By exploiting different spatial modes arising in few-mode fiber (FMF), mode-division multiplexing (MDM) has increased the capacity of fiber transmission systems to more than 22 Pb/s, breaking the capacity crunch of standard single-mode fiber (SSMF)^[1,2]. Although the FMF nonlinearity is weaker than that of SSMF, nonlinear impairments arising in FMF can bring a performance penalty to MDM transmission^[3]. On the other hand, nonlinear interactions arising in FMF have been utilized for several all-optical signal-processing applications, such as few-mode parametric amplification, intermodal wavelength conversion, and optical cross connection^[4–6].

To manifest the FMF nonlinearity, the precise measurement of the nonlinear coefficient is indispensable. Generally, there exist two types of FMF nonlinear coefficients: intramodal and intermodal nonlinear coefficients. In contrast with the SSMF, the intermodal nonlinear coefficient is a new parameter occurring among various guided modes^[5,6]. Therefore, the characterization of the intermodal nonlinear coefficient is different from its intramodal counterpart. Several measurement schemes of intramodal (or SSMF) nonlinearity that explore various nonlinear effects, including intramodal self-phase modulation (SPM) based on either pulse^[7,8] or continuous-wave (CW)^[9,10], intramodal cross-phase modulation (XPM) based on either pulse^[11] or CW^[12,13], and intramodal four-wave mixing (FWM)^[14], have been investigated previously. In comparison with the CW scheme, the characterization of a nonlinear coefficient based on the pulse scheme suffers from the pulse width variation^[15]. Moreover, in contrast with the characterization of the intramodal nonlinear coefficient, there have been few studies on the characterization of the intermodal nonlinear coefficient. One reason is that the intermodal nonlinear effect often co-occurs with the intramodal nonlinear effect, leading to a difficult discrimination from the intermodal nonlinear coefficient. Meanwhile, the polarization perturbation along the FMF, including random mode coupling and polarization rotation, makes the intermodal nonlinear coefficient measurement a tough task^[16]. By replacing the intramodal effective area with the intermodal effective area and exploiting the equivalent nonlinear refractive index obtained from the intramodal measurement result, the intermodal nonlinear coefficient has been estimated in previous research^[5,17], but without experimental verification. Recently, in order to measure the intermodal nonlinear coefficient, a novel method^[18] that compares the spectral broadening of probe pulse induced by the intramodal SPM and the intermodal XPM has been reported. Since the pulse width is used for comparing different nonlinearities, one must use a special source that maintains the same full width at half-maximum (FWHM) for different spatial modes. Meanwhile, the dispersion of two involved modes is required to be equal, so that the output spectrum of the LP11 mode can be calibrated with the spectral broadening of the LP01 mode. Moreover, the information on the intramodal nonlinear coefficient is necessary for the successful measurement of the intermodal nonlinear coefficient. Since an optical device^[19] is versatile and the intermodal nonlinear coefficient plays an important role in MDM systems, a convenient direct measurement scheme is necessary.

In this Letter, we experimentally demonstrate a direct measurement of the intermodal nonlinear coefficient based on the intermodal CW-XPM. The proposed method does not require any priori information on the intramodal nonlinear coefficient or special light source, allowing the FMF under test with different dispersion characteristics.

For two mode groups a and b arising in FMF, the intermodal nonlinear coefficient can be written as γab=γafab, where γa is the intramodal nonlinear coefficient of mode a, fab=Aaeff∬|Fa|2|Fb|2dxdy∬|Fa|2dxdy∬|Fb|2dxdy is the normalized overlap integral between modes a and b, Aaeff is the effective area of mode a, and Fa and Fb are the spatial distributions of modes a and b, respectively^[18]. When it comes to a real FMF, coupling among all guided modes is inevitable, because of cabling, bending, twisting, and longitudinal variation of FMF geometry parameters. The birefringence fluctuation along the FMF reduces the corresponding fiber nonlinearities. Taking the rapid and random birefringence variation along the FMF into account, Manakov equations averaging all possible states of polarization (SOPs) have been theoretically presented and numerically verified^[20]. For intermodal nonlinearities among various mode groups, the average over random birefringence fluctuation reduces the factor of 2 that is associated with XPM to 4/3^[20,21].

Assume that an intensity-modulated pump with a modulation frequency of ωm at mode a and a CW probe light at mode b co-propagate along the FMF with low differential group velocities. The intensity-modulated pump will transfer an intermodal XPM-induced nonlinear phase shift (NPS) to the CW probe, which is^[16]ΦIM−XPM(z,t)=κγabPa(z,t)Leff,(1)where Leff is the effective length of the FMF, Pa is the pump intensity that introduces the harmonic generation of the CW probe, and κ is the intermodal XPM factor depending on both the relative SOP of pump and the spatial overlap between the two modes. When the double sideband (DSB) modulation is implemented, the pump intensity is^[22]PaDSB(0,t)=Pa0+Pamsin(ωmt),(2)where Pa0 is the power of the direct current (DC) component and Pam is the power of the sinusoidal modulation that contributes to the harmonic generation of the probe. Taking Eq. (2) into Eq. (1), and then expanding the probe with Fourier transform, we can determine the intermodal XPM-induced NPS by the relative intensity ratio of the probe and the generated harmonics, which can be expressed as I0I1=J02(ΦIM−XPM)J12(ΦIM−XPM),(3)where I0 and I1 are the intensities of the zero- and first-order harmonics of the probe, respectively, as shown in Fig. 1(a). Jn is the first kind of Bessel function of the nth order. On the other hand, when the carrier-suppressed DSB (CS-DSB) modulation is applied to the pump, its intensity is PaCS−DSB(0,t)=PaCS−DSB[1−cos(2ωmt)]2.(4)

Therefore, with the CS-DSB modulation scheme, the intermodal XPM-induced NPS only introduces even-order harmonics of the probe. Taking Eq. (4) into Eq. (1) and expanding the probe spectral with the Bessel function, we obtain I0I1=J02(ΦIM−XPM2)J12(ΦIM−XPM2).(5)

In case the carrier is sufficiently suppressed, the CS-DSB-modulated pump can be treated as a beat signal^[10]. However, when either the carrier is not sufficiently suppressed or the CW probe is not weak enough, there can exist odd-order harmonics for the CS-DSB configuration, resulting from the unnecessary interaction among the carrier, the DSB, and the probe, as shown in Fig. 1(b). In such a case, the NPS generated from the intermodal XPM becomes weak, due to the unnecessary consumption of pump power.

Note that the above derivation occurs when the intermodal XPM is efficiently stimulated, while the intramodal XPM is successfully suppressed. Meanwhile, since different mode groups have different relative inverse group velocities (RIGVs) with respect to the operation wavelength, the intermodal walk-off effect that reduces the intermodal XPM efficiency can be eliminated by properly setting the wavelength spacing between pump and probe lasers. This is quite different from the intramodal XPM case, where the intramodal walk-off is challenging to be zero, when the pump and the probe are set at different wavelengths^[22]. Therefore, different β2 values of spatial modes do not decrease the intermodal CW-XPM efficiency^[16,22]. For FMF supporting more than two mode groups, the intermodal nonlinear coefficient between different mode groups can be obtained, by measuring the intermodal XPM-induced NPS between the involved mode groups.

The experimental setup for direct measurement of the intermodal nonlinear coefficient is shown in Fig. 2. The tunable laser source (TLS) together with a variable optical attenuator (VOA) is used to generate the probe. A synthesizer is used to modulate the pump. After the intensity modulation, the pump is amplified by a high-power erbium-doped fiber amplifier (EDFA) and followed by a bandpass filter (BPF) to suppress the amplified spontaneous emission (ASE) noise. The mode-division multiplexer (MMUX), based on the phase plate (PP) and the beam splitter (BS), is used to realize the mode selective conversion and multiplexing, with an insertion loss of 5.9 dB for the LP01 mode group and 8 dB for the LP11 mode group over the C-band. The modal cross talk of the MMUX between the LP01 and LP11 mode groups is around −20dB^[18]. The modulated pump is launched into the LP11 mode group, while the probe is kept in the LP01 mode group. In order to observe the relative intensity ratio of the probe and the generated harmonics unambiguously, the FMF output is fed into the ultrahigh resolution optical spectral analyzer (UHR-OSA, AP208x-C), whose spectral resolution can be as high as 0.04 pm. The FMF used in the experiment is a 1.9-km long graded-index FMF, supporting LP01 and LP11 mode groups, with a mode spacing Δneff of about 1.3×10−3^[17]. Figure 3 shows the measured RIGVs and the β2 values for the FMF under test. The RIGVs of LP01 and LP11 mode groups over the C-band is characterized by the time-of-flight technique. Based on the measured RIGV, the β2 values of the LP01 and LP11 mode groups are derived. Since there exists an intersection point between the RIGV of the LP01 and LP11 mode groups, various wavelength spacings between the pump and probe to realize a zero intermodal walk-off can be achieved.

We first investigate the XPM-induced NPS under either DSB or CS-DSB schemes for the FMF under test, when the pump is modulated with a 10-GHz sinusoidal signal. The carrier suppression ratio of the CS-DSB scheme is about 20 dB in order to mitigate the unnecessary interactions among the carrier, the DSB, and the probe. The wavelength of the modulated pump at the LP11 mode group is initially set at 1552 nm, while the probe at the LP01 mode group is 1553.7 nm. For a fair comparison, the XPM-induced NPS of the probe from either the DSB or CS-DSB-modulated pump is normalized to fit the same Bessel function. Figures 4(a) and 4(b) show the optical spectra of the LP01 mode probe with the XPM-induced NPS, which are captured directly by the UHR-OSA when the LP11 mode pump is modulated with DSB and CS-DSB schemes, respectively. Meanwhile, Fig. 4(c) shows the normalized NPS of the probe at the LP01 mode group with respect to the pump power under both DSB and CS-DSB configurations, which is relevant to the theoretical results. As shown in Fig. 4, the normalized NPS of the probe induced by the CS-DSB modulated pump is smaller than that under the DSB-modulated pump. In comparison with DSB modulation, the CS-DSB modulation scheme leads to a deviation of 9.5% from the theoretical result of the intermodal nonlinear coefficient. We infer that the generation of odd-order harmonics reduces the pump power contributed to the intermodal XPM interaction, resulting in the underestimation of the intermodal nonlinear coefficient. Since the generation of odd-order harmonics is inevitable for the CS-DSB pump scheme, we choose the DSB scheme for the following experimental verification.

Note that even though the pump and the probe are launched at different mode groups, intramodal XPM can exist because of the linear mode coupling in FMF and the narrow wavelength spacing between the pump and probe. To characterize the intermodal nonlinear coefficient, we must efficiently stimulate the intermodal XPM, while suppressing the intramodal XPM. Therefore, we start to investigate the impact of wavelength spacing between the pump and probe on the intermodal XPM, according to the measured RIGV of the FMF. When the intermodal walk-off turns into zero, the intermodal XPM efficiency reaches its maximum^[16]. Figure 5 shows the XPM-induced NPS with respect to the wavelength spacing Δλ between the pump and probe. The launch power of the pump at the LP11 mode group is 14.6 dBm, while it is −14.9dBm for the probe at the LP01 mode group. When the pump is fixed at 1552 nm, the operation wavelength of the probe to provide the same group velocity is about 1551 nm, corresponding to a wavelength spacing Δλ=1nm. Since a narrow wavelength spacing Δλ between the pump and probe facilitates intramodal nonlinear interaction, both the intramodal and intermodal XPMs contribute to the NPS generation. As shown in Fig. 5(a), the XPM-induced NPS increases as the wavelength spacing decreases. Meanwhile, when Δλ is more than 3.8 nm, the NPS induced by the intramodal and intermodal XPM becomes negligible. To mitigate the impact of the intramodal XPM, a wider wavelength spacing is necessary. Figure 5(b) shows the XPM-induced NPS with respect to the wavelength spacing Δλ between the probe and pump when the pump laser is operated at 1557 nm. According to Fig. 3, to maximize the intermodal XPM efficiency, the probe wavelength at the LP01 mode group under this situation is about 1552 nm, with a wavelength spacing of about 5 nm, leading to a successful suppression of the intramodal XPM interaction. As shown in Fig. 5(b), with the growing wavelength spacing Δλ, the XPM-induced NPS increases first, which is in contrast to the intramodal interaction. After reaching a maximum, indicating a zero intermodal walk-off point for the inter-modal XPM, the XPM-induced NPS decreases as Δλ further increases.

Figure 6(a) shows the intermodal XPM-induced NPS versus the pump power when the pump wavelength at the LP11 mode group and the probe wavelength at the LP01 mode group are 1557 and 1552 nm, respectively. According to Eq. (1), the average value of the intermodal nonlinear coefficient can be obtained by the linear fitting. The measured intermodal nonlinear coefficient is in good agreement with the theoretical calculation result of about 2.8 W⁻¹ km^−1[17] when the factor of κ is chosen as 4/3. The fluctuation of the measurement result is about 10%, mainly due to the uncertainty of power measurement and the mode coupling caused by longitudinal fluctuations. Since the FMF under test is not a typical polarization-maintaining one, more than 15 measurements are carried out to further examine the factor of κ under different pump modulation frequencies. Figure 6(b) shows the number of counts versus the factor of κ under the pump modulation frequencies of 8, 9, and 10 GHz, respectively. Most experimental results located at κ=4/3 agree well with the weakly coupled Manakov equation to describe the average effect of random birefringence fluctuations for intermodal nonlinearities^[20].

We have demonstrated the direct measurement of an intermodal nonlinear coefficient using an intermodal XPM-induced NPS. The proposed method relaxes the requirement of both equal dispersion among involved modes and a priori knowledge of the intramodal nonlinear coefficient and intermodal effective area. Based on the intermodal CW-XPM arising in the FMF under test, the intermodal nonlinear coefficient of a 1.9-km FMF supporting LP01 and LP11 mode groups has been successfully characterized. Meanwhile, the 4/3 factor accounted for the birefringence fluctuation in the weakly coupled Manakov equation is experimentally verified.