1. Introduction

Fibers supporting several higher-order modes have recently received considerable attention in nonlinear fiber optics, sensing, optical communication, quantum optics, and fiber lasers because of their increased flexibility compared to single-mode fibers. For example, a wide range of new nonlinear phenomena have been observed in two- and few-mode fibers (FMFs) [1–3], such as supercontinuum generation [4,5], soliton propagation [6], and intermodal four-wave mixing (IFWM) [7–10], including cross-polarization four-wave mixing [10]. FMFs have also been studied as transmission links with capacity increased by spatial mode division multiplexing [11–13]. In this context, FMF-based amplifiers have recently been developed [13]. Moreover, FMFs are used in numerous devices such as dispersion compensators [14], polarization filters [15], high-power lasers [16], multi-parameter sensors based on intermodal interactions [17], and mode converters [18–20].

For practical applications of FMFs, it is important to selectively excite the required spatial and polarization modes. The most popular method of higher-order mode excitation is to focus an offset or tilted beam on the fiber core. Although such an approach is very simple, it suffers from a lack of selectivity. Increased efficiency and selectivity of excitation of specific modes can be achieved by imprinting a phase mask on the fiber end-face [21,22] or by inscribing an in-fiber long-period grating [19,23]. Such methods are applicable to high power sources but, unfortunately, they are non-tunable (neither spatially nor spectrally) because of required permanent modification of the fiber structure. Methods that allow switching between different spatial modes, however still spectrally limited, include using mode converters [24–26], stress-induced gratings [27], free space phase masks [28], and dynamic beam shaping by spatial light modulators (SLMs) [29–31]. Recently, SLM and Wollaston prism (WP) have been used in combination to generate free-space vortex beams [32–34], however, this method has not yet been used to excite optical fibers.

In this study, we propose a simple method for free-space selective excitation of different combinations of fundamental (LP₀₁) and first-order (LP₁₁) modes in a birefringent fiber. The input system consists of a Wollaston prism, rotatable polarizer, and achromatic half-wave plate, and allows for the excitation of individual modes, pairs, or larger groups of spatial LP₀₁ and LP₁₁ modes in parallel or orthogonal polarizations. It is worth to mention that the free-space coupling system employing a three-surface prism has been proposed in [35] for high capacity MDM/WDM transmission using LP₀₁ and LP₁₁ modes, however, this approach does not allow to control polarization of excited modes. The main advantages of our method over other solutions known from the literature are the possibility of high-power operation, low wavelength dependence, and tunability, allowing for excitation of almost all possible combinations of the LP₀₁ and LP₁₁ modes. These features make the proposed method suitable for numerous applications involving LP₀₁ and LP₁₁ modes, such as studying intermodal and cross-polarization nonlinear phenomena, broadband mode conversion, intermodal interference sensors [17], and broadband measurements of chromatic dispersion of individual modes [31,36], as well as intermodal dispersion for selected pairs of modes [37,38]. The input field distribution can easily be modified to match LP₁₁ modes in fibers with different geometries and core diameters by longitudinally shifting the WP, which changes the distance between the two light spots split by the WP and focused on the fiber end-face. The selectivity of the proposed method was experimentally confirmed by the high suppression ratio (exceeding 20 dB) of the unwanted modes with respect to the targeted mode.

As an example application of the proposed excitation method, we show the generation of Raman sidebands in different modes and gain tunability of intermodal four-wave mixing in a highly birefringent polarization-maintaining fiber. We also experimentally demonstrate a broadband operation of the method by converting the supercontinuum light beam from the fundamental LP₀₁ to the LP₁₁ mode in the spectral range from 550 to over 1100 nm. Moreover, we present specific alignments of the input system allowing the excitation of different groups of modes and the optimal system parameters determined by numerical simulations that ensure the highest possible coupling efficiencies.

2. Principle of operation of the Wollaston prism-based input system

The proposed input system is shown in Fig. 1(a), with indicated example azimuths of the polarization elements used for the selective excitation of one of the LP₁₁ modes (LP₁₁^ye), one of the fundamental modes (LP₀₁^x) or both, LP₀₁^x and LP₁₁^ye, modes. The polarization azimuth of a linearly polarized input beam from the light source LS (either a laser or a broadband source, such as a supercontinuum) is set at a 45° angle with respect to the WP polarization axes. The input beam is split by the WP at an angle of 2θ_WP into two orthogonally polarized beams (Fig. 2). The polarization azimuths of the two beams are then rotated by a half-wave plate λ/2 adjusted at an angle of α_λ/2 = ±22.5° with respect to the x-axis of the coordinate system. The linear polarizer P with a transmission azimuth set vertically (y-axis) blocks the x-polarized components of the two beams, which are further focused on the input fiber end-face by a microscope objective MO. The field distribution created this way at the fiber input is polarized in the y-direction and is antisymmetric in phase, similar to the LP₁₁^ye mode shown in Fig. 1(b). If the splitting angle of the Wollaston prism 2θ_WP as well as the numerical aperture and focal length of MO are properly adjusted, this input field distribution allows the selective and efficient excitation of the LP₁₁^ye mode. By rotating the polarizer by 90°, one can block the antisymmetric (y-polarized) component of the input beam and excite only the symmetric, x-polarized fundamental mode LP₀₁^x. Removing the polarizer from the input beam allows the simultaneous excitation of both modes LP₁₁^ye and LP₀₁^x.

 

Fig. 1. (a) Input system used for selective excitation of the LP₀₁ and LP₁₁ fiber modes:

LS – light source (laser or broadband source), WP – Wollaston prism, λ/2 – half-wave plates, P – linear polarizer (removed to excite specific mode combinations), MO – microscope objective, F – fiber, OSA – optical spectrum analyzer, Δt_WP, Δl_WP – transverse and longitudinal displacement of the WP from its initial position. The amplitude distributions of the input beam after transformations by successive system elements and on the fiber end-face for different settings of the polarizer P: α_P = 90° – excitation of LP₁₁^ye mode, α_P = 0° – excitation of LP₀₁^x mode, with the polarizer removed – excitation of LP₁₁^ye and LP₀₁^x modes simultaneously. (b) Field structure in x- and y-polarized symmetric and antisymmetric modes from the LP₀₁ and LP₁₁ groups in birefringent fiber with stress-applying elements.

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Fig. 2. (a) Schema illustrating the operation principle of the Wollaston prism and the effect of different system parameters on light distribution at the fiber input; (b) x- and y-polarized components of the field distributions at the fiber end-face for different phase shifts δ₀ between the split beams introduced by transverse displacement of the WP.

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The structure of the field distribution focused on the fiber end-face can also be modified by varying other parameters of the input system, such as the transverse (t_WP) and longitudinal (l_WP) positions of the WP and its angular orientation (α_WP). By transversely shifting the WP by Δt_WP, one can control the initial phase difference δ₀ between the split beams and tune the relative power division between the fundamental and first-order modes of the same polarization. For non-collimated input beam (Fig. 2), the longitudinal shifting of the WP with respect to the focusing objective by Δl_WP changes the spatial separation of the split beams on the fiber end-face and increases the coupling efficiency. Finally, by changing the azimuth α_WP of the WP, the split light spots can be rotated with respect to the fiber end-face and excite different combinations of spatial LP₁₁ modes. These features ensure that the proposed input system is highly flexible and allow for the excitation of almost every combination of LP₀₁ and LP₁₁ modes in fibers with different geometries. Specific system configurations for exciting individual modes or group of modes are discussed in detail in Section 3.

The effect of the WP alignment on the resulting field distribution at the fiber input end-face is schematically illustrated in Fig. 2. The WP split angle 2θ_WP is related to the apical angle γ_WP of the two constituent crystal prisms, as follows:

The phase shift δ₀ between the split beams can be continuously tuned by the lateral displacement of the WP according to the following equation:

For a collimated input beam (l = ∞), the longitudinal position of the WP (l_WP) has no effect on the field distribution at the fiber input. In the case of a divergent/convergent beam, the separation 2D of the light spots on the fiber end-face can be tuned by longitudinally shifting the WP. It should be noted that the two light spots at the fiber end-face arise through imaging of two virtual foci of the illuminating beam located at a distance l from the MO and separated by 2d = (l – l_WP)tan(2θ_WP). By applying the thin lens equation, it is possible to derive a simple relation linking the light spot separation 2D on the fiber end-face with the longitudinal position of the Wollaston prism l_WP, the position of the input beam focus l, and the focal length f of the microscope objective:

The above equation shows that by changing the distance l_WP, the light spots separation 2D on the fiber end-face can be continuously tuned from zero (for l = l_WP) to the maximum value reached at l_WP = 0. This allows the size of the input field distribution to be matched to the targeted mode in fibers of different constructions and maximizes the coupling efficiency.

3. System alignments for excitation of specific modes

To analyze the performance of the proposed input system and identify its parameters allowing the excitation of specific individual modes or groups of modes, we conducted numerical simulations based on the Jones matrix formalism. In this approach, polarization transformation by successive elements, as well as focusing properties of the microscope objective are considered. This finally allows us to link the initial polarization state of the input beam and the input system parameters with the distribution of the complex amplitude on the fiber end-face E_i(x,y). The initial polarization state of the beam incident on the WP (i.e. after the first half-wave plate shown in Fig. 1) is represented by the following Jones vector:

As previously explained in Section 2, the separation distance 2D between the two light spots on the fiber end-face depends on the system parameters according to Eq. (3). Without loss of generality, we assumed in our simulations that the incident beam is collimated (i.e., l = ∞), and in this case:

Finally, the distribution of the field E_i focused on the fiber end-face can be expressed as:

For the input system without the polarizer P, T_P becomes the identity matrix.

Using Eqs. (4)-(10), we calculated the complex field distributions on the fiber end-face for specific alignments of the polarization elements (with and without the polarizer P) while keeping the other system parameters constant: λ = 1064 nm, 2θ_WP = 0.01°, and a microscope objective with 5× magnification (i.e., f = 33 mm and NA = 0.12). The calculated x- and y-polarized components of the complex amplitude E_i were then decomposed into three constituents:

We also confirmed numerically that the component that is antisymmetric with respect to both the x- and y-axes, matching the modes LP₂₁^xo/ye, is null, therefore, these modes cannot be parasitically excited in the spectral range below their cut-off.

All possible combinations of modes that can be excited using the proposed method and the corresponding system alignments are described in Table 1. The azimuth angles specified in Table 1 were counted from the positive x-semiaxis. The only pairs of modes that cannot be excited selectively using the proposed input system are LP₁₁^xe+LP₁₁^ye and LP₁₁^xo+LP₁₁^yo because the mode fields required to excite each of these pairs also match other LP₁₁ modes, and thus, all four LP₁₁ modes are excited, as shown in Table 1, row (h). The pairs LP₁₁^xe+LP₁₁^yo and LP₁₁^xo+LP₁₁^ye, as well as the combination of all four LP₁₁ modes, can be excited; however, as indicated in Table 1, rows (e) and (h), in these cases, an additional half-wave plate with azimuth α_λ/2 = 22.5°, placed after the polarizer P, is necessary. An unavoidable drawback of the system configurations requiring the polarizer P is that the power of the input beam is reduced by half. Therefore, in some applications, such as the generation of nonlinear processes, a stronger light source may be required.

Table 1. Parameters required to excite specific modes.

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Finally, we determined the optimal values of NA and D by maximizing the overlap integrals between the input field distributions E_i(x,y) and fiber modes E_q(x,y) versus these parameters using the following relation:

The optimization was performed for λ = 1064 nm and the geometry of the fiber used in the experiments presented in the next section (i.e., core diameter of 8.5 µm and numerical aperture of 0.125). The maximum overlap between the input field and the modes from the LP₀₁ group occurs for NA = 0.10 and D = 2.2 µm and is equal to 86%, which corresponds to 43% coupling efficiency with reference to the initial power of the input beam (a loss of 3 dB is caused by the polarizer P). For the modes from the LP₁₁ group, the maximum overlap was achieved for NA = 0.12 and D = 4.0 µm and varied between 72% and 76% depending on the mode symmetry and polarization, corresponding to a coupling efficiency of 36% to 38%. The parameters NA = 0.12 and D = 3.0 µm ensured almost the same coupling efficiency for all six individually excited modes (ranging from 34% to 36%). This means that for the system configurations described in Table 1, row (a), which allow for the excitation of six individual modes, up to 36% of the input power can be coupled into the targeted mode. For excitation of pairs of LP₀₁ and LP₁₁ modes with use of a polarizer P, as shown in Table 1, rows (b) and (e), up to 18% of the input power can be coupled into each of the two targeted modes. For the system configurations with the removed polarizer described in Table 1, row (d), which allow for the excitation of two modes in orthogonal polarizations, approximately 35% of the input power can be coupled into each mode.

For the optimized values of NA and D, we also calculated the overlap integrals for the input system with the Wollaston prism oriented at α_WP = 45° with respect to the fiber polarization axes (Table 1, row (c)), which allows the excitation of pairs of x- or y-polarized symmetric and antisymmetric LP₁₁ modes (LP₁₁^xe+LP₁₁^xo or LP₁₁^ye+LP₁₁^yo). For these system configurations, the coupling efficiency varies between 17% and 18%, indicating that the targeted LP₁₁ modes are excited roughly evenly within each pair.

According to Eq. (9), for the microscope objective with 5× magnification (NA = 0.12, f = 33 mm), for the collimated input beam the value of D = 3.0 µm (ensuring the highest coupling efficiency for six individually excited modes) corresponds to 2θ_WP = 0.01°. For the fixed split angle of the WP, a convergent input beam can be used according to Eq. (3) to allow tuning the value of D by longitudinally shifting the WP and matching the modal field distributions to fibers of different geometries.

In our simulations, we focused on specific alignments of the input system that yielded symmetric field distributions required for the even excitation of two or more modes (δ₀ = 0 or δ₀ = π/2 and polarization azimuths set along the fiber axes or at a 45° angle). If necessary, the power division between the targeted modes can be tuned by the transverse displacement of the WP, as shown in Fig. 2(b), or by rotating the respective polarization elements.

4. Experimental validation

We verified the effectiveness and selectivity of the proposed excitation method in nonlinear experiments involving LP₀₁ and LP₁₁ modes, such as observation of Raman sidebands for selectively excited individual modes and tuning the gain of intermodal four-wave mixing for selectively excited pairs of modes. We also experimentally proved the broadband operation range of the proposed method by converting the supercontinuum light beam from the fundamental LP₀₁ to the LP₁₁ mode in the spectral range from 550 to over 1100 nm. To conduct these experiments, we used a 12 m long highly birefringent polarization-maintaining fiber with stress-applying elements (core diameter of 8.5 µm, NA = 0.125). The cut-off wavelengths of the LP₁₁ and LP₂₁ modes in this fiber were 1300 and 850 nm, respectively. For the light source, we used a 1064.3 nm Nd:YAG pulse laser with a repetition rate of 20 kHz, pulse duration of 1 ns, average power of 140 mW, and beam divergence of 0.35°. The split angle of the quartz Wollaston prism was 2θ_WP = 0.04°. To focus the input beam on the fiber end-face, we used a 5× microscope objective with NA = 0.12 and f = 33 mm. Because the split angle of the used WP was not optimal, we took advantage of the slight divergence of the laser beam and adjusted the position of the laser and WP with respect to the microscope objective to obtain the greatest possible coupling efficiency (for l = 0.3 m and l_WP = 0.2 m, the optimal split angle is 2θ_WP = 0.035°). Thus, it was possible to couple over 30% of the laser beam into each individually excited mode for the setup configuration shown in Table 1, row (a) and approximately 60% for pairs of modes excited in the setup without the polarizer (Table 1, row (d)).

The selectivity of each mode excitation was confirmed in the generation of Raman and intermodal four-wave mixing bands. For the setup configurations shown in Table 1, row (a), only the Stokes (1117.5 nm) and anti-Stokes (1016.8 nm) bands were observed in the output spectrum, with no trace of peaks related to intermodal four-wave mixing (Fig. 3). Using a linear polarizer, a diffraction grating, different gray filters, and a camera, we were able to register the intensity distributions at the fiber output for the pump and the Raman sidebands, which confirmed that the sidebands were generated in the targeted modes.

 

Fig. 3. Spectra registered at the fiber output for the input system configurations that excite individual modes (Table 1, row (a)), as well as images of respective modes for the pump and Raman sidebands registered using a diffraction grating at the fiber output.

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In the next step, by transversely shifting the WP, we gradually tuned the power division between the LP₀₁ and LP₁₁ modes of the same polarization, which resulted in the appearance of the IFWM peaks in the output spectrum. As shown in Fig. 4, for the initial selective excitation of the LP₁₁^yo mode (corresponding to the system alignment specified in Table 1, row (a)), the transverse displacement of the WP from its central position introduced a non-zero phase shift δ₀, which resulted in a gradual excitation of the LP₀₁^y mode. This led to the appearance of the IFWM peaks at the wavelengths of 1038.2 (in LP₀₁^y mode) and 1091.3 nm (in LP₁₁^yo mode), shown in Fig. 4. The IFWM peaks reached the maximum gain for δ₀ = π/2, which ensured almost even power division of the pump between the LP₀₁^y and LP₁₁^yo modes and gradually decayed as δ₀ approached π. For δ₀ = π, the IFWM peaks completely vanished because for this system alignment, only the fundamental LP₀₁^y mode is excited at the pump wavelength.

 

Fig. 4. Spectra registered for different initial phase shifts δ₀ between the split beams tuned by transverse displacement of the Wollaston prism with clearly visible IFWM peaks. The starting (δ₀ = 0) and end (δ₀ = π) input system alignments are detailed in Table 1, row (a), whereas the alignment ensuring even power division between the two modes (δ₀ = π/2) is shown in Table 1, row (b).

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As shown in Fig. 4, for the starting (δ₀ = 0) and end (δ₀ = π) input system alignments, when only one mode is excited, the IFWM sidebands are suppressed by 52-60 dB compared to the maximal gain. This indirectly proves the high excitation selectivity of the targeted modes. It is worth mentioning that other IFWM sidebands arising at different wavelengths can be generated by exciting the respective pairs of the LP₀₁ and LP₁₁ modes.

Finally, to confirm the broadband operation range of the proposed excitation method, we present the results of the conversion of the supercontinuum beam propagating in a single-mode fiber (spectral range 400-2300 nm) into the LP₁₁ mode of the birefringent polarization-maintaining fiber used in the IFWM experiments. To purely excite the LP₁₁ mode (LP₁₁^xo as an example), we used the input system configuration specified in Table 1, row (a). The modal structure of the converted supercontinuum was observed using a color camera (operation range of 400-800 nm) and a monochromatic camera (400-1100 nm) with different narrowband filters (FWHM = 10 nm). The top row in Fig. 5 shows the intensity distributions at the output of the fiber registered using a monochromatic camera and different interference filters. The two bottom rows show the converted supercontinuum at the fiber output diffracted in the horizontal direction by a reflective grating (600 lines/mm).

 

Fig. 5. Intensity distribution in the supercontinuum beam converted into the LP₁₁^xo mode registered by a camera with narrowband filters (top row) and the beam diffracted in the horizontal direction by a reflective grating (middle and bottom rows).

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At a wavelength of 550 nm, the fiber supports the following groups of modes: LP₀₁, LP₁₁, LP₂₁, LP₀₂, LP₃₁, and LP₁₂; however, the registered intensity distribution is characteristic of the pure LP₁₁ mode. The same pattern is visible in almost the entire spectral range (550-1100 nm), which can be analyzed with the cameras used. Higher-order modes are parasitically excited only at wavelengths below 500 nm because of the increasing mismatch between the input field distribution and the LP₁₁ mode caused by the small spot size, residual variation of δ₀ with wavelength, and chromatic aberrations of the microscope objective. A drop in the intensity above 1000 nm is caused by the decreasing sensitivity of the camera.

The coupling efficiency was measured by comparing the power at the input (before the polarizer) and the output of the birefringent fiber for different wavelengths selected by narrowband spectral filters. The power relation showed minor wavelength dependence varying between 17% and 20% in the range from 450 to 900 nm and increased to 27% at longer wavelengths. We believe that the greater loss in the shorter wavelength range was caused mostly by chromatic aberration of the microscope objective and spectral dependence of the WP split angle.

Additionally, we estimated the selectivity of mode excitation by comparing the output spectrum for the supercontinuum beam coupled into the LP₁₁^xo mode and the LP₀₁^x mode of the birefringent fiber after transverse displacement of the Wollaston prism (Fig. 6). The intensity drop visible above 1300 nm (i.e., above the cut-off wavelength of the LP₁₁ mode) from -50 to -72 dBm proves that for the selectively excited LP₁₁ mode, the contribution of the fundamental mode is at least 20 dB smaller.

 

Fig. 6. Output spectrum registered with OSA for the supercontinuum beam coupled into LP₀₁^x (blue line) and LP₁₁^xo (red line) modes of the birefringent two-mode fiber.

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5. Conclusions

We proposed a simple method for the selective excitation of individual modes and different combinations of LP₀₁ and LP₁₁ modes in a birefringent optical fiber and identified specific system alignments allowing for the excitation of different groups of modes. Based on the numerical simulations, we showed that the coupling efficiency may reach up to 36% (with a loss of 3 dB caused by the polarizer) for each individually excited mode, whereas for the pairs of LP₀₁ and LP₁₁ modes of orthogonal polarization (using the input system without a polarizer; see Table 1, row (d)), up to 36% of the initial power is coupled into each mode. We validated the method experimentally and proved its feasibility in nonlinear experiments, such as the generation of Raman sidebands in all six selectively excited polarization modes. We also demonstrated the possibility of fine tuning of the intermodal four-wave mixing gain by precisely controlling the pump power division between the LP₀₁ and LP₁₁ modes of the same polarization. Moreover, we experimentally confirmed the broadband operation range of the proposed method by converting the supercontinuum beam from the LP₀₁ mode to the LP₁₁ mode in the spectral range from 550 to over 1100 nm with modal selectivity exceeding 20 dB. The demonstrated broadband coupling of the supercontinuum beam into specific modes (or pairs of modes) can be further exploited for chromatic dispersion (or intermodal dispersion) measurements using white-light interferometry techniques.

The advantages of the proposed excitation method over other methods employing spatial light modulators, mode couplers, and phase plates are low wavelength dependence and applicability to high-power beams, while maintaining the possibility of switching between different modes. The main drawback of the proposed method is a power loss of 3 dB related to the use of a linear polarizer, however, in many applications, such a loss can easily be compensated by increasing the power of the light source. Moreover, the proposed method can be extended to the excitation of selected modes from the LP₂₁ group by using two crossed Wollaston prisms. Because all components of the coupling system can be miniaturized, we expect that it has potential as a compact, tunable broadband mode converter.