Intermodal-vectorial four-wave mixing processes involving LP 01 , LP 11 , LP 02 and LP 21 modes of birefringent fibers

. We present the complete (analytical, numerical and experimental) analysis of intermodal-vectorial four-wave mixings proccesses in birefringent fibers. We analyze phase-matching condition and overlap coe ffi - cients to indicate possible processes. Then, we demonstrate multiple four-wave mixing processes in LP 01 and LP 11 modes numerically and experimentally. Finally, we extend theoretical analysis to account higher-order modes: LP 02 and LP 21 .


Introduction
One of the frequency conversion processes observed in the nonlinear fibers is a four-wave mixing.In few-mode fibers (FMFs) and multimode fibers (MMFs) different modes can be involved in this process.The four-wave mixing (FWM) processes were observed in different spatial modes of non-birefringent fibers [1][2][3] and in different polarization modes of birefringent fibers [4,5].Typically, the two modes (spatial modes or polarization modes) of the fiber are excited at the pump wavelength and two spectrally separated bands (signal and idler) appear due to the FWM.Each of the generated bands is in one of the two excited modes of the fiber.The phase-matching conditions and the modes overlaps are determined by the fiber properties and depend on the fiber design.The fiber properties and excitation condition determine which modes are involved in the FWM and what are the spectral positions of signal and idler bands.

Selective excitation
Recently presented method [6] enables selective excitation of different combinations of LP 01 and LP 11 polarization modes in a birefringent optical fiber.With this method, it is possible to excite numerous combinations of modes including the following pairs of modes: (i) different spatial modes of the same polarization, (ii) orthogonal polarization modes of the same spatial mode, (iii) orthogonal polarization modes of different spatial modes.
In the first two cases the well-known FWM processes can be observed: the intermodal FWM in the first case, and the vectorial FWM in the second case.The third case enables variety of FWM processes including two simultaneous intermodal-vectorial FWMs as described in our recent work [7].* e-mail: karol.tarnowski@pwr.edu.pl

Phase-matching condition and overlap coefficients
The phase-matching condition for the four-wave mixing process involving two pump modes (l and m), signal mode (p) and idler band (n) -where the propagation constants were written as a Taylor series up to the second ordercan be noted as follows [7]: where: ∆β (p,l) 0 and ∆β (m,n) 0 are the propagation constants differences at the pump wavelength; β (p,n) 1 is related to signal/idler group refractive index difference β(p,n) 2 is the average chromatic dispersion of the signal and idler modes; and Ω is the signal angular frequency detuning.
If only two modes are involved (p = l and m = n), then the free term zeros and Eq. ( 1) has one nontrivial solution However, in general case the free term does not vanish and up to two pairs of signal/idler bands can be generated [7].
The nonlinear overlap coefficient S K for N considered modes form a fourth-order tensor with N 4 elements [8].The coefficient for given mode combination must be nonzero to enable four-wave mixing process.The nonlinear overlap factors depend on the field distribution of the modes (F i ): where superscripts p, l, m, n indicate the mode combination.

LP 01 and LP 11 modes
First, we used a commercially available birefringent fiber (Nufern PM1550B-XP) that supports LP 01 and LP 11 modes at 1064.3 nm.The fiber cross-section is schematically shown in Fig. 1(a).The fiber was experimentally characterised in terms of chromatic dispersion of each mode as well as intermodal group and phase refractive indices differences.The numerical model was used to calculate the electric field distributions of the guided modes shown in Fig. 1(b) and to evaluate the overlap coefficients with Eq. (2b).
Knowing the fiber properties, we calculated analytically the positions of signal and idler bands for different excitation conditions.Additionally, we used the same parameters in numerical simulations based on the system of coupled nonlinear Schrodinger equations to simulate the intermodal-vectorial four-wave mixing processes.Finally, we measured the polarization-resolved spectra generated for different combinations of excited modes.The theoretical predictions based on analytical calculations and numerical simulations are in perfect agreement with experimentally observed spectra [7].
Example spectra generated when the pump excites LP y 01 and LP xe 11 pair are presented in Fig. 2. In this case, we observe signal and idler pair of bands generated in LP y 01 , LP xe 11 modes (in the two-mode process) and two pairs of bands genereted in intermodal-vectorial four-wave mixing processes in LP x 01 , LP ye 11 modes.In the next step, we performed analysis of intermodalvectorial processes involving additionally LP 02 and LP 21 modes.We considered the step-index fiber with stress applying elements (core diameter: 8.2 µm and refractive index core-cladding difference: 5 × 10 −3 ).The calculated electric field distributions of all 12 guided modes at 650 nm presented in Fig. 3 were used to calculate overlap coefficients for intermodal-vectorial four-wave mixing.

Figure 1 .
Figure 1.(a) Schematical cross section of the birefringent fiber.(b) Calculated electric field of the guided modes at 1064.3 nm.

Figure 2 .
Figure 2. Polarization resolved spectra revealing three pairs of signal/idler bands generated for LP y 01 /LP xe 11 excitation.

Figure 3 .
Figure 3. Calculated electric field of the guided modes at 650 nm.