We demonstrate by means of numerical simulations of the generalized Nonlinear Schrödinger Equation that the variation of the diameter of a tapered fiber along the fiber axis can be used as a new degree of freedom to tailor the spectrum generated by ultrashort laser pulses. We show that, apart from the cross-section geometry of the fiber and the materials used for the core, cladding, and surrounding medium, the diameter profile along the fiber axis crucially influences the soliton dynamics, the temporal and spectral evolution as well as the generation of a supercontinuum. As an example, we have investigated a few centimeters long conical waists, which reveal large differences of the output spectra depending on the incoupling direction. For a decreasing fiber diameter, we find that, keeping the pulse energy constant, a lower input peak power may generate a broader supercontinuum. We attribute this result to the dynamics of higher-order solitons. A comparison of the simulated spectra to experimentally measured ones shows excellent agreement.
© 2010 Optical Society of America
In the recent years, tapered optical fibers have been used successfully as a nonlinear medium in combination with a femtosecond laser pump source for various studies on soliton dynamics and frequency conversion processes, such as Raman effect and supercontinuum generation [1–8]. These processes become more efficient or even just possible due to the unique properties of tapered fibers: on the one hand their reduced diameter - decreased down to a few micrometers - leads to a very small effective mode area A eff yielding a high nonlinearity described by the nonlinear parameter γ, which is inversely proportional to A eff. On the other hand, the dispersion of the fiber can be shifted such that the commonly used pump wavelengths lie in the anomalous dispersion regime [9, 10]. This property in combination with nonlinear effects like the self-phase modulation (SPM) enables the formation of optical solitons , whose break-up due to perturbation by higher-order dispersion and Raman effects may trigger an octave-spanning supercontinuum [12, 13], if the parameters of the fiber and pump source are carefully adjusted. An excellent overview over the mechanisms of the supercontinuum generation is given in  for the case of Photonic Crystal Fibers (PCFs) , which can be applied for tapered fibers as well since the underlying physics is the same.
Recently we published a method which allows us to fabricate tapered fibers from existing single-mode fibers with desired non-homogeneous waists , for example inclining or declining cones. The propagation of ultrashort laser pulses in such structures is highly interesting, as they exhibit different z-dependent optical parameters. In particular, since the dispersion and the nonlinear parameter crucially depend on the actual diameter of the tapered fiber , the properties of a soliton launched into this kind of fibers such as its soliton number, peak power, and width will change in a different manner compared to the propagation in a fiber with a constant diameter. Hence, the design of the diameter profile along the fiber axis offers new possibilities to tailor the temporal and spectral output of ultrafast laser pulse coupled into these structured fibers, in addition to the possibilities already given by the choice of the fiber cross-section geometry and used materials.
Previous work has already demonstrated that the special case of tapered PCFs with a decreasing zero dispersion wavelength can be used to influence the spectral output [18–22]. However, either large fiber lengths were needed (up to 200 m) or the investigation mainly focused on the evolution of a once generated supercontinuum such as the subsequent blue expansion of the spectrum. For us, the initial generation of the supercontinuum is of higher importance in this paper. Another method connects different types of fibers to affect the propagation dynamics .
In this work we present simulations of ultrafast pulse propagation in a few centimeters long linearly increasing or decreasing fiber waists, concentrating on the initial generation of supercontinua. One of our results is that the direction at which the pulses are coupled into the cone dramatically influences the dependence of the generated spectral width on the input peak power. We analyze this behavior in detail and find that the key to its understanding is the propagation distance where the break-up of the launched higher-order soliton is triggered at. This distance is determined by the interplay between the soliton number, soliton period, and the input pulse parameters. The local dispersion and nonlinearity given by the diameter at this very position set the phasematching conditions, e.g., which wavelength the generated non-solitonic radiation is located at, and hence the spectral features of the evolving supercontinuum.
As another example, we show how the soliton dynamics can be influenced by a change of the fiber diameter, which leads to a subsequent rise of the soliton number of a fundamental soliton generated by the fission of a second-order soliton. This new higher-order soliton can again decay in a second-stage fission.
Finally, we compare measured spectra generated by fibers with conical and sinusoidal waists to the simulated ones and find an excellent agreement.
2. Simulation method
The generalized Nonlinear Schrödinger Equation (NLSE)  has been proven to accurately describe the propagation of femtosecond laser pulses in microstructured fibers, both for tapered fibers or PCFs. The optical effects governing the pulse propagation in these types of fibers differ only in details such as the effective refractive index of the holey region of a PCF; so the results presented here are applicable for both types of fibers. We choose to work with tapered single mode fibers (SMF) rather than with PCFs because the structuring of the diameter profile in case of the PCFs is more complicated due to the possible collapse of the air-filled photonic structure [24, 25]. The NLSE we implemented reads as
Here, A is the envelope of the electric field, z the propagation distance, and T the time in the co-moving reference frame. D̂ denotes the dispersion operator as discussed later. γ = n 2 ω 0/(cA eff) stands for the nonlinear parameter with n 2 being the nonlinear index of refraction, ω 0 the central frequency of the pulse, c the vacuum speed of light and A eff the effective mode area, which is also discussed below. The delayed response of the fiber material mainly due to the Raman effect is included by R(t). This response function is well known for fused silica  and can be parametrized as
with fR = 0.18 being the Raman fraction, τ 1 = 12.2 fs, and τ 2 = 32 fs. As a simplification, we neglect the influence of noise.
As the basis fiber we choose the widely used step-index single-mode fiber SMF-28 made of fused silica, which has a core diameter of 8.2 µm and a cladding diameter of 125 µm; the refractive index difference is 0.36%. As the surrounding medium we assume air. For an untapered fiber the light is guided as a core mode, and the influence of the cladding-air interface is negligible. This changes dramatically when the fiber is tapered down to cladding diameters of a few micrometers. Now the cladding mode is dominant and the cladding-air interface becomes important, whereas the impact of the core-cladding interface almost vanishes. To take this change into account, we derive the propagation constant β, which is strongly connected to the dispersion coefficients βk via based on a three-layer model as demonstrated in  with the layers core-cladding-air instead of the two-layer model commonly used with the layers cladding-air in case of thin waists of tapered fibers. With the boundary conditions that the tangential and z-components of both the electric and magnetic field are continuous at both interfaces core-cladding and cladding-air, we obtain a linear system of 8 equations, which can be represented by an 8-by-8 matrix. Setting the determinant of this matrix equal to 0, we obtain an eigenvalue equation for β, which is solved numerically. The largest root corresponds to the propagation constant of the fundamental mode, which we use for the further calculations. This procedure delivers β in dependence of the cladding diameter d and the frequency ω. Knowing β, we can calculate the effective mode area, also in dependence of d and ω, using an equation similar to the one given in , adapted to the three-layer model.
To solve Eq. (1) we implement the symmetrized split-step Fourier method (SSFM) , which is well suited for the pulse durations and spectral widths we expect for the results. Furthermore, since the propagation is simulated step-by-step, we have the opportunity to adjust the dispersion parameter as well as the nonlinear parameter according to the actual diameter of the fiber. This very possibility allows us to simulate non-homogeneous diameter profiles. Here, attention has to be paid that the step size in z-direction ∆z is small enough to warrant a quasicontinuous change of the simulation parameters. For the fibers we investigate, ∆z is chosen to be smaller than 10 µm.
Solving the NLSE using the SSFM, the dispersion is treated in the frequency domain. The Fourier-transformed dispersion operator D̃ can be written as
with Δω = ω − ω 0. Since the calculation of β described above yields the full functional dependence β(ω,d), we are able to circumvent the commonly used Taylor expansion for β(ω) with its problematic choice of the highest order. β 0 and β 1 necessary for our ansatz can easily be derived from β(ω) with a high precision. For the treatment of the nonlinear operator in the time domain, we apply the convolution theorem and the 4th-order Runge-Kutta algorithm. As the initial pulse we choose a sech-pulse given by
where P 0 denotes the peak power, T 0 the pulse width and C the chirp parameter, which imposes a linear chirp if not set to 0. As the central wavelength we set λ 0 = 773 nm for the following simulations. In the following section the given values for the diameter refer to the cladding diameter.
3. Simulation results
3.1. General considerations
If a higher-order soliton is launched into a thin fiber in the anomalous dispersion regime, we find that in the absence of disturbance due to higher-order dispersion or Raman effects, the temporal evolution of the soliton over one soliton period zs first exhibits a strong temporal compression with a simultaneous increase of the peak power up to a maximum value. Propagating further, the peak power decreases again, and the soliton undergoes its well-known ‘breathing’ behavior . For us in particular the propagation distance z MPC, which we define as the propagation distance at which the first local maximum of the peak power is achieved, is of importance, because in the presence of disturbing effects the break-up of the soliton occurs at this very position. This distance is closely connected to the so-called fission length L fis , which can be expressed in the absence of disturbing effect as L fis ≈ √2zs(πN)−1 with N being the soliton order .
To quantify the finding described above we plot in Fig. 2(a) the simulated position of the maximal pulse compression z MPC relative to the soliton period versus the soliton number N, defined by
where Ep is the pulse energy, which is connected to the peak power and pulse width by Ep = 2P 0 T 0 for a sech2-pulse. If we translate the results of this plot to the longitudinal position of maximum pulse compression z MPC, we find that for an increasing peak power of the input pulse z MPC decreases, regardless if T 0 or Ep is kept constant. Notably, this holds not only for an ideal soliton (blue line) but also for the pulses launched into conical fibers, although in these cases the curves change slightly for different input and fiber parameters (red and black lines, parameters are given later). The results are in good agreement with the theoretical approximation L fis/zs as defined in Ref.  (dashed blue line).
As stated above, the fiber diameter at the position z MPC determines the important optical parameters, i.e., dispersion and nonlinearity, which in turn define main features of the generated supercontinuum such as the wavelength of the non-solitonic radiation (NSR). Figure 2(b) shows the dependence of the nonlinear parameter γ on the diameter. In the inset, we plot γ for a constant λ = 773 nm, our pump wavelength. We note that γ increases significantly for smaller diameters, since the effective area is reduced.
Using these results and a fiber profile varying its diameter longitudinally, we are able to choose the very diameter whose corresponding optical parameters control the generation of a supercontinuum with certain desired features by setting the input peak power such that at the corresponding z MPC the profile of the fiber just fits that diameter. In other words, scanning the input peak power while Ep is kept constant translates to a scanning of the diameter profile. For a quantitative analysis we assume a conically shaped fiber with a diameter changing linearly from 0.9 µm to 2.7 µm over a distance of 60 mm. The input peak power P 0 is varied while Ep is fixed to 1 nJ. Figure 3 shows the corresponding results, which we are going to discuss in detail in the following paragraph.
Depending on the incoupling direction we obtain very different output spectra [Fig. 3(a) and 3(b)]: in the case of the increasing cone, with an increasing input peak power the final spectrum becomes broader. As expected the increasing input peak power leads to a smaller z MPC. This implies that also the diameter at this position becomes smaller [Fig. 3(c)], which leads to a higher nonlinearity as seen above [Fig. 3(e)]. The opposite is true for the decreasing cone: Here again a higher input peak power yields a smaller z MPC, but now the diameter corresponding to a smaller z MPC rises (middle right of Fig. 3), leading to a lower γ at this position (bottom right of Fig. 3). We unexpectedly obtain narrower output spectra beyond P 0 = 5 kW, although the input peak power still is increased. For a more detailed analysis we therefore compare the pulse evolution of the increasing cone with the decreasing one at this particular input peak power of 5 kW in the following.
3.2. Increasing cone
Figure 4 shows the temporal and spectral evolution of the pulse in the increasing cone. As described above the pulse self-compresses until at z MPC ≈ 5.5 mm the break-up of the launched higher-order soliton occurs and the supercontinuum formation is triggered. During the decay NSR is generated at a wavelength according to the phasematching condition [27, 28]
where ωs is the central frequency of the soliton, Ps its peak power, and ω NSR the central frequency of the generated NSR. Figure 4(f) visualizes this phasematching condition. With the diameter of ≈ 1.1 µm corresponding to z MPC ≈ 5.5 mm and Ps = √2NP 0 ≈ 163 kW  we obtain the phasematched wavelength λ NSR ≈ 350 nm, which fits well to the simulated spectrum.
We have added the path of the dominant soliton during its evolution as well as the corresponding group-velocity-matched path on the short wavelength side to the group velocity map shown in Fig. 4(d). The wavelengths of the latter path are far away from λ NSR, which means that the soliton and the NSR do not co-propagate and hence not interact. This leads to the isolation of the NSR in the temporal as well as in the spectral domain. This finding is supported by the XFROG trace  demonstrated in Fig. 4(c), which shows that the NSR is delayed by several picoseconds with respect to the solitonic part of the spectrum.
Since the diameter increases while γ decreases for longer propagation distances, the phasematching condition does not allow the generation of components with even smaller wavelengths. Hence, this isolation of the NSR leads to sharp edges at the short wavelength side of the output spectrum, whereas the long-wavelength solitons keep redshifting during the propagation due to intra-pulse Raman scattering.
This behavior also holds for the other input peak powers. The only difference is that the phasematching condition becomes valid for smaller λ NSR leading to broader output spectra, since the diameter where the NSR is generated becomes smaller for higher input peak powers.
3.3. Decreasing cone
The case of the decreasing waist is demonstrated in Fig. 5. Here we find z MPC ≈ 30.5 mm corresponding to a diameter of 1.75 µm. With Ps ≈ 64 kW the phasematching condition [Eq. (6)] leads to λ NSR ≈ 480 nm, which again is in good agreement to the initially generated NSR shown in the spectral evolution [Fig. 5(b)].
However, in contrast to the case of the increasing cone the group velocities of the solitons and the corresponding NSR are nearly matched as can be inferred from Fig. 5(d). This leads to the trapping of the NSR by the solitons  and hence an enhanced redshift of the solitons. Since now NSR and solitons are co-propagating they can interact and create new wavelength components by four-wave-mixing resulting in a further broadening of the spectrum during the propagation. The temporal pairing of solitons and the corresponding NSR can be clearly seen in the XFROG trace [Fig. 5(c)].
Again, we observe this behavior also for the other input peak powers. The hort-wavelength edge almost does not change due to this generation of new wavelengths, although λ NSR increases from 400 nm to 500 nm when increasing the input peak power in the given interval.
The effect described above is more effective upon close matching of the group velocities of solitons and NSR. The optimum matching is achieved for the input peak power of 5 kW, resulting in the broadest spectrum, while for higher input peak powers this effect becomes less effective. In combination with the higher λ NSR, the output spectra becomes narrower again.
3.4. Subsequent soliton fission
Apart from the conically shaped fibers, we want to give another example of controlling the soliton dynamics by a change of the fiber diameter (Fig. 6): A second-order soliton is launched into a thin fiber (d = 2 µm) and decays into two fundamental solitons while emitting NSR. After a propagation length of 200 mm, which allows the two solitons to separate temporarily and spectrally, we increase the diameter of the fiber over a small distance linearly to 3 µm such that the soliton number of the dominant soliton with the larger wavelength rises again to 2. This is possible since according to Eq. (5) N is not only dependent on the pulse parameters but also on the diameter-determined β 2 and γ. This new second-order soliton decays again into two fundamental solitons and a new dispersive wave, which shows a different central wavelength when compared to the one generated by the first decay in agreement with Eq. (6). As a second effect of the increased diameter, the wavelength of the short-wavelength soliton of the first decay shifts from the anomalous to the normal dispersion regime, turning the soliton into a dispersive wave. The temporal and spectral evolution of this process as well as a simulated XFROG trace  of the output pulse are shown in Fig. 6. Table 1 quantifies the finding described above by giving the relevant values at the different evolution steps. P 0 and T 0 have been extracted by fitting a sech2-function to the solitonic part of |A(z,T)|2 for the given propagation distances.
This technique of subsequent fission processes might be used, e.g., for the generation of multiple wavelength bands by choosing the diameters of the homogeneous fiber pieces such that the spectral positions of the subsequently generated NSR, which are determined by phasematching condition Eq. (6), are tuned to the desired wavelengths.
4. Experimental results and conclusion
To verify our simulation method we have fabricated and measured two tapered fibers with conical waists as well as a fiber with a sinusoidally modulated waist as proposed in . As the pump source we use a mode-locked Ti:sapphire oscillator delivering pulses with a duration of 181 fs FWHM and a central wavelength of 773 nm at a repetition rate of 80 MHz. The fiber parameters and the results are shown in Fig. 7. For the propagation simulation, the full fiber profile has been taken into account, including the ≈ 10 cm long fiber pigtails at both ends of the fiber as well as the ≈ 35 mm broad taper transition region. Since the laser pulses have been precompressed by a prism sequence, we approximate this by setting the chirp parameter C in Eq. (4) to −3.
The simulated spectra show excellent agreement to the measured ones. The spectral width is predicted precisely as well as the wavelengths of intra-pulse features such as the solitons at the long-wavelength side. Both simulation and measurement support the findings described above, e.g., the sharp edges on the short-wavelength side in case of an increasing profile in contrast to the case of a decreasing diameter.
In conclusion, we have demonstrated that the variation of the diameter of a tapered fiber along its axis serves well as additional tool for engineering the spectral output of femtosecond pulses launched into the profiled fiber. We could attribute the differences of the spectra generated by an increasing and decreasing waist to the dynamics of the higher-order soliton, in particular to the propagation distance of its maximum self-compression. We have shown how the order of a soliton can be raised after a certain propagation to trigger subsequent fission processes. Finally, measured output spectra of profiled tapered fibers pumped with femtosecond pulses confirmed our simulation method to yield realistic results.
The authors would like to thank the Landesgraduiertenförderung of Baden-Württemberg for support of this work.
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