × 10−5 m−1 for microscopic heterogeneities is much smaller than the coefficient of 8.4 × 10−4 m−1 for microbending.
Fig. 2 Power coupling coefficients hij as a function of the propagation constant difference Δβ for mode coupling due to (a) microscopic heterogeneities and (b) microbending.
For the parabolic refractive-index profiles, the optical pulse response of GI POF can be understood by considering mode couplings between the principal mode groups. Figures 3(a) and 3(b) show the average coupling coefficient HMN between mode groups with the principal mode numbers of M and N calculated for microscopic heterogeneities and microbending, respectively. It should be noted that H11 and H22 have no values because mode groups 1 and 2 have only one guided mode each. The results show that forward scattering by microscopic heterogeneities can contribute to both the intragroup (M = N) and intergroup (M ≠ N) mode couplings, whereas only nearest-neighbor couplings can be induced by microbending. Note that intrinsic mode coupling due to microscopic heterogeneities is stronger for lower-order mode groups owing to higher degrees of mode-field intensity overlap [10]. However, coupling coefficients for the microbending-induced coupling become larger for higher-order group pairs, as shown in Fig. 3(b).
Fig. 3 Average power coupling coefficients between mode groups with principal mode numbers M and N for random mode coupling due to (a) microscopic heterogeneities and (b) microbending.
3. Efficient group-delay averaging
Since the parabolic index profile is not optimum for PMMA-based GI POF [1], random mode couplings can affect pulse spreading owing to group delay averaging. Using the power coupling coefficient hij and the modal delay per unit length τi, we investigate the optical pulse response of GI POF at a wavelength of 650 nm in an overfilled launch condition with homogeneous mode-power distribution using the numerical analysis in Eq. (2) with consideration of LP mode degeneracy. In the present study, only the x-polarized LP mode couplings are evaluated based on the assumption that the optical source is linearly polarized, and the GI POF has no structural or material anisotropies. Also, the attenuation is not taken into account in the numerical calculation because it has no significant effect on pulse response in the case without mode-dependent attenuation. Actually, the radiative mode coupling or the scattering loss due to the microscopic heterogeneities, which is the origin of the attenuation, has no strong dependence on the mode number.
Figures 4(a)–4(c) show the output-pulse waveform from GI POF with microscopic heterogeneities, with microbending, and without any perturbations calculated for different fiber lengths. The temporal pulse shape of the incident light is Gaussian with a full width at half maximum (FWHM) of 83 ps. Owing to intermodal dispersion, the optical pulses are broadened with propagation in all GI POFs. For the parabolic profile of GI POF, higher-order mode groups should have shorter group delays, which are approximately in inverse proportion to the group order except for some groups near the cutoff. Thus, the output pulse waveform from the ideal GI POF without any perturbations has an approximately linear decay with time in the pulse trailing edge [Fig. 4(c)], which reflects the fact that a greater number of guide-modes exist in higher-order mode groups. The pulse waveforms are changed by perturbation through group delay averaging owing to random mode coupling. For the GI POF with the microbending, the mode coupling effect is pronounced only in the pulse build up because the nearest-neighbor coupling become stronger for higher-order mode groups, as shown in Fig. 3(b). However, the microscopic heterogeneities affect the whole pulse waveform, and the pulse is broadened with little change of the pulse shape. Figures 4(d)–4(f) show the corresponding frequency response of GI POF with microscopic heterogeneities, with microbending, and without any perturbations for the different fiber lengths. These results show that the random mode couplings increase the −3dB bandwidth of the GI POF. Note that microscopic heterogeneities result in much stronger bandwidth enhancement than that due to microbending regardless of their much smaller average coupling coefficient as mentioned before. This indicates that microscopic heterogeneities can efficiently average the group delays of all the guided modes because of the random power transition.
Fig. 4 Output pulse-waveform for lengths of (a) 50 m, (b) 100 m, and (c) 200 m, and the relative frequency response for lengths of (d) 50 m, (e) 100 m, and (f) 200 m. The pink, light blue, and black lines correspond to GI POFs with microscopic heterogeneities, with microbending, and without any perturbations, respectively.
Figure 5 shows pulse broadening as a function of fiber length for GI POF with microscopic heterogeneities, with microbending, and without any perturbations. The pulse broadenings were obtained by calculating the root-mean-square (rms) widths of the impulse responses. The ideal GI POF has a linear increase in pulse broadening with fiber length because of the linear length dependence of the maximum delay difference [12]. In GI POF with microscopic heterogeneities, intrinsic mode coupling can decrease pulse broadening through group delay averaging. As shown in Fig. 5, pulse broadening is approximately proportional to L0.65 for sufficiently long fiber lengths to obtain equilibrium power mixing whereas the linear dependence (σ ∝ L) is observed for shorter lengths with little mode coupling effect. The resultant coupling length is ~100 m where the two asymptotes (σ ∝ L and σ ∝ L0.65) intersect. However, microbending hardly affects pulse broadening in the evaluated fiber lengths, and the resultant coupling length is much longer, although it has a much higher average coupling coefficient than that of microscopic heterogeneities.
Fig. 5 Pulse broadening in GI POFs with microscopic heterogeneities (pink), with microbending (light blue), and without any perturbations (dark gray). The dotted line is the fitted curve of σ ∝ L0.65 for the equilibrium mode coupling of GI POF with microscopic heterogeneities.
7. Conclusion
In conclusion, we numerically demonstrated that microscopic heterogeneous structures in GI POF core material can cause a much stronger mode coupling effect in short-haul communication networks than glass MMFs. These results showed that random power transition between all guided modes results in more efficient group-delay averaging and stronger bandwidth enhancement than microbending-induced nearest-neighbor coupling. This suggests that the mode coupling effect can be controlled using core polymer materials with different spatial correlation characteristics of microscopic heterogeneous structures. Now, we are comprehensively investigating the optical pulse response of GI POFs with various core materials to quantitatively correlate mode coupling with microscopic material structure.
Acknowledgment
This research is supported by the Japan Society for the Promotion of Science (JSPS) through its “Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program).”
References and links
1. Y. Koike and K. Koike, “Progress in low-loss and high-bandwidth plastic optical fibers,” J. Polym. Sci. B 49(1), 2–17 (2011). [CrossRef]
2. A. Polley and S. E. Ralph, “Mode coupling in plastic optical fiber enables 40-Gb/s performance,” IEEE Photon. Technol. Lett. 19(16), 1254–1256 (2007). [CrossRef]
3. S. E. Golowich, W. White, W. A. Reed, and E. Knudsen, “Quantitative estimates of mode coupling and differential modal attenuation in perfluorinated graded-index plastic optical fiber,” J. Lightwave Technol. 21(1), 111–121 (2003). [CrossRef]
4. W. R. White, M. Dueser, W. A. Reed, and T. Onishi, “Intermodal dispersion and mode coupling in perfluorinated graded-index plastic optical fiber,” IEEE Photon. Technol. Lett. 11(8), 997–999 (1999). [CrossRef]
5. R. F. Shi, C. Koeppen, G. Jiang, J. Wang, and A. F. Garito, “Origin of high bandwidth performance of graded-index plastic optical fibers,” Appl. Phys. Lett. 71(25), 3625–3627 (1997). [CrossRef]
6. R. Olshansky, “Propagation in glass optical waveguide,” Rev. Mod. Phys. 51(2), 341–367 (1979). [CrossRef]
7. R. Olshansky, “Mode coupling effects in graded-index optical fibers,” Appl. Opt. 14(4), 935–945 (1975). [CrossRef] [PubMed]
8. Y. Koike, S. Matsuoka, and H. E. Bair, “Origin of excess scattering in poly(methyl methacrylate) Glasses,” Macromolecules 25(18), 4807–4815 (1992). [CrossRef]
9. Y. Koike, N. Tanio, and Y. Ohtsuka, “Light scattering and heterogeneities in low-loss poly(methyl methacrylate) glasses,” Macromolecules 22(3), 1367–1373 (1989). [CrossRef]
10. A. Inoue, T. Sassa, K. Makino, A. Kondo, and Y. Koike, “Intrinsic transmission bandwidths of graded-index plastic optical fibers,” Opt. Lett. 37(13), 2583–2585 (2012). [CrossRef] [PubMed]
11. P. Debye and A. M. Bueche, “Scattering by an inhomogeneous Solid,” J. Appl. Phys. 20(6), 518–525 (1949). [CrossRef]
12. D. Marcuse, Theory of Dielectric Optical Waveguide (Academic Press, 1974).
13. K. Kitayama, S. Seikai, and N. Uchida, “Impulse response prediction based on experimental mode coupling coefficient in a 10-km-long graded-index fiber,” IEEE J. Quantum Electron. 16(3), 356–362 (1980). [CrossRef]
