Nonlinear phenomena in multimode and few-mode fibers have been studied extensively and continue to garner attention, particularly for the following reasons: (i) expectations related to the application of spatial division multiplexing in telecommunication networks [1]; (ii) new possibilities for observing nonlinear phenomena provided by compact and powerful laser sources [2]. The newly observed phenomena are multimode solitons [3], beam self-cleaning [4,5], geometric parametric instability [6], and discretized conical emission [7–9]. In addition, other nonlinear phenomena such as Raman scattering [10] and four-wave mixing [11,12] have also been revisited.

Recent research on intermodal four-wave mixing (IM-FWM) [12–16] has addressed various aspects such as supercontinuum generation, cascaded FWM, and mode-multiplexing. These studies reported the observance of IM-FWM involving two spatial modes of a nonbirefringent fiber. Nonlinear conversion occurs from two excited modes to signal and idler bands, each generated in a different mode from the set of excited modes. For specific fiber and excitation conditions, the phase-matching condition and overlap coefficients determine: (i) the modes that are involved in the FWM; (ii) the positions of the generated bands; and (iii) the conversion efficiency. For example, IM-FWM was observed for the following pairs of modes: $\mathrm {LP}_{01}^{}$-$\mathrm {LP}_{02}^{}$, $\mathrm {LP}_{01}^{}$-$\mathrm {LP}_{11}^{}$, and $\mathrm {LP}_{11}^{}$-$\mathrm {LP}_{21}^{}$ [12]. Moreover, conversion may also occur for other pairs involving higher-order spatial modes of nonbirefringent fibers [11]. In birefringent fibers, the FWM involves polarization modes [17]. In particular, phase matching between two polarization modes of a single (typically fundamental) spatial mode can be achieved [18]. This type of vectorial FWM has been investigated in the context of sensing [19] and entangled photon pairs generation [20,21]. However, in a few-mode birefringent fiber, more complex FWM processes involving intermodal-vectorial interactions are also possible. Garay-Palmett et al. [22] presented an experimental and theoretical analysis of this type of FWM. In their experiments, pairs of polarization modes (limited to a single polarization direction) were excited, resulting in the observation of orthogonally polarized signal and idler bands.

In this study, various IM-FWM processes occurring in step-index birefringent fibers supporting $\mathrm {LP}_{01}^{}$ and $\mathrm {LP}_{11}^{}$ spatial modes are investigated. Furthermore, the experimental differentiation of distinct IM-FWM processes is possible employing the recently described method of exciting different combinations of polarization/spatial modes using a Wollaston prism [23]. In particular, pairs of different spatial modes of orthogonal polarizations can be purely excited. This enabled for the first time the experimental investigation of the intermodal-vectorial FWM processes in a two-mode birefringent fiber. In contrast to the excitation of only one spatial mode of orthogonal polarizations (vectorial FWM) or two spatial modes of the same polarization (intermodal FWM), which results in the generation of one pair of signal and idler bands (in different polarizations or different spatial modes, respectively), the mixed intermodal-vectorial FWM process produces up to three pairs of signal/idler bands simultaneously in different spatial modes and orthogonal polarizations. Furthermore, this study explains that the difference in the phase modal birefringence in the $\mathrm {LP}_{01}^{}$ and $\mathrm {LP}_{11}^{}$ spatial modes is responsible for the observed sidebands multiplication.

The phase-matching condition for the FWM process can be expressed as $\beta ^{(l)} + \beta ^{(m)} = \beta ^{(p)} + \beta ^{(n)}$, where $\beta ^{(i)}$ is the wave vector of the mode $i$. The superscripts $l$ and $m$ denote pump modes, while superscripts $p$ and $n$ denote the signal and idler modes, respectively. Assuming that both excited modes are at a single frequency, and expanding the propagation constants in the Taylor series up to the second order at that frequency, the following relation is obtained:

If only two modes are involved, as in the case of vectorial or intermodal FWM, $p = l$ and $m = n$, respectively. Consequently, the free term in Eq. (3) vanishes, giving one trivial solution $\Omega = 0$ and one non-trivial solution $\Omega = -\Delta \beta _1^{(l,m)}\big /\bar \beta _2^{(l,m)}$ [24]. In the general case of intermodal-vectorial FWM, the pump and generated band modes are of different orders and orthogonal polarizations. Thus, to focus on this, it is assumed that the pump modes $(m,l)$ are the $\mathrm {LP}_{01}^{y}$ and $\mathrm {LP}_{11}^{xe}$ polarization modes (see Fig. 1). In this case, intermodal-vectorial FWM can occur in two different ways. In the first scenario, which is always possible, the pump modes $\mathrm {LP}_{01}^{y}$ and $\mathrm {LP}_{11}^{xe}$ are converted to the same pair of modes and the spectral positions of the signal/idler bands are determined by the ratio of the difference in their group effective indices to the average dispersion, as previously explained. In the second scenario, which is possible if Eq. (3) has real solutions, four different modes are involved in the conversion process according to scheme $(m,l)\rightarrow (p,n)$. Thus, for a two-mode birefringent fiber and the assumed excitation, it is equivalent to the ($\mathrm {LP}_{01}^{y}$, $\mathrm {LP}_{11}^{xe}$)$\,\rightarrow \,$ ($\mathrm {LP}_{01}^{x}$, $\mathrm {LP}_{11}^{yo}$) process. In this case, the free term in Eq. (3), representing the difference in phase modal birefringence for the respective spatial modes, is nonzero, resulting in the signal/idler bands being doubled.

 

Fig. 1. (a) SEM image of the cross section of the birefringent fiber with stress-applying elements (Nufern PM1550B-XP). (b) Electric field of the $x$- and $y$-polarized modes in the $\mathrm {LP}_{01}^{}$ and $\mathrm {LP}_{11}^{}$ groups.

Download Full Size | PDF

However, phase matching is not the only prerequisite for intermodal FWM. In addition, the mode overlapping coefficients $S^{(plmn)}_K$ and $S^{(plmn)}_R$ for the interacting modes must be nonzero. These coefficients appear in the coupled nonlinear Schrödinger equations (CNLSE) [25]:

The nonzero overlap coefficients for the Nufern PM1550B-XP birefringent fiber used in the experiment were calculated. The manufacturer specifications state the diameter and numerical aperture of the fiber core as 8.5 µm and 0.125, respectively. The SEM image of the fiber cross section and the normalized electric field of the $\mathrm {LP}_{01}^{}$ and $\mathrm {LP}_{11}^{}$ modes at pump wavelength 1064.3 nm are shown in Fig. 1. Moreover, the figure explains the shortened notation ($x$, $y$, $xo$, $ye$, $xe$, and $yo$) used to denote the respective polarization modes. The mode distributions shown, calculated with the COMSOL Multiphysics mode solver, were used to obtain the overlap factors using Eqs. (4a) and (4b). The coefficient $S_K^{(x,x,x,x)}$ is the inverse of the effective mode area of the fundamental mode. The magnitudes of the overlap factors are normalized to $S_K^{(x,x,x,x)}$; $f^{(plmn)} = \left |S_K^{(plmn)}\right |\big /S_K^{(x,x,x,x)}$ show the relative strength of nonlinear processes in reference to the intramodal nonlinearities of the fundamental mode [26]: $f^{(pppp)}$ coefficients are used to describe intramodal nonlinearites of each mode, such as self-phase modulation, and $f^{(ppnn)}$ and $f^{(pnpn)}$ correspond to cross-phase modulation and two-mode FWM, respectively. The remaining coefficients govern other intermodal and/or vectorial FWM processes.

The normalized overlap factors for the intramodal and two-mode FWM presented in Table 1 indicate that the effective nonlinearities in the $\mathrm {LP}_{11}^{}$ modes are lower than those in the $\mathrm {LP}_{01}^{}$ modes. Furthermore, the efficiency of the cross-polarization nonlinear process in a particular spatial mode is $2/3$ that of the intramodal nonlinearity. Finally, two-mode FWM is possible even between two orthogonally polarized modes of different spatial groups. The normalized overlap coefficients for all intermodal-vectorial FWM processes, that can occur by exciting orthogonally polarized modes of different spatial mode groups are presented in Table 2. There exist six such pairs; however, four of them can be excited under the applied excitation scheme [23]: $(y,xo)$, $(x,ye)$, $(y,xe)$, and $(x,yo)$. For such an FWM process, the free term in the phase-matching condition [Eq. (3)] is nonzero, and two additional pairs of signal/idler bands are expected if phase matching is fulfilled.

Table 1. Normalized Overlap Coefficients for Intramodal and Two-Mode Processes^a

View Table | View all tables in this article

Table 2. Normalized Overlap Factors for Selected Intermodal-Vectorial FWM^a

View Table | View all tables in this article

To predict the positions of the signal/idler bands generated by the intermodal, vectorial, and intermodal-vectorial FWM processes for the pump at 1064.3 nm, the fiber was experimentally characterized. Table 3 summarizes the properties of the modes supported by the fiber. The table lists the chromatic dispersion ($D^{(p)}$) of the mode $p$ measured with the white-light interferometry technique in a setup applying a spatial light modulator [27], the difference in group refractive indices ($\Delta N^{(p,y)}$) of the mode $p$ and mode $y$, and the difference in phase refractive indices of two polarization modes of a single spatial mode ($\Delta n$) measured using spectral interference with the lateral point force method [28]. Thus, knowing the processes that are allowed in terms of overlap factors, $\Omega$ was calculated by solving Eq. (3) for all cases. Consequently, the predicted spectral positions of the signal and idler bands ($\lambda _\mathrm {t}^{(p)}$ and $\lambda _\mathrm {t}^{(n)}$), respectively, were obtained, as presented in Table 4.

Table 3. Measured Modes’ Characteristics at 1064.3 nm

View Table | View all tables in this article

Table 4. Comparison of Theoretical ($\lambda _\mathrm t [\mathrm{nm}]$) and Experimental ($\lambda _\mathrm e [\mathrm{nm}]$) Positions of Signal/Idler Bands Generated in Different FWM from a 1064.3 nm Pump^a

View Table | View all tables in this article

The theoretical analysis focused on the FWM conversion directly from the pump. Subsequently, the nonlinear light propagation in the considered fiber was simulated to study the cascaded processes. The CNLSE solver based on the software implemented by Wright et al. [29] was used. The measured modes properties, given in Table 3, were then used to calculate the linear terms in the simulations. Further, the fiber nonlinearity $n_2 = {2.6\times10^{-20}}{\mathrm{m}^2/\mathrm{W}}$ and overlap factors, obtained using Eqs. (4a) and (4b), were used to calculate the nonlinear terms. The simulated spectral positions of the bands generated by the direct FWM were consistent with the theoretical predictions for all excitation cases. Figure 2(a) shows the simulated mode-resolved spectra generated in a 12 m long piece of the fiber from a CW pump (1064.3 nm, 1500 W) coupled equally in the $y$ and $xe$ modes. Different intermodal-vectorial FWM processes occur simultaneously for this combination of excited modes. Primarily, there are: (i) the FWM process $(y,xe)\rightarrow (y,xe)$ that involves only two excited modes, which generates a single signal/idler pair in the same modes; (ii) the two paths of FWM process $(y,xe)\rightarrow (x,yo)$ that involve four modes and generate the two signal/idler pairs. Subsequently, there are four cascaded stimulated FWMs: $(yo,x)\rightarrow (y,xe)$, $(yo,y)\rightarrow (y,yo)$, $(y,xe)\rightarrow (y,xe)$, $(y,yo)\rightarrow (y,yo)$. Figure 2 shows the bands involved in all FWM processes with their spectral positions and modal state.

 

Fig. 2. (a) Mode-resolved spectra obtained in simulations for CW excitation of $y$ and $xe$ modes at 1064.3 nm. (b) Diagram indicating the spectral positions of the bands in distinct spontaneous (top three) and cascaded (bottom four) FWMs. (c) Polarization-resolved spectra observed experimentally. Insets show images of the pump and generated bands in both polarizations captured by a camera placed at the fiber output following a diffraction grating (600 lines/mm)[23].

Download Full Size | PDF

Finally, experiments were performed to confirm the predictions of the theoretical analysis and numerical simulations. A 1064.3 nm Nd:YAG laser with a pulse duration of 1 ns, repetition rate of 19 kHz, and an average power of 140 mW was employed as the pump. The selective excitation of all the considered combinations of modes was carried out as described in [23]. Example spectra registered for the excitation of the $y$ and $xe$ modes are shown in Fig. 2(c), with the field distribution of involved bands as evidence for the participation of the given polarization mode in the phase-matching condition. The measured spectral positions of the signal/idler bands ($\lambda _\mathrm {e}^{(p)}$ and $\lambda _\mathrm {e}^{(n)}$) for all the possible excitations are listed in Table 4. However, the position of the single band is missing for the $(y,xe)\rightarrow (y,xe)$ process because the expected peak is covered by a broad Stokes Raman band.

As shown in Table 4, the registered spectra generated by spontaneous FWM were consistent with the simulation results for all possible combinations of excited modes. Moreover, the measured positions of the bands generated in the cascaded processes were also consistent with the simulations (Fig. 2). The experimental results of this study further demonstrate that when pairs of two spatial modes with orthogonal polarizations are excited, separate intermodal-vectorial FWM processes can be initiated, leading to the generation of multiple signal/idler bands.

In summary, this study presents the various FWM processes that are possible between six polarization modes in a birefringent step-index fiber. Subsequently, the role of phase birefringence in the phase-matching condition for intermodal-vectorial FWM processes that occur when two spatial modes of orthogonal polarizations are excited by the pump is highlighted. The obtained results show that the selective excitation of spatial/polarization modes allows the position of FWM bands to be controlled over a wide range, which can aid in extending space-division multiplexing schemes with polarization degree and in entangled photon pairs sources.