1. Introduction

The generation of hollow optical potentials has found extensive applications in both life and physical sciences. Hollow beam is a light field whose intensity along the central axis is the minimum or vanishes in the ideal case along the beam axis. Various proposals have been carried out to generate this type beam. Such as axicon [1–3], hollow optical systems [4–9], multimode light guides [10, 11], waveguide with metal rods [12], and mode convertors [13, 14]. An impressive performances have been obtained by using multi-mode holey fiber [15], suggesting attractive fiber optic solutions. Recently it was shown that the optical vortex beam can be generated by using the coherent-superposition of multi-beams in a radially symmetric configuration [16–18]. However, it is impossible to keep the initial phase of the laser arrays stable in practical engineering excepting that the phase noise is corrected [19, 20]. The difficulty of this correction inspires us to explore the realization of laser beam array in the multi-core PCF.

Meanwhile, some biophotonics special applications need combining the hollow beam with various spectroscopic techniques, which can simultaneously manipulate and characterize a single trapped cell by gathering spectral information [21, 22]. Taking advantage of the design freedom of photonic crystal fibers, we demonstrate a GeO₂ doped triangular-core (three core) enhanced nonlinear PCF. With the femtosecond laser pulse is coupled into the triangular-core PCF, supercontinuum can be generated and a hollow beam can be produced in far-field region through the supermode superposition. Unlike a narrowband laser, supercontinuum hollow beam has an additional degree of freedom. Through mapping of different wavelengths to different positions by diffractive–dispersive optical elements, it is possible to create a chromatic optical trap that can bring new capability and versatility to manipulation.

2. Fiber fabrication and properties

PCFs are widely investigated both theoretically and experimentally due to their flexible management of dispersion and nonlinearity [23, 24]. The multi-core PCFs we designed is formed by three missing central neighboring air holes and made by the standard stack-and-draw process. A scanning electron microscope (SEM) image of the GeO₂-doped core PCF is displayed in Fig. 1 . The fiber cladding lattice has a pitch (hole spacing, Λ) of 3.7 μm and a hole diameter to pitch ratio (d/Λ) of 0.81. Generally, GeO₂ is always doped into the fiber core as a means of additional method to enhance nonlinearity and engineer the dispersion. Here we show that the GeO₂ dopant can obviously influence the field distribution of PCF, resulting in a novel characteristic of PCF for mode control. The GeO₂ doped regions (depicted by the white circles in Fig. 1) are oval with a dimension of 1.5 μm × 1 μm and have a relative refractive index difference (Δ) of 0.35%. This three-fold symmetry structure promises special polarization properties and far field distributions.

 

Fig. 1 Scanning electron micrograph of PCF end face, and white circles depict GeO₂ regions.

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The supermode$\tilde{E}$of the triangular-core PCF can be considered as a coupling superposition of the individual modes [25]

In addition, the full-vectorial finite-difference frequency-domain (FV-FDFD) method is employed to numerically calculate the group-velocity dispersion (GVD) and group index (Fig. 2(a) ), and the effective mode areas (Fig. 2(b)) of the fiber. The triangular-core structure displays a relative large effective area while the zero dispersion wavelength (ZDW) shifts toward short wavelength. The ZDWs of the two modes (the in-phase supermode is labeled with blue solid line and the degenerate out-phase supermode is labeled with green dashed line) are 992 nm and 869 nm and their effective areas are around 11 μm² and 12 μm² at pump wavelength of 1040 nm, respectively. That the effective mode area profile of the in-phase supermode is strikingly flat is induced by the coupling strength depending on the wavelength [26]. At short wavelengths, such as 500 nm, the modes are well guided in each cores, extend a little fraction into the adjacent cores and poorly overlap (see the pattern shown in Fig. 2(b) inset 1). A relative large effective mode area is 11.7 μm² at wavelength of 500 nm. As the wavelength is increased, the evanescent portion of the fields extends further away from the GeO₂ doped core, the mode overlap increases, and so does the coupling coefficient. And then the mode field is drawn to the central of the core (Fig. 2(b) inset 2), resulting in a smaller field area 10.9 μm² at 1040 nm. The fiber was slightly birefringent, allowing the sensitivity of the mode pattern to the input polarization. The fundamental modal birefringence is 4 × 10⁻⁵ at pump wavelength.

 

Fig. 2 GVDs and group index, that is speed of light divided by the group velocity (a), and wavelength dependences of effective mode area (b) of the in-phase supermode (blue solid line) and the out-phase supermode (green dashed line). The red dotted lines in (a) indicate the group-velocity matched wavelengths of the out-phase experiment. Insets in (b) depict the near field distributions of 0.5 μm (1) and pumping wavelength (2) in the in-phase supermode.

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More detailed supermode field profiles of the PCF are calculated in Fig. 3 . The near field distribution of the in-phase supermode (Fig. 3(a)) features a strong coupling strength between the adjacent cores, and the three fields nearly coupled together due to in-phase condition. The phase of each core is identical, which corresponds with the situation of in-phase coupling A₁ = [1,1,1] , and Gaussian-shaped supermode is obtained in far-filed [Fig. 3(d) ]. Furthermore, there are two out-phase supermodes which are depicted in Fig. 3(b) and Fig. 3(c), the field phases of the cores are consistent with the eigenvectors of the coupled equation A₂ = [-1, 1, 0], A₃ = [-1, 0, 1] respectively. One far field profile of them is also given in Fig. 3(e), what is noteworthy is that the intensity in center almost vanishes. Generally, hollow-beam modes arise as a superposition of two or more modes [15, 27].When propagating in the fiber, the effective refractive index of these involved modes should be close enough for co-excitation and have comparable intensities. For the PCF considered here, both of the out-phase supermodes will contributed to the hollow-beam modes, and the effective refractive index difference of the two modes is around 8 × 10⁻⁵ at wavelength of 600 nm. The normalized superposition of their far filed is displayed Fig. 3(f), apparently a good-shape hollow-beam.

 

Fig. 3 Calculated near (top) and far field (bottom) distribution of the in-phase ((a), (d)) and out-phase ((b), (c), (e), (f)) mode, where (e) is the far field of out-phase supermode (b) and (f) is superposition of out-phase supermode (b) and (c) .

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3. Experiments and discussion

The fabricated PCF was pumped by a femtosecond Yb-doped PCF laser oscillator working in a dissipative mode-locking regime [28]. Nonlinear polarization rotation technique assisted with a spectral-filtering effect enables the generation of 10 W of average power at a 49 MHz repetition rate at a central wavelength of 1038 nm. A transmission grating compressor dechirped the output pulses down to 95 fs. An optical isolator is placed at the output of the oscillator to prevent light feedback into laser system. The pump power is controlled by the combination of a half-wave plate and a polarizer and the polarization of the input pulses is controlled with another half-wave plate placed in front of the microscope objective. Laser radiation in our experiments was coupled into the fiber with a coupling efficiency up to 65%. It is worth mentioning that the input beam is aligned exactly along the axis of the PCF without tilting in the whole experiment. The output beam was coupled into a high-resolution spectrometer (ANDO 6315A) to record spectrum.

All experiments were performed in the anomalous dispersion regime of a 40-cm PCF, the nonlinear process is dominated by soliton-related nonlinear propagation: Higher-order solitons are formed from the input pulses and then, because of perturbations such as higher-order dispersion and/or Raman scattering, separate into fundamental solitons, which is well-known as soliton fission. The same perturbations also modify the subsequent evolution of the ejected solitons, leading to the shedding of the solitons energy via the generation of dispersive wave (DW) in the normal dispersion regime. The DW will blue-shift due to the “trapping effect” driven by the soliton self-frequency shift [29].

As mentioned in section 2, the triangular-core PCF was slightly birefringent, so laser polarization was aligned to parallel to one of the principal axis using the half-wave plate and then laser field propagated in the fundamental in-phase supermode within the PCF. The visible laser field nearly uniformly distributed in the three fiber cores (Fig. 4(a) , recorded at pump power of 1.04 W). The discrepancy from Fig. 3(d) should be caused by measurement. A typical six-fold symmetry pattern called in-phase supermode (Fig. 4(b)) was obtained in far-field recorded by a CCD camera. In addition, there was a annular violet outside the pattern which was not expected from our design and was probably due to a Cherenkov radiation generation in a higher mode [30]. Figure 5(c) shows the spectrum evolving with the input power up to 1.04 W. A supercontinuum is obtained in the output of fiber thanks to the small effective mode area (Fig. 2(b)) and high peak power. The spectrum is characteristic of red-shifted soliton and trapping dispersive waves, lining with group-velocity matching. A relative high power on short-wavelength side of the pump (occupying about 56% of the total energy at pump power of 1.04 W) is due to that the pump pulses are close to the ZDW, allowing more spectral component of the solitons to cross ZDW and transform to dispersive wave.

 

Fig. 4 Results of experiment for first order mode. (a) the near field profile of the PCF output, (b) the far field profile of the PCF output and (c) the spectral intensity of the PCF output as a function of the wavelength and the average power of the input field.

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Fig. 5 Experiment results of the out-phase supermode superposition. (a) the far field profile of the PCF when one mode’s output is visible, (b) the far field profile of the superposed mode and (c) the spectral intensity of the PCF output with a input power of 1.04 W.

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The out-phase supermodes were excited by adjusting the input polarization. An intense nonlinear spectra broadening similarly took place. As is well-known, the short-wavelength edge of supercontinuum is mainly originate from DW whose initial frequency${\omega }_{DW}$is determined by a phase-match condition$\beta ({\omega }_{DW})={\beta }_{sol}({\omega }_{DW})$($\beta$is the wave number of linear wave and${\beta }_{sol}$is the host soliton wave number). The phase-match condition strongly depends on the dispersion of the fiber. The slight difference in dispersion between the two out-phase modes (that is not shown in Fig. 2(a)) results in a different DW frequency around 625 nm (the nonlinear phase is neglected) and the subsequent blue-shift of DW and new frequency generation [30]. At a suitable low pump power, only one mode extended its spectrum into the visible region. The recorded far-field is presented in Fig. 5(a) that is well agreed with the calculated result (Fig. 3(e)). As we increased pump power, two out-phase supermodes appeared and overlapped into a hollow beam pattern (Fig. 5(b)), which supports the symmetry of the PCF cross-section structure and is consistent with the calculated result (Fig. 3(f)). The output was passed a polarizer to analysis the state of polarization. The power behind the polarizer varied little when we rotated the polarizer. Combining the theory of superposition hollow beam [15, 27] and previous calculation, we infer that the output should be radial or spiral polarization pattern. When increasing pump power up to 1.04 W, we recorded the output spectrum (Fig. 5(c)) spanning from 540 nm to 1540 nm. The total average output power was 670 mW and the continuum appeared bright reddish yellow to the eye. The average spectral powers were around 400 nW/nm in the visible spectral region, which is large enough for many applications. The bandwidth, power and flatness of the continuum can be improved by increasing the fiber length and input power, and further improved by carefully designing the fiber and pump conditions. The well-resolved interference fringes manifest the coexistence of more than one mode. Two distinguishable solitons located at the long-wavelength edge contribute the DW that blue-shifts from original wavelength to short-wavelength and then forms the blue edge of the supercontinuum. This interaction between soliton and DW is well-known as “trapping effect” [31] and is promoted by the GeO₂ doped in the core [32]. Two mechanisms have been suggested for this interaction: Cross-phase modulation (XPM) [33] and a type of four-wave mixing (FWM) [30]. For both of the two methods, one feature of this effect is group-velocity matching [34] which is marked with red dotted lines in Fig. 5(c), as well as red dotted lines in Fig. 2(a). The theoretical calculation and the experimental observation agree extremely well on the wavelength position of the dispersive wave.

4. Conclusion

In conclusion, we experimentally demonstrated a hollow beam supercontinuum generation in a GeO₂ doped triangular-core PCF. The GeO₂-doping assists in structuring the guiding region: Each doped region can act as an independent waveguide and its phase can be controlled by the incident polarization. The hollow beam is a superposition of out-phase supermode but can be easily excited with a suitable incidence polarization. In this case, the PCF behaves as a simple mode convertors along with nonlinear-optical transformation. In addition, an in-phase supermode was obtained when the laser was polarized parallel to one of the main axis of the PCF. The high pump peak power gives rise to supercontinuum generations both in hollow mode and in-phase supermode, and the spectra can be controlled by input power due to the inherent feature of nonlinear broadening. The output wideband free-space hollow beam supplies a flexible laser source in both life and physical sciences.