1. Introduction

Femtosecond supercontinuum (SC) generation setups in optical fibers aimed at obtaining maximal spectral width are typically based on soliton dynamics like soliton fission and soliton self-frequency shift [1, 2]. The major defining features of these dynamics are the dispersion characteristics of the waveguide and the relationship to the input pump wavelength. For the generation of optical solitons, anomalous dispersion is essential. Due to the high flexibility they offer for designing the dispersion characteristics by hole diameter and hole distance, photonic crystal fibers (PCFs) are the fibers mostly used for supercontinuum generation [1], but other fiber-based waveguides like fiber tapers and nanofibers have been successfully applied as well [3–6]. Supercontinua generated by anomalous dispersion dynamics share a complex temporal profile and substantial fluctuations in intensity and phase, and therefore are unsuited for many applications that require a highly coherent broadband source or a single ultrashort pulse in the time domain [7, 8]. These are essential demands, e.g., for time-resolved measurements, applications where parts of the SC pulse are amplified in parametric processes, or for few-cycle pulse generation.

Recently it has been shown that PCFs with an all-normal dispersion profile are promising waveguide structures for the generation of octave-spanning coherent single-pulse supercontinua [9, 10]. In these waveguides, the pump pulse is initially symmetrically broadened by self-phase modulation (SPM) until this process is disrupted by optical wave braking (OWB). OWB induces degenerated four-wave mixing (FWM) that adds additional spectral components at both ends of the spectrum. According to simulations, all these processes take place in a stable temporal sequence and result in a coherent, single broadband pulse [10].

Since the desired dynamics are due only to the all-normal dispersion profile and not to the photonic crystal cladding itself, all other waveguides with an all-normal dispersion profile are, in principle, also suited for coherent single-pulse SC generation [11, 12].

We show that appropriate optical nanofibers and suspended-core fibers with sub-micron diameters exhibit all-normal dispersion profiles and therefore are applicable for pulse-preserving octave-spanning SC generation. For efficient input coupling, a fiber taper configuration is considered. In addition to PCFs with an utilizable all-normal dispersion maximum in the near-infrared and visible wavelength range as investigated in [9,10,13], tapered suspended-core fibers and optical nanofibers offer the possibility to access wavelength ranges in the visible and even ultraviolet spectral regions. Therefore, these structures are an important complement of all-normal dispersion PCFs and allow the tailoring of broadband coherent SC generation to the requirements of specific applications.

While tapered optical nanofibers are promising candidates for coherent SC generation in the ultraviolet spectral region, experimental realization is difficult mainly due to the relatively long required taper transitions, which can alter the properties of the input pulse significantly before it reaches the nanofiber waist [6]. A more favorable behavior is expected in tapered suspended-core fibers. Their initial core diameter in the range of a few microns allows easy input coupling and short transitions with negligible influence on the input pulse. In addition, their higher mechanical stability, due to their larger overall size and the prevention of surface contamination, eases experimental success.

We present SC generation in an all-normal dispersion tapered suspended-core fiber with a dispersion maximum around 630 nm. The smooth and stable spectrum covers the entire visible spectral region and spans over more than one octave from 370 nm to almost 900 nm. Numerical simulations reveal the dynamics of the SC generation process and are used to examine coherence and temporal properties.

2. Dispersion of submicron fiber waveguides

The combination of chromatic and waveguide dispersion has been modeled by a rigorous full-vectorial finite element method (FEM) mode solver, and the dispersion parameter D is deduced from the calculated wavelength- and geometry-dependent propagation constant. While the nanofiber geometry can be easily defined by the fiber diameter, the suspended-core fiber has to be defined in a more complex manner. As an example, we consider a suspended-core fiber with a trigonal symmetry (Fig. 1a ). If we take the contact point of three adjacent hexagons with rounded corner points as the basic geometry, the suspended-core fiber is clearly defined by the radius r of curvature of the hexagon corner points and the distance d between the adjacent hexagon walls.

 

Fig. 1 a) Geometric construction of a suspended core as the centre of three hexagons with rounded corner points. b) Evolution of dispersion profiles D with core diameter of a suspended-core fiber and a bare silica nanofiber.

Download Full Size | PDF

Exemplarily for such a typical suspended-core structure, the fiber design shown in Fig. 2a ) drawn in-house at the IPHT has been modeled and scaled down to a smaller geometry to imitate the results of a tapering process. The initial untapered fiber can be described by a radius of curvature of r = 5200 nm and a wall distance of d = 360 nm, which leads to an incircle diameter of 2000 nm for the solid core. The outside cladding diameter of the fiber is 125 µm. For a better comparison between a nanofiber and a suspended-core fiber with accordingly tapered core, we will hereinafter characterize each suspended-core fiber by the incircle diameter of the core.

 

Fig. 2 Scanning electron microscopy (SEM) images of a suspended-core fiber a) before and b) after tapering.

Download Full Size | PDF

Modeling of the dispersion characteristics as shown in Fig. 1b) reveals remarkable differences in the position of the maximum of the all-normal dispersion profile, which is situated around 490 nm wavelength for nanofibers and around 630 nm for suspended-core fibers. It was shown in [10] that pumping close to the maximum dispersion wavelength results in the broadest bandwidth of the generated SC. Therefore, the two types of optical fibers give access to different wavelength ranges for coherent SC generation.

Apart from the shift in the all-normal maximum dispersion wavelength, the corresponding fiber diameters show significant deviations. While the diameter of the freestanding nanofiber has to be below 480 nm to achieve all-normal dispersion behavior, an incircle diameter of 570 nm is sufficient to obtain all-normal dispersion in suspended-core fibers. Therefore, the suspended-core fiber has to be tapered only by a factor of roughly 1:3.5 in order to reduce the incircle diameter from the initial 2000 nm to the required size, which can be implemented with a taper transition of just a few millimeters. By contrast, the freestanding nanofiber requires a reduction in diameter by a factor of about 1:250 if a standard optical fiber with initially 125 μm outer diameter is used. This requires a significantly longer taper transition, which can lead to dispersive and nonlinear broadening of the input pulse and negatively affects the bandwidth of the generated SC spectrum [6].

Although the physics of dispersion is well understood, and therefore calculation of dispersion profiles is routinely done nowadays, uncertainty and idealization in the geometry boundaries and refractive index distribution cause deviations between theory and practical experience. For this reason, experimental verification of the proposed dispersion profiles is preferable. We conducted dispersion measurements with the help of a Mach-Zehnder interferometer. Due to the low transmission intensity level caused by input coupling challenges we were not able to check the dispersion of the nano-scaled all-normal dispersion waveguides, but had to switch to slightly different ones with sufficiently large core cross-sections for input coupling of light. Figure 3 shows, in comparison, the theoretical and experimental dispersion both of a nanotaper with a diameter around 850 nm including transitions, and a short untapered fiber section at both ends as well as of the initial suspended-core fiber we intended to taper. The good agreement between theory and experiment in these cases demonstrates that the idealized geometry well represents the experimental data. Hence, good agreement is expected also in the case of the all-normal dispersion waveguides.

 

Fig. 3 Comparison between theory and experimental data of the dispersion profile for the initial suspended-core fiber with an incircle core diameter of 2000 nm, and for a fiber nanotaper with a waist diameter of 850 nm including exponential transitions and a small section of untapered fiber on both sides of the waist.

Download Full Size | PDF

3. Tapering of optical fibers

We used a CO₂ laser heating method described in detail in [14] for producing the suspended-core fiber taper. A CO₂ laser is focused on the fiber from two opposite sides to maintain cylindrical symmetry, and scanned along a fixed section while two translation stages pull the fiber in opposite directions. In order to prevent the collapse of the holes of the suspended-core fiber, the fiber was pressurized with 0.8 bar excess pressure during the tapering process. This pressure value was derived from the formula of the equilibrium pressure p = σ/r of a gaseous cylinder with radius r in a liquid with surface tension σ. The value for σ of the silica/air interface was taken as 0.415 N/m. The initial suspended-core fiber with a hole radius of approximately 10 µm and an incircle core diameter of 2000 nm requires an excess pressure of 0.4 bar. This fiber was scaled down by a factor of 1:3.7 to achieve all-normal dispersion at an incircle diameter of 540 nm. The final fiber therefore needs an excess pressure of 1.3 bar. The tapering process worked well if we kept the excess pressure fixed between these two values, i.e. at 0.8 bar. Scanning electron images of the suspended-core fiber before and after the tapering process are shown in Fig. 2 and confirm that the geometry of the fiber was well conserved.

The final suspended-core fiber taper geometry consists of a 70 mm long waist with 3 mm taper transitions at both ends and 1-2 mm of untapered fiber at the input and output ends. The taper transitions have been shaped to be adiabatic, based on the adiabaticity criteria as introduced in [15]. In this case, no propagation losses are expected for the fundamental mode. The transmitted light was transferred to an optical spectrum analyzer via a large mode area fiber to prevent further spectral changes.

4. Numerical model

Pulse propagation in the suspended-core fiber was simulated using the Runge-Kutta in the interaction picture integration method (RK4IP) to solve the generalized nonlinear Schrödinger equation [16]. The conservation quantity error adaptive step size method was employed in order to improve computational speed [17]. The nonlinear coefficient and the dispersion profiles were calculated by a full vectorial finite element method. A single-mode polarization-maintaining fiber pumped along one polarization axis was considered for simplicity.

For the numerical simulation, the experimental taper profile was modeled by starting the pulse propagation in 2 mm of untapered fiber, instantaneously followed by the taper waist and the final untapered fiber section. Due to its short length, the transition between the untapered fiber and the waist section was neglected. The results from simulation are compared with the experimental results in the following section.

5. Experiments and discussion

The pump we used was an ultrafast optical parametric amplifier OPerA Solo from Coherent with an output pulse duration of about 50 fs and a repetition rate of 1 kHz.

The suspended-core fiber taper was pumped at 625 nm wavelength close to the expected all-normal dispersion maximum. The maximum transmitted pulse energy measured at the taper end before material ablation at the cleaved input fiber end face was 1.3 nJ. At this output energy we observed two distinctly different SC spectra depending on the pump light coupling conditions (Fig. 4 ). These results are now compared with a simulation where the fiber taper was simulated as a combination of a leading untapered fiber followed by the tapered fiber waist with reduced core diameter.

 

Fig. 4 Supercontinuum spectra for an output pulse energy of 1.3 nJ, a) generated mainly in the waist section at 1.3 nJ everywhere along the fiber and b) generated mainly in the untapered introductory part of fiber at 10 nJ, assuming appropriate losses down to 1.3 nJ at the taper transition.

Download Full Size | PDF

The spectrum in Fig. 4a) corresponds well to a simulation assuming 1.3 nJ in the untapered fiber and a complete energy transfer to the fiber taper waist. Due to the relatively low intensity, nonlinear effects are not strong enough to reach significant spectral changes in the short untapered introductory part of the fiber, as seen in the simulated spectrum entering the waist (Fig. 4a) and the simulated spectral evolution (Fig. 5a ). Nonlinear processes become significant not until the optical pulse reaches the taper waist with its reduced mode field diameter and enhanced nonlinear parameter. The broadening dynamics are therefore dominated by the all-normal dispersion properties of the waist, where SPM and OWB lead to a flat and smooth octave-spanning spectrum. Ranging from 370 nm to 895 nm, a spectral width of 525 nm or 1.27 octaves at the −20 dB level was achieved. The peak around the pump wavelength could not be reproduced by simulations and is assumed to originate from leakage of the pump through the experimental setup.

 

Fig. 5 Spectral evolution during propagation a) for the narrow but flat top spectrum mainly generated in the waist and b) the wide spectrum mainly generated in the 2 mm part of untapered input fiber.

Download Full Size | PDF

The spectrum in Fig. 4b) can be explained by a higher pulse energy of 10 nJ in the untapered fiber and losses occurring before the start of the fiber taper waist. The value of 10 nJ equals the maximum of energy we were able to transmit through the suspended-core fiber without any tapered section. This high energy of 10 nJ leads to an extraordinary broadening already in the short untapered piece of the leading fiber, as indicated by the simulated spectrum entering the waist (Fig. 4b) and the simulated spectral evolution (Fig. 5b). When this broadband pulse reaches the waist and losses are taken into account by scaling down the pulse energy, almost all nonlinear dynamics stop despite some additional spectral smoothing. Therefore, the SC generation dynamics are dominated by the properties of the untapered suspended-core fiber with its zero dispersion wavelength (ZDW) around 780 nm. The asymmetric spectrum ranges from 350 nm to 1150 nm at the −20 dB level. Thus, on the long wavelength side it is broader than the symmetric spectrum, but it does not exhibit a smooth and flat profile and shows significant spectral variations.

The spectral evolutions shown in Fig. 5 reveal that, in general, the steady state of the spectral broadening is already reached after a few mm of propagation distance. This indicates that short transitions are important when the spectrum is to be defined by the properties of the all-normal dispersion taper waist and not by the input coupling section.

Until now we have only considered light propagation in the fundamental mode, although FEM simulations suggest that two modes could be allowed in the taper waist and 13 modes in the untapered suspended-core fiber. When the simulations where performed using the dispersion profiles of the 2nd mode of both the untapered and the tapered fiber, much lower agreement was observed between simulation and experimental results. That indicates that the fundamental mode dispersion profile is essential for obtaining the observed spectra. Therefore, simulations suggest that only the fundamental mode is responsible for the observed broadening and has been excited within the fiber. There is no conclusive explanation why there are different losses of the fundamental mode depending on the guided energy. Since the taper transition is adiabatic concerning the fundamental mode, no losses should occur, and different transmitted energies should be observed. Nevertheless we are sure about the fact that there is no way to obtain the long wavelengths within the waist at any appropriate energy or to suppress them in the untapered input coupling fiber section at high energies around 10 nJ.

In general, the results obtained indicate that it greatly depends on the pulse energy whether an optical fiber taper is suited for input coupling and all-normal dispersion spectral broadening. The employed suspended-core fiber taper with an initial fiber diameter of 2.0 µm seems to be well suited for low pulse energies around 1 nJ but fails at higher pulse energies around 10 nJ. This may be explained by the circumstances that the pulse has to propagate through a short section of untapered fiber and that the following adiabatic transition is longer than the ideal minimum length due to experimental constraints. Since the shortest adiabatic transition is in the range of 0.01 mm in the present case, a nearly ideal input coupling geometry without any section of untapered suspended-core fiber and an adiabatic taper transition well below the applied 3 mm in length would be well suited even for higher energies above 10 nJ. This highlights the necessary and inevitable efforts it takes to assemble appropriate input coupling taper transitions to optical nanofibers for coherent UV SC generation where high pulse energies are essential [6].

The broadening effect within the taper transition may be circumvented to some extent by using longer pulses. This slows down nonlinear dynamics, and longer propagation distances would be needed to broaden the pulse before the taper waist. A longer input pulse duration would better conserve the spectral pulse characteristics in a given input taper transition geometry. However, nonlinear dynamics in the taper waist are slowed down as well, and longer taper waists are required in this case [6].

6. Additional properties of the generated SC

Since the spectrum in Fig. 4a) is mainly generated in the all-normal dispersion waist of the taper, it should exhibit properties similar to spectra generated in the all-normal dispersion PCFs discussed in [9, 10]. In particular, high coherence and the conservation of a single ultrashort pulse in the time domain are expected. This is confirmed in the calculated spectrogram as shown in Fig. 6a ). Rapid nonlinear dynamics ensure that the SC is produced completely within a propagation length of only a few millimeters, and guarantee a low temporal pulse broadening if a suitable fiber length is applied. Each spectral component has a clearly identifiable position within the pulse and a fixed relative phase. Thus, these pulses are suitable for time-resolved applications and pulse compression. The pump wavelength has no outstanding intensity in comparison to the generated wavelength. This allows spectroscopic investigations not only near but even directly at the pump wavelength, which must be filtered out by additional optical components otherwise in coherent bulk-generated SCs as usually employed in spectroscopy [18]. In the entire spectral range the intensity varies by only a few dB. Spectral characterizations using such light sources could permit continuous investigations at a uniform signal-to-noise ratio.

 

Fig. 6 a) Calculated spectrogram and b) calculated coherence function of the 1.3 nJ SC pulse after 10 mm propagation distance. A single pulse is maintained in the time domain, and the spectrum is highly coherent over the entire bandwidth.

Download Full Size | PDF

Since the SC generation process involves only deterministic processes such as SPM and OWB, no pulse-to-pulse noise in intensity and phase is expected, and hence the spectrum should be highly coherent over its entire bandwidth, provided that there is adequate pump stability. This property is verified by including input pulse shot noise and spontaneous Raman noise in the numerical simulation and computing the complex degree of first-order coherence $\left|g_{12}^{(1)}\right|\left(\lambda \right)$ from an ensemble of 20 independently generated spectra according to the procedure explained in [10]. The result is shown in Fig. 6b). As expected, $\left|g_{12}^{(1)}\right|\left(\lambda \right)=1$is valid for the entire bandwidth, which corresponds to perfect coherence.

There is no principal limit to spectral broadening from a theoretical point of view as long as materials with negligible attenuation are considered. Increasing the input pulse energy always results in an additional broadening of the generated spectrum. Experimental limits are set by the amount of energy coupled into the fiber via a taper transition without significant spectral changes and by the steep edges of the dispersion profile, since nonlinear dynamics stop when all spectral components are spatially separated and arranged according to velocity.

7. Summary

Short adiabatic fiber downtapers sufficiently preserve the spectral and temporal properties of light and the guided power, and therefore are well suited for coupling of pulsed light into sub-micron waveguide structures. Suspended core fibers with initial incircle core diameters in the micrometer range allow short and adiabatic taper transitions and exhibit all-normal dispersion if adequately tapered. They are a viable tool for flat-top and octave-spanning broadband supercontinuum generation in the visible and near-infrared wavelength range. The same concept should be applicable for freestanding optical nanofibers, with a blue-shifted dispersion profile extending the generated SC into the UV.