We have studied the excitation of higher-order modes and their role in supercontinuum generation in a three-hole silica suspended-core fiber, both experimentally and numerically. We find that pump coupling optimized to highest transmission can yield substantial excitation of higher order modes. With up to about 40% of the pump power coupled to higher order modes, we have studied supercontinuum generation in this fiber. In agreement with experiments, simulation results based on the multimode generalized nonlinear Schrödinger equation confirm that the spectral width is determined by spectral broadening in the fundamental mode, whereas the numerical analysis reveals that intermodal nonlinear interactions are strongly suppressed.
© 2013 Optical Society of America
During the last decade, supercontinuum light sources have successfully made their way into a number of applications in fields such as optical metrology, spectroscopy, telecommunications and imaging . This development has become possible particularly thanks to the introduction of microstructured optical fibers (MOFs) [2, 3]. These fibers are characterized by a cladding structure of air holes running along the length of the fiber, which enables the tailored design of the linear and nonlinear optical guiding properties over a broad range. For instance, by the increase of the relative hole size, i.e. the air-filling fraction, the effect of waveguide dispersion can be enhanced to shift the zero-dispersion wavelength towards shorter wavelengths. This has enabled moving the onset of anomalous dispersion in fused silica fibers down to the visible range , providing a much greater flexibility in the selection of suitable pump laser sources. Typically, MOFs designed with a large air-filling ratio are multimoded when the core size has not been scaled accordingly . Although the fundamental mode can in practice be quite efficiently excited and thus a high beam quality achieved, recent developments have actually shown an increased interest in studying the role of higher-order modes (HOMs) in supercontin-uum generation (SCG) [6–17].
These studies have been motivated, on the one hand, by the steady demand for broadband light sources with increased output powers, which inevitably requires larger core sizes and can thus involve multimode operation. On the other hand, HOMs offer access to distinct dispersion properties, which are not obtainable by the fundamental mode. In particular, this has been utilized to efficiently extend the short-wavelength edge of the supercontinua into the ultraviolet [6, 16, 17].
One particular design of MOFs that achieves an air-filling fraction of nearly 100% are suspended-core fibers (SCFs), first reported with lead silicate glass . The characteristic cladding structure of these fibers is described by a single ring of large air holes. The small core is thus only attached through thin walls to the outer solid cladding. Large fractions of the mode energy can reach into the air holes, which has made these fibers particularly attractive for sensing applications [19–22].
Compared to other MOF designs, SCFs offer the potential to reach much higher numerical apertures and thus enable smaller core sizes with strong confinement of the light . This has been taken advantage of for efficient SCG in silica SCFs . Furthermore, by increasing the number of supporting walls while simultaneously decreasing their width, SCFs ultimately approach the dispersion characteristics of optical nanofibers . In this limit all-normal dispersion can be tailored with minimum dispersion wavelengths located down to the ultraviolet spectral range for coherent SCG [25, 26]. Benefiting from the transmission characteristics in soft glass SCFs, impressive spectral broadening has also been demonstrated in the mid-infrared spectral range [27, 28].
The numerical analyses complementing these experimental results [26–28] assumed pulse propagation in the fundamental mode only, as light guidance in HOMs was found to be suppressed under the particular conditions . This is not necessarily the case in untapered silica SCFs, where in principle numerous HOMs can be guided. Their initial excitation is enhanced by tuning the pump laser beam from the fiber core center , which, however, is typically also expected to go along with a decrease in the total light transmission.
In this paper, we demonstrate that depending on the size of the pump laser beam relative to the fiber geometry substantial HOM excitation can be realized in a straightforward way by simply optimizing the light transmission. This approach has enabled us to couple up to about 40 % of the laser power into HOMs and study their nonlinear pulse propagation, both experimentally and numerically. With the simulations based on the multimode generalized nonlinear Schrödinger equation (MM-GNLSE) we have been able to analyze the strengths of nonlinear intermodal processes and their contribution to spectral broadening in this fiber, reaching good agreement with the experimental results.
2. Fiber properties
In the present study an air-filled, fused-silica three-hole SCF was used. A scanning-electron microscopy (SEM) image of the fiber’s end facet is shown in Fig. 1(a). The image was taken after coating the end facet first by a 1.5 nm thick titanium layer and then by a 20 nm thick gold layer to avoid charge build-up and improve the image quality. From the SEM we accurately determined the fiber geometry. In the close vicinity of the core, the fiber geometry can be described by three parameters: the in-circle core radius, r = 875 nm, the curvature radius of the air holes near the core, R = 3350 nm, and the wall thickness near the core, d = 600 nm, as indicated in Fig. 1(a).
In order to take the complete information and thus also small discrepancies from this ideal fiber geometry into account, we converted for the analysis of the linear optical guiding properties the SEM image into vector format and then properly scaled it to the fiber dimensions. Based on this information we performed a full vectorial finite-element method (FEM) analysis utilizing COMSOL Multiphysics® software and found out that the fiber can in principle guide up to 12 modes in the vicinity of 780 nm, which is in the operating wavelength range of the Ti:Sapphire laser used in the experiments. Of those, we calculated for the six lowest-order modes (labeled M1 – M6) the wavelength dependent propagation constants which yield the group-velocity dispersion parameters shown in Fig. 1(b). We could locate the zero-dispersion wavelength (ZDW) of the fundamental mode M1 at 760 nm, whereas the ZDWs of the HOMs are blue-shifted to about 600 nm. Together with the corresponding mode-intensity distributions presented in Fig. 1(c), we find that modes M1 and M2 as well as modes M4 and M5 form degenerate pairs, differing only by their polarization state. We note that despite their similarities in the mode-intensity distributions, modes M3 and M6 differ by the wavelength dependence of the group-velocity dispersion parameters. The high index contrast at the core-air interface results in a high numerical aperture (NAf = 0.68), which tightly confines the light within the core.
3. Experimental results
We studied experimentally the nonlinear spectral broadening in the SCF by pumping the fiber with 150 fs long light pulses emitted from a wavelength-tunable Ti:Sapphire mode-locked laser operating at a repetition rate of 82 MHz. Optimized for highest transmission, the light pulses were coupled through a microscope objective (NAo = 0.65) into the fiber core. A tunable neutral density filter placed in front of the objective was used to vary the pump power. The light emitted from the SCF was either spectrally analyzed by an optical spectrum analyzer (ANDO AO-6315E) or imaged through a bandpass filter onto a CCD camera for recording the near-field images at different wavelengths.
The spectra measured at the output of a 33.5 cm long piece of the fiber are shown in Fig. 2. For these measurements, three different pump wavelengths, λ = 742 nm, 780 nm and 800 nm, and coupled pump peak powers P ranging from 5.3 kW to 17.7 kW were selected. The spectral width increases with increasing pump power but also the further the pump wavelength is located in the anomalous dispersion regime. At high enough pump powers, the spectra appear mostly smoothed with nearly the entire spectral range being contained within ≈ 10 dB from the maximum spectral power. For wavelengths shorter than the pump wavelength, the emitted powers are slightly enhanced, leading to a visibly white emission. Also in the vicinity of the pump wavelength, a clear peak appears in all spectra.
We selected exemplarily one of the pumping conditions (λ = 780 nm; P = 10.6 kW) and studied also the near-field profiles at discrete wavelengths of the generated supercontinuum. The recorded near-field images together with the analyzed spectrum are shown in Fig. 3. We observe, particularly from the near-field images at the pump wavelength but also from those taken at wavelengths up to ≈ 100 nm further into the IR, that HOMs are not only excited but also experience spectral broadening.
4. Numerical analysis and discussion
In order to gain deeper insights into the SCG process and to study the role of the HOMs in particular, we compare the experimental results obtained at a pump wavelength of λ = 780 nm and a pump power of P = 10.6 kW with simulations. The simulations are based on numerically solving the multimode generalized nonlinear Schrödinger equation (MM-GNLSE) as presented in . This approach enables us to include besides nonlinearities, self-steepening, polarization and higher-order dispersion effects, also wavelength-dependent intermodal processes for an, in principle, unlimited number of modes . Here, we restrict the simulations to a maximum number of 6 modes as the modes of higher-order differ in their mode-intensity distributions from the measured near-field images.
Following , we first assume that by the optimization of the fiber transmission we have perfectly aligned the pump beam with the fiber core in the measurements. Consequently, only the fundamental mode is expected to be initially excited which, in turn, requires the presence of the HOMs observed in the experiments to originate from intermodal nonlinear energy transfer. We verify this possible route by simulating the propagation of a hyperbolic-shaped pump pulse with peak power of P = 10.6 kW at a wavelength of 780 nm, which is initially completely coupled into the fundamental mode. Still, in the simulation all six modes are initially excited by quantum noise to allow for possible intermodal interaction during pulse propagation, as discussed in . The resulting smoothed overall spectrum of the single-shot simulation, shown in Fig. 4 together with its modal contributions, demonstrates that under these pumping conditions the spectral broadening is determined by the fundamental mode only and no power is transferred to HOMs, except to the degenerate mode M2. Furthermore, the simulated spectrum is broader than in the measurements indicating that the actual pulse propagation dynamics were different in the measurements.
We gain a better understanding for the suppression of the intermodal nonlinear energy transfer found in the simulation by studying the mode-coupling coefficients . In the MM-GNLSE each of these coupling coefficients represents a nonlinear process in which from modes l and m each one photon is extracted to generate, under total energy conservation, two new photons in modes p and n, respectively . The magnitude of the mode-coupling coefficient determines the strength of the respective intermodal process. For instance, we find that for the present fiber geometry the highest values have only those coefficients which are attributed to intramodal nonlinearities and intermodal cross-phase modulation (XPM). These processes, however, will not transfer energy to an initially empty mode. Intermodal four-wave mixing (FWM), on the other hand, can in principle be a very efficient route for intermodal energy transfer, but it is here, except between the degenerate modes M1 and M2, strongly suppressed due to the large phase mismatch. In fact, we identify this process as the main route for the observed power transfer from mode M1 to mode M2 by which two photons from mode M1 are transformed into two nearly identical photons, one in M1 and the other in M2.
Since intermodal nonlinear interaction between the fundamental and any non-degenerate HOM is found to be prevented, we conclude that the HOMs observed in the measured near-field images have to originate from the coupling of the pump laser beam. We can estimate the initial mode excitation by calculating the overlap integral  between the mode fields and the incident pump beam, which we approximate by an ideal, linearly polarized Gaussian beam. In general, we find, in accordance with , that for a pump beam radius exceeding the fiber core radius, the highest coupling efficiency is obtained when the pump beam is aligned with the core center. In Fig. 5(a) we show that with this alignment and for a beam waist radius of 1000 nm the pump power is basically entirely coupled to the degenerate mode pair of the fundamental mode. The polarization angle ϕ of the pump beam relative to the fiber geometry thereby determines the relative excitation of the two modes. When, however, the beam waist radius is reduced towards the diffraction limit, the point of maximum coupling efficiency shifts away from the core center. For instance, a beam waist radius of 500 nm is actually expected to yield at three locations, each misaligned by about 500 nm from the core center, almost equal maximum coupling efficiency (see Fig. 5(b)). Aligning the pump beam to one of the points of maximum coupling efficiency will result in launching up to about 50% of the total pump power into modes M3 to M6. Although under these conditions the total coupling efficiency will remain almost independent of the pump beam polarization, the individual excitation of these modes will strongly vary.
For the laser beam used in the here presented experiments we estimate the 1/e - beam waist radius with a laser beam quality parameter M2 = 1.2, λ = 780 nm and the numerical aperture NAo of the focusing microscope objective. From the accordingly calculated overlap integrals we combine the mode excitations as a function of the beam position to yield the coupling maps shown in Fig. 6(a). Each of the coupling maps corresponds to a polarization angle aligned along one of the fiber struts. We find that, independent of the polarization angle, maximum coupling can be achieved only when the pump beam is misaligned from the core center. Furthermore, due to a slight asymmetry of the fiber core only one location turns out to yield maximum coupling for the present beam size. At this beam position the relative mode excitation strongly depends on the polarization angle of the pump beam, as illustrated in Fig. 6(b).
In the experiments we have aligned the pump laser beam to yield maximum transmission but we did not have control on the fiber rotation angle. We therefore select here exemplarily the polarization angle ϕ = 90° and simulate the SCG with the pump power being distributed among the six modes according to the relative fractions listed in Table 1. The resulting spectrum from the single-shot simulation, represented by the coherent sum of the modal contributions, is given in Fig. 7(a). Figure 7(b) shows that by receiving about 57 % of the input power, the degenerate mode pair of the fundamental mode determines the overall spectral width, in agreement with the experimental observations. Once initially excited, the HOMs do also contribute to the overall spectral evolution. The spectral broadening in these modes is mainly the result of red-shifting solitons (Fig. 7(c)), which are produced through a soliton fission process that does not induce the creation of dispersive waves as pumping occurs with insufficient power far in the anomalous dispersion regime. Spanning from ≈ 760 nm to ≈ 950 nm (−20 dB/nm level), the simulated HOM spectra cover yet a wavelength range similar to where also HOMs have been identified in the measured near-field images. Further analysis of the simulation results shows that despite initial excitation of the HOMs, by which also intermodal XPM becomes possible, the short walk-off distances prevent efficient nonlinear interaction. We estimate the walk-off distances between the fundamental mode and the HOMs as well as between non-degenerate HOMs to be ∼2.5 mm and ∼10 mm, respectively, whereas the respective nonlinear lengths are ∼ 3 mm and ∼ 20 mm. Only within the degenerate mode pairs (M1, M2) and (M4, M5) the walk-off lengths become sufficiently long to enable nonlinear interactions. Similar as for the degenerate mode pair (M1, M2), phase-matching is also well satisfied for the degenerate mode pair (M4, M5) and thus leads to efficient energy transfer through intermodal FWM as observed in the simulated modal spectra. We note, that for a proper comparison of the experimental and simulated spectra shot-to-shot fluctuations in the spectral evolution have to be taken into account by averaging over an ensemble of simulated spectra. This is particularly for simulations based on the MM-GNLSE and including the full interaction between six modes computationally very demanding and would go beyond the scope of this publication. We expect, however, that as a result of the ensemble average the resulting simulated spectrum will become smoothed to more closely resemble the experimental results, while still preserving the key characteristics discussed above.
Generally, in few-mode fibers higher-order modes can be excited by misaligning the pump beam from the fiber core center. This usually also results in a lower overall coupling to the fiber core. Here we find from a detailed analysis of the initial mode excitation by the pump laser beam that in three-hole suspended core fibers highest overall coupling to the fiber core can actually be achieved at an off-center alignment, sensitively depending on the size of the pump beam. This means that in experiments, where the laser beam is usually aligned to maximum transmission, higher-order modes can in fact become strongly excited.
In this way, we have coupled in the here presented experiments up to about 40 % of the laser power into higher-order modes and studied the supercontinuum generation, both experimentally and numerically. Utilizing the multimode generalized nonlinear Schrödinger equation, we identified that the spectral width of the measured spectrum is determined by pulse propagation in the degenerate pair of the fundamental mode. The analysis also showed that spectral broadening in the higher-order modes evolves mostly independent as nonlinear intermodal processes are strongly suppressed, in particular between non-degenerate modes. The here presented route towards efficient excitation of higher-order modes in suspended-core fibers opens up new potentials and is thus believed to trigger further interest when choosing this fiber for applications in supercontinuum generation and fiber-based sensing.
This work has been financially supported by the Academy of Finland as part of the ”Photonics and Modern Imaging Techniques” research programme (project 134857). IS wishes to thank the Graduate School of Modern Optics and Photonics. The authors also acknowledge Aleksandr Kravchenko for his help in taking the SEM images of the fiber and Kay Schuster for providing the fiber sample.
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