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There exists a fundamental Shannon capacity limit due to fiber nonlinearity in single-mode fiber (SMF) optical systems [1], [2], [3]. Recent demonstrations show that on SMF platform, this capacity limit has been quickly approached within some practical engineering margin [4], [5], [6]. As a result, multicore fiber (MCF) and few-mode fiber (FMF) were proposed to further improve the fiber capacity [7], [8], [9], [10], [11], [12]. In MCF, the optical signals are carried on several individual cores [11], [12], and the intercore nonlinear interaction is minimized. Therefore, the system capacity is expected to be increased by the number of cores. For FMF based transmission, there is strong field overlapping between different modes, and therefore, it is important to analyze the FMF systems, taking into account potential intermodal nonlinearity [13]. In this paper, we extend our previously derived closed-form expression for channel capacity [2], [3] from SMF to FMF systems. We then conduct some experiments to measure these nonlinear coefficients for a customized two-mode fiber (TMF) (actually containing three spatial modes including one Formula$LP_{01}$ mode and two degenerate Formula$LP_{11}$ modes, Formula$LP_{11}^{a}$ and Formula$LP_{11}^{b}$). It is found that despite strong spatial overlapping of the three modes in TMF, the channel capacity approaches to three times of that of SMF. There may be an advantage of using FMF as opposed to MCF in terms of higher integration density and power efficiency. Our work show that by densely packing optical signals into a few modes does not significantly reduce the link capacity limit in comparison with much loosely packed MCF systems.

The paper is organized as follows: Section 2 introduces densely-spaced multicarrier systems for TMF and analyzes fiber nonlinearity impact on such systems. A closed-form expression for link capacity is presented in this section. Section 3 describes the experiments for TMF nonlinear coefficient measurement. Then, the experimental results, as well as theoretical analysis, are discussed in Section 4. Finally, the conclusion is drawn in Section 5.



The channel capacity can be estimated by filling all the bandwidth with densely-spaced orthogonal frequency-division multiplexing (OFDM) subcarriers as shown in Fig. 1. When the spectrum is continuously occupied, the nonlinear effects such as self-phase modulation (SPM), cross-phase modulation (XPM), and four-wave mixing (FWM) can be considered as FWM [2], [3]. Due to the FWM effect, the interaction of subcarriers at the frequencies of Formula$f_{i}$, Formula$f_{j}$, and Formula$f_{k}$ produces a mixing product at the frequency of Formula$f_{g} = f_{i} + f_{j} - f_{k}$. The magnitude of the FWM product for one span of a fiber link is given by [3] Formula TeX Source $$c_{m_{g}, g}^{\prime} = \gamma_{ijkg}\left[\left(c_{m_{k}, k}^{+}c_{m_{i}, i}\right)c_{m_{j}, j} + \left(c_{m_{k}, k}^{+}c_{m_{j}, j} \right)c_{m_{i}, i}\right]e^{-\alpha L/2 - i\beta_{g}L}{1 - e^{-\alpha L}e^{-j\Delta\beta_{ijk}L} \over j\Delta \beta_{ijk} + \alpha}\eqno{\hbox{(1)}}$$ where Formula$c_{m_{i}, i}$ is the optical field in Jones vector form for the frequency Formula$f_{i}$ in the mode Formula$m_{i}$, Formula$\beta_{m_{i}, i}$ is the wavenumber for the frequency Formula$f_{i}$, Formula$\Delta\beta_{ijk} \equiv \beta_{m_{i}, i} + \beta_{m_{j}, j} - \beta_{m_{k}, k} - \beta_{m_{g}, g}$ represents the phase mismatch for the FWM product, and Formula$L$ is the length of the fiber. The subscript for mode at times is dropped for brevity. Without loss of generality, we assume all the modes have the same loss coefficient of Formula$\alpha$. Formula$\gamma_{ijkg}$ is the nonlinear coefficient between the four waves, expressed as Formula TeX Source $$\gamma_{m_{i}m_{j}m_{k}m_{g}} = {n_{2}\omega_{0} \over A_{m_{i}m_{j}m_{k}m_{g}}c}, \quad A_{m_{i}m_{j}m_{k}m_{g}} = {\left[\left\langle\vert F_{m_{i}}\vert^{2}\right \rangle \left\langle\vert F_{m_{j}}\vert^{2}\right\rangle \left\langle\vert F_{m_{k}}\vert^{2}\right\rangle \left \langle\vert F_{m_{g}}\vert^{2}\right\rangle \right]^{1/2} \over \left\langle F_{m_{i}}^{\ast}F_{m_{j}}^{\ast}F_{m_{k}}F_{m_{g}}\right\rangle}\eqno{\hbox{(2)}}$$ where “Formula$\langle\ \rangle$” stands for integration over the cross section of the fiber, Formula$n_{2}$ is the intrinsic nonlinear coefficient of fiber equal to Formula$2.41 \times 10^{-20}\ \hbox{m}^{2}/\hbox{W}$, Formula$\omega_{0}$ is the center frequency of the FWM, Formula$c$ is the speed of light, Formula$A_{m_{i}m_{j}m_{k}m_{g}}$ is the effective area (EA) of the nonlinear interaction, and Formula$F_{k}$ is the mode profile for the frequency component Formula$k$. From (1), for FWM to be effective, it is very critical that Formula$\Delta\beta_{ijk}$ is maintained to be small on the order of fiber loss coefficient of Formula$\alpha$. There are two mechanisms that greatly reduce the FWM impact:

  1. Modal wavenumber mismatch. Assume that Formula$\beta_{i}$ and Formula$\beta_{j}$ are in the same mode, but Formula$\beta_{k}$ and Formula$\beta_{g}$ are in the other modes; then, Formula$\Delta\beta_{ijk} = \Delta\beta$, where Formula$\Delta\beta$ is the wavenumber mismatch between the Formula$\hbox{LP}_{01}$ and Formula$\hbox{LP}_{11}$ modes. Formula$\Delta\beta$ is a few orders of magnitude larger than Formula$\alpha$ [7] and, therefore, makes the FWM effects insignificant.
  2. Modal group delay mismatch. Assume that Formula$\beta_{i}$ and Formula$\beta_{k}$ are in the same mode, e.g., in Formula$\hbox{LP}_{01}$ mode, but Formula$\beta_{j}$ and Formula$\beta_{g}$ are in the other mode, Formula$\hbox{LP}_{11}$ mode. This is equivalent to classical XPM. Then, we have Formula TeX Source $$\Delta \beta_{ijk} \equiv \beta_{m_{i}, i} + \beta_{m_{j}, j} - \beta_{m_{k}, k} - \beta_{m_{g}, g} = \Delta \omega\left(1/ \upsilon_{g}^{m_{i}} - 1/\upsilon_{g}^{m_{j}}\right)\eqno{\hbox{(3)}}$$

where Formula$\Delta\omega$ is the angular frequency difference between frequency component Formula$i$ and Formula$k$. Formula$\upsilon_{g}^{m_{i, j}}$ is the group velocity for mode Formula$i$ or Formula$j$. The group delay mismatch between two Formula$\hbox{LP}_{01}$ and Formula$\hbox{LP}_{01}$ modes gives a value of a few Formula$ps/m$ [7], indicating that as long as Formula$\Delta\omega$ is larger than several tens of megahertz, the FWM or nonlinear interaction is also negligible.

Figure 1
Fig. 1. Schematic of densely spaced subcarriers with four-wave mixing products.

The modal group delay mismatch between two Formula$\hbox{LP}_{11}$ modes are less than that between Formula$\hbox{LP}_{11}$ and Formula$\hbox{LP}_{01}$. However, it is still measured at a few hundreds ps/km for our customized fiber, implying that as long as Formula$\Delta\omega$ is larger than a few hundreds of MHz, the FWM or nonlinear interaction is also negligible [13]. The wavenumber mismatch is larger than group delay mismatch. All these indicate the nonlinear interactions between two Formula$\hbox{LP}_{11}$ modes are also negligible [13]. Now we can conclude that to have significant nonlinear effects, all the four waves should be in the same mode. Otherwise, the produced FWM is insignificant due to modal wavenumber or group delay mismatch. Based on these observations, the spectral efficiency (SE) for three modes Formula$(S_{t})$ can be the summation of SE from three individual modes Formula$S_{i}$. Formula$S_{i}$ is given by [3] Formula TeX Source $$\eqalignno{S_{t} = &\, \sum_{i = 1}^{3}S_{i}, \quad S_{i} \cong 2\log_{2}\left({1 \over 3}\left(8\pi\alpha_{i}\vert\beta_{i}\vert\right)^{1/3}\left(3 \gamma_{i}^{2}n_{0}^{2}N_{s}h_{e}\ln(B/B_{0})\right)^{-1/3}\right)\cr n_{0} = &\, N_{s}N_{0}, \quad N_{0} = (G - 1)n_{sp}h\upsilon \cong 0.5e^{\alpha_{i}L}h\upsilon \cdot NF, \quad B_{0} = \alpha_{i}/\left(2\pi^{2}\vert\beta_{i} \vert B\right)&\hbox{(4)}}$$ where Formula$i$ of 1, 2, and 3 stands for modes of Formula$LP_{01}$, Formula$LP_{11}^{a}$, and Formula$LP_{11}^{b}$, respectively, Formula$\gamma_{i} = \gamma_{m_{i}m_{i}m_{i}m_{i}}$ is the nonlinear coefficient for the Formula$i$th mode, Formula$n_{0}$ is the accumulated optical noise, Formula$N_{s}$ is number of spans, Formula$G$ is the amplifier gain, Formula$NF$ is the noise figure of amplifier, Formula$h$ is Planck constant, Formula$\upsilon$ is the light frequency, Formula$B$ is the overall optical bandwidth, and Formula$h_{e}$ is noise enhancement factor given by (36) in [3]. Derivation of (4) is shown in [3] using (31), (36), and (40) from the reference. Additionally, although the derivation of (4) assumes densely-spaced CO-OFDM signals, as is shown in [3], the closed-form expression can apply to densely-spaced single-carrier systems such as Nyquist WDM systems [14].



We construct an experimental setup to measure the nonlinear coefficient of Formula$\hbox{LP}_{11})$ and Formula$\hbox{LP}_{01})$ modes by means of using degenerate FWM products where frequency Formula$i$ and Formula$j$ in (1) are identical. The Formula$\hbox{LP}_{01}$ (or Formula$\hbox{LP}_{11})$ nonlinear coefficient is measured by launching two optical pumps in Formula$\hbox{LP}_{01}$ (or Formula$\hbox{LP}_{11}$) mode. As shown in Fig. 2, cw-light from two tunable ECL lasers are separately amplified, and launched into a free-space mode coupling systems consisted of two collimator input ports, free-space combiner, and collimator output port that is connected to the input of a customized TMF with detailed parameters shown in Table 1. To generate two Formula$\hbox{LP}_{11}$ modes, two long period fiber grating (LPFG) based mode converters are used to convert Formula$\hbox{LP}_{01}$ to Formula$\hbox{LP}_{11}$ [7]. We choose TMF of 10 m and 50 m for the measurement. The reason of utilizing short fiber is to improve the stability and repeatability. With the increase of fiber length, the spatial mode patterns become hard to maintain due to any mechanical disturbance, making it difficult to identify different nonlinear products. As a result, we use 10 meters and 50 meters TMF in this paper. The disadvantage of shorter fiber is reduced strength of FWM product. Nevertheless, we manage to obtain FWM products with these short fibers at least 10 dB above the sensitivity floor of the optical spectrum analyzer (OSA) used. The launch powers of two pumps into the TMF are between 14 to 16 dBm, producing FWM of between −50 to −80 dBm, depending on launched wavelengths and modes. At the output of the TMF, the right-side FWM product is filtered before fed into OSA for power measurement to avoid dynamic range limitation of OSA. For Formula$\hbox{LP}_{11}$ FWM products measurement, a free-space coupler is used to couple Formula$\hbox{LP}_{11}$ mode into Formula$\hbox{LP}_{11}$-to- Formula$\hbox{LP}_{01}$ mode converter. The loss from the output fiber to the OSA is measured and used to calibrate the data obtained from the OSA.

Figure 2
Fig. 2. Experimental setup for fiber nonlinear coefficient measurement. PC: polarization controller; MS: mode stripper; MC: mode converter; FBS/C: free-space splitter/combiner; TMF: two-mode fiber; SMF: single-mode fiber; OSA: optical spectrum analyzer.
TABLE 1 Parameters of the custom-designed two-mode fiber

Fig. 3 shows the measurement of the FWM power after calibration as a function of the two pump wavelength spacing for both Formula$\hbox{LP}_{01}$ and Formula$\hbox{LP}_{11}$ nonlinear coefficient measurement. For Formula$\hbox{LP}_{11}$ mode, besides 10 m TMF, we also measured the nonlinear coefficient for fiber length of 50 m. Insets show the spectra for Formula$\hbox{LP}_{01}$ and Formula$\hbox{LP}_{11}$ FWM products measured after a filter centered at the right-side FWM with residual pumps Formula$(\lambda_{1}, \lambda_{2})$ and ASE noise. Theoretical simulation is also conducted using (1) with the following parameters based on our customized TMF design: chromatic dispersion (CD) of 22.1 ps/nm/km for Formula$\hbox{LP}_{01}$ and 17 ps/nm/km for Formula$\hbox{LP}_{11}$. EA is 94.7 Formula$\hbox{um}^{2}$ for Formula$\hbox{LP}_{01}$ and 99.9 Formula$\hbox{um}^{2}$ for Formula$\hbox{LP}_{11}$. We have relatively good agreement of nonlinear coefficient of Formula$\hbox{LP}_{01}$ mode measured at 1.15/W/km, as opposed to theoretical 1.03/W/km. However, there is a big mismatch for Formula$\hbox{LP}_{11}$ nonlinear coefficient measured at 0.52/W/km for 10 m TMF measurement and 0.58/W/km for 50 m TMF measurement, respectively, as opposed to the theoretical value of 0.98/W/km. We contribute the error of Formula$\hbox{LP}_{11}$ measurement to possible excitation of all the true modes, Formula$\hbox{TE}_{01}$, Formula$\hbox{TM}_{01}$, and two degenerated Formula$\hbox{EH}_{21}$ in the TMF [14], which degrades the overall FWM efficiency of Formula$\hbox{LP}_{11}$. It is worth noting that we also measure FWM i) between Formula$\hbox{LP}_{01}$ and Formula$\hbox{LP}_{11}$ modes, and ii) between Formula$LP_{11}^{a}$ and Formula$LP_{11}^{b}$ modes. In case i), we find no measurable product confirming insignificant nonlinearity between Formula$\hbox{LP}_{01}$ and Formula$\hbox{LP}_{11}$ modes. In case ii), at 2-nm pump wavelength spacing, it is about 16 dB down below the FWM product between Formula$LP_{11}^{a}$ and Formula$LP_{11}^{a}$ modes. This verifies that the nonlinear interaction between Formula$LP_{11}^{a}$ and Formula$LP_{11}^{b}$ modes is also insignificant. We leave it to a future work to further improve the measurement accuracy of Formula$\hbox{LP}_{11}$ nonlinear coefficients by launching either true fiber modes or using even shorter TMFs.

Figure 3
Fig. 3. FWM product measurement. Inset: Spectra after filter centered at right-side FWM product for (i) Formula$\hbox{LP}_{01}$ and (ii) Formula$\hbox{LP}_{11}$. Theo.: Theoretical results; Exp.: Experimental results.


Fig. 4 shows the numerical calculation of SE for TMF systems using (1). The span length is assumed to be 100 km, and the noise figure of optical amplifier is set to be 6 dB. The chromatic dispersion and nonlinear coefficients are as follows: CD is 22.1 (17) ps/nm/km for Formula$\hbox{LP}_{01}\ (\hbox{LP}_{11})$. EA is 94.7 (99.9) Formula$\hbox{um}^{2}$ for Formula$\hbox{LP}_{01}\ (\hbox{LP}_{11})$. These are the data based on the design and simulation of the customized TMF. Because of the CD and EA of Formula$\hbox{LP}_{01}$ are both larger than those of SSMF (CD of 17 ps/nm/km, fiber loss of 0.2 dB/km, EA of 80 Formula$\hbox{um}^{2}$), we expect a slight nonlinear enhancement of link capacity for TMF. The current fiber loss is measured at 0.26 dB/km in our TMF. However, there is an expectation that the loss for both modes should approach those of SSMF if properly designed [13]. For the fiber loss of 0.26 dB/km using current design and for 10-span transmission, the capacity for TMF is 23.6 bit/s/Hz less than three times of 9.5 bit/s/Hz for SSMF. If we use 0.2 dB/km for the TMF, the capacity is increased to 29.5 bit/s/Hz, slightly better than three times of that of SSMF. Similar ratio can be found with different number of spans. We conclude that although the three modes are densely packed and overlapped with each other, TMF nonlinearity does not additionally decrease the SE. However, the fiber loss is a critical parameter for TMF to attain the full capacity of factor of 3 time SE enhancement compared with SSMF.

Figure 4
Fig. 4. Spectral efficiency (SE) for TMF systems. SE of SSMF is also drawn for comparison.


We have measured fiber nonlinear Kerr coefficient and presented discussion of system capacity for TMF. The system capacity approaches three times of that of SMF even though there is strong spatial overlapping between the modes in TMF. Our future work will be to improve the measurement accuracy of Formula$\hbox{LP}_{11}$ nonlinear coefficients by launching either true fiber modes or using even shorter TMF fibers.


Corresponding author: X. Chen (e-mail:


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Xi Chen

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An Li

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Guanjun Gao

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Abdullah Al Amin

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William Shieh

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