A simple method is described for efficient, asymmetric and coherent continuum generation in the mid-infrared region based on the dynamics of a stabilized soliton in the vicinity of a second dispersion zero of a nonlinear fiber. The mechanism involves nonlinear soliton compression, Raman self-frequency shift and resonant emission of a dispersive (Cherenkov) wave in a non-uniformly tapered ZBLAN fluoride fiber pumped by a low-power compact femtosecond laser at 1.55 μm. The fiber taper features a continuous shift of the second zero dispersion wavelength, which facilitates the progressive shift in the wavelength of the dispersive wave generated by the stabilized soliton. Numerical solution of the generalized nonlinear Schrödinger equation, which accounts for the exact wavelength dependence of dispersion and nonlinear coefficients, shows robust generation of near-octave continuum spanning 1.5–3 μm wavelength range.
©2009 Optical Society of America
Coherent mid-infrared broadband sources have a wide range of potential applications in metrology, coherent control, non-destructive testing, spectroscopy, and optical coherence tomography. Coherent mid-infrared light has been generated by using optical parametric oscillators, quantum cascade lasers and fiber lasers . Recent experiments have demonstrated mid-infrared supercontinuum generation in ZBLAN fluoride fiber based on multiple soliton dynamics in fibers [2, 3]. The continuum generation in those experiments, however, required high power pump lasers and coherence of the continuum is largely degraded due to the nature of modulation instability (MI), which initiates the breakup of the pump pulse into multiple solitons.
It is well known that continuum generation in a fiber pumped in the region of anomalous dispersion relies exquisitely on soliton dynamics. Irrespective of the input pulse duration, either one of the two competing processes can initiate the breakup of a pump pulse into fundamental solitons: noise-seeded MI with a characteristic length LMI = LNL/2 , or deterministic soliton fission with an empirical characteristic length Lfiss ≈ LD/N , where LNL and LD are the nonlinear and dispersion lengths, respectively, and N = √LD/LNL is the soliton number. The ratio of the soliton fission and MI characteristic lengths, which is proportional to N, determines the degree of coherence. Therefore, a critical value of the soliton number exists (Ncrit ≈ 10) above which soliton fission will be overcome by MI thus changing the nature of the continuum from coherent to incoherent .
Here we present a new concept for coherent continuum generation in the mid-infrared by using a non-uniform fluoride fiber taper. Unlike most previous schemes, the continuum is generated asymmetrically with high efficiency by fundamental solitons propagating in tapered fiber through dispersive wave (DW) generation in a stabilized regime in the vicinity of a second zero dispersion wavelength (ZDW), where dispersion slope dD/dλ < 0. Importantly, we use realistic parameters in our design for the pump pulse based on commercially available compact femtosecond fiber lasers at 1.55 μm central wavelength. Also, the low input soliton number (N ≪ Ncrit) used in our design ensures high level of continuum spectral coherence as opposed to continuum generated in the multiple soliton regime.
Supercontinuum generated in uniformly tapered or photonic crystal fibers (PCF) near the first ZDW (dD/dλ > 0), as is usually done [6, 7], is nearly symmetric with respect to the pump wavelength, with soliton fission and Raman self-frequency shift responsible for the long-wavelength components and DW generation accounting for the short-wavelength part of the resulting spectrum . Considering that most of the commercially available pump lasers are available only in the visible and near-infrared, an asymmetric continuum generation scheme is preferred to efficiently generate broad spectra in the mid-infrared region. By using PCFs or tapered fibers that have a second ZDW, injection of a soliton into the anomalous group-velocity-dispersion (GVD) region can lead to DW generation at longer wavelengths across the second ZDW via a phase-matched interaction between the soliton and the DW [8, 9]. This DW generation occurs under favorable conditions in which the soliton is stabilized near the second ZDW through a balance between the Raman self-frequency shift and spectral recoil effect . With the flexibility of dispersion engineering in PCFs and tapered fibers, it is possible to use a non-uniform taper design to provide a continuous shift of the second ZDW along the fiber, thus progressively shifting the phase-matched wavelength of the generated DW. As a result, the generated DW sweeps a range of wavelengths thus forming a chirped continuum, which can be compressed, if desired. This is in sharp contrast to the previously demonstrated mid-infrared continuum generation schemes where the long-wavelength part of the continuum consists of a large number of solitons with overlapping spectra and random delays , thus rendering such a continuum intrinsically incompressible.
2. Non-uniform fiber taper concept and design
The general idea of the proposed asymmetric continuum generation, illustrated in Fig. 1, can be considered to consist of two stages. In the first stage the input pulse undergoes nonlinear soliton compression (and moderate soliton fission if the input pulse soliton number N is greater than 2), followed by the Raman self-frequency shift toward longer wavelengths. Because the Raman shift of a fundamental soliton is inversely proportional to the fourth power of its duration , the initial soliton compression is very important to achieve rapid Raman shifts. In the second stage the soliton approaches the second ZDW of the fiber with a negative dispersion slope and begins to emit a DW at longer wavelengths, which in turn stabilizes the soliton in the vicinity of the second ZDW halting further Raman shift . The fiber taper is designed to have its second ZDW continuously shifting toward shorter wavelengths so that the phase-matching condition for the emission of the DW is continuously adjusted and the wavelength of the DW emitted by the soliton is progressively getting shorter. The soliton is “pushed” by the ZDW toward shorter wavelength until nearly all of its energy is converted to a DW continuum. At this point the soliton no longer follows the ZDW shift and leaks through into the normal dispersion region.
Certain features of the proposed scheme are worth mentioning. In particular, only relatively low pulse energies, corresponding to a near-fundamental soliton are required to generate the continuum. The energy of a N = 1 soliton is E = 2|β 2|/γT 0, where β 2 and γ are the dispersion and nonlinear parameters, and T 0 is the soliton duration. Typical energies for realistic fibers would thus be in a sub-nanojoule range. Appropriate selection of the specific fiber structure allows certain degree of flexibility in selection of the input pulse duration and energy to either maximize the continuum power or to use a compact low-energy pump source. Another important implication of the controlled nature of the continuum generation via phase-matched DW emission by a soliton is that nearly 100% of the pump power can be transferred to the continuum and only into the wavelength range determined by the design of the ZDW wavelength as a function of propagation distance. Although only one simple profile λZDW (z) is studied below, more complex dependences may yield spectral shapes unachievable by other means.
For the design, we choose ZBLAN fluoride glass as the material for the fiber taper because of its transparency in the mid-infrared wavelength region. ZBLAN also has rather favorable material dispersion properties with its first ZDW near 1.6 μm and relatively flat dispersion profile D(λ), thus requiring only mild (e.g. compared to silica or IRG2) waveguide contribution to the total dispersion in order to form a second ZDW at a longer wavelength. A core-cladding index difference of Δn = 0.09 is chosen for the fiber. A Δn value much smaller is not able to provide the required second ZDW because contribution from the waveguide dispersion becomes too small. While the choice of such a large core-cladding index difference is hardly realistic for a conventional step-index fiber, it is certainly attainable through PCF designs. The GVD β 2 as a function of wavelength for various core radii is plotted in Fig. 2(a). Wavelength-dependent values for β 2 (and the mode-propagation constant β) are obtained by numerically solving the eigenvalue equation for the fundamental HE11 mode of a step-index fiber, with material dispersion of ZBLAN included in the calculation. As shown in Fig. 2(a), the second ZDW shifts to shorter wavelengths with decreasing core radius, although no second ZDW is available for core radius larger than ~ 2.4 μm. Fig. 2(b) shows the nonlinear coefficient γ for various core radii, which is numerically calculated using 
Here n 2 = 2.2×10-20m2/W is the nonlinear index of refraction for ZBLAN, and Sz is the longitudinal component of the Poynting vector, numerically calculated from the field distribution. For the same wavelength γ increases with decreasing core size because of a reduced effective mode area.
The designed taper structure consists of a 600-cm non-uniform ZBLAN fiber taper with an exponentially decreasing core radius from 4.0 μm to 1.5 μm, r(z) = r 0 exp(-z/611.73), where r 0 = r(z = 0) = 4 μm is the initial core radius and z is the propagation distance in cm. We note that similar lengths of silica PCFs tapers have been recently drawn for different applications [12, 13]. Because of the essentially identical dispersion and nonlinearity characteristics between PCF and fiber taper [14, 7], our ZBLAN fiber taper can be easily applied to a PCF design. The relatively long length of the fiber taper is mainly needed to maximize the soliton red-shift and approach the second ZDW, although the DW continuum is generated near the end of the fiber taper so that in principle shorter fiber tapers and other r(z) dependencies may be used. Additionally, one may also consider tapers in which other parameters vary along the length, such as Δn, material composition, hole size and shape in PCFs, etc. as control parameters for achieving ZDW and thus DW phase-matching wavelength variation along the fiber.
3. Numerical simulation
To monitor the nonlinear propagation in the designed fiber taper, we have numerically solved the generalized nonlinear Schrödinger equation (GNLS) under the slowly varying envelope approximation [4, 15]:
In Eq. (2), A(z, T) is the complex envelope of the electric field in the reference frame moving with the group velocity of the input pulse. The left-hand side of Eq. (2) represents linear propagation, where a is the fiber linear loss, and β m is the mth order dispersion coefficient in a Taylor series for the mode-propagation constant β(ω). The right-hand side describes the nonlinear effects, with γm = ∂mγ/∂ ωm the mth order derivative of the nonlinear coefficient γ(ω), and the response function R(T) = (1 -fR)δ(T) + fR hR(T) describing both the instantaneous electronic and delayed Raman contributions.
We consider an input pulse of a hyperbolic-secant shape, with a center wavelength at 1.55 μm and a full-width-half-maximum (FWHM) duration of 100 fs. With the numerical values of β(ω) available over the frequency range involved, we include all orders of dispersion in the calculation by replacing the dispersion operator in the frequency domain with β (ω) - β 0 - ω β 1 . Such approximation-free approach is necessary due to the broad spectral range and complex wavelength dependence of dispersion involved. Similarly, nonlinear coefficient γ(ω) to all orders is also included numerically, although we find that inclusion of only the zeroth and first order derivative of γ(ω) is sufficient to yield similar results, as γ is approximately linear to frequency for the chosen taper parameters. It is, however, essential to include spectral dependence of γ (resulting from variation of the modal effective area with wavelength, in addition to the self-steepening term) in the calculation in order to correctly describe the nonlinear propagation, with the continuum spectrum generated in mid-infrared largely affected by such spectral dependence . We also include in the calculation the experimentally measured Raman gain spectrum for ZrF4-BaF2 fiber , and choose the fractional contribution of Raman response fR to be 0.2 (similar to fused-silica glass). The numerical calculation is carried out with a symmetrized split-step Fourier method , in which the linear propagation part of Eq. (2) is solved completely in the frequency domain, while the nonlinear part is solved partially in the time domain and partially in the frequency domain. The convolution product between Raman response and field intensity in Eq. (2) is calculated as a product in the frequency domain. The nonlinear and dispersion characteristics at each step are determined based on the taper core radius at that step. To cover both the temporal and spectral spans of the generated continuum, a temporal window of 60 ps and spectral window of 270 THz are chosen, which corresponds to 214 sampling points.
4. Results and discussion
The spectral evolution with propagation distance and the output spectrum are shown on logarithmic scale in Fig. 3. The output continuum extends over 3.0 μm, with a 10 dB spectral width of ~ 1.5 μm (an octave). The spectrum shown in Fig. 3 is obtained for an input pulse energy of 1.0 nJ (a peak power of 8.8 kW), a realistic parameter for a compact femtosecond laser source.
The soliton number, N = √γP 0 T 0 2/|β 2|, at the fiber input equals to 2.3. Because N > 1, the input pulse undergoes initial spectral broadening and temporal compression, followed by soli-ton fission process. A fundamental soliton, taking most of the input pulse energy and having a shorter duration, is ejected from the input pulse, and starts to Raman self-frequency shift toward longer wavelengths. As shown in Fig. 3, the fundamental soliton propagates for a distance of 480 cm before DW generation begins at the long wavelength end. A long fiber taper is required for sufficient soliton self-frequency shift because of the small soliton number involved. The non-uniform fiber taper helps further compress the fundamental soliton and achieve rapid soliton self-frequency shift. Unlike in a dispersion decreasing fiber, such soliton compression results from a combination of enhancement of nonlinearity and decrease of the absolute value of the GVD (because soliton is approaching the second ZDW) during propagation.
After 480 cm of propagation the fundamental soliton approaches the second ZDW (λ zdw = 2.4 μm for a taper core radius of 1.83 μm) with a center wavelength of 2.2 μm and a FWHM duration of 25 fs and begins to emit the DW at ~ 3.0 μm. This is in good agreement with the DW wavelength calculated through the phase-matching condition between soliton and DW . For a uniform fiber taper, the soliton wavelength would have stabilized near the second ZDW when the Raman self-frequency shift is canceled by the spectral recoil effect. However, in our core-reducing fiber taper, because of the continuous shift of the second ZDW the soliton center wavelength is also “pushed” toward shorter wavelength, while the wavelength of the emitted DW becomes progressively shorter as well. This process lasts until the energy transfer from the soliton to DW is nearly complete. The rate at which the second ZDW shifts determines the bandwidth of the DW generated. Shifting the ZDW too fast will result in a non-adiabatic process in which ZDW passes through the soliton and the DW generation seizes prematurely. On the other hand, an overly slow shift will create a rather narrow-band DW as soliton loses most of its energy during the initial stage of ZDW shifting. We note that fringes between 1.5 and 2.3 μm in the spectrum is a result of spectral interference between DW’s generated from the first fundamental soliton and the second fundamental soliton, which has less energy and smaller Raman shift. This is confirmed by spectral filtering in the numerical simulation.
In Fig. 4(a), we plot the output spectra obtained for several input pulse energies that are close to 1 nJ. The spectra look relatively similar and stable, indicating that the continuum possesses high spectral coherence away from the pump laser shot noise. In fact, given that DW generation from coherent soliton is a coherent process, an input soliton number of N < 10 will ensure high level of spectral coherence , unless random modulations are present on the pump pulse capable of seeding the MI and uncontrolled pulse breakup even for small N . To check the spectral coherence of the generated continuum, we numerically calculate the complex degree of first-order coherence at each wavelength, which is defined as 
Here E 1(λ) and E 2(λ) are continuum spectra generated by successive input pulses, or, as in our simulation, spectra generated independently from input pulses with random noise seeds. The angular brackets denote an ensemble average over a large number of such continuum spectra. In our calculation, we consider only the input pulse quantum noise, and ignore pump laser intensity fluctuations and noise that is associated with spontaneous Raman scattering. The input pulse shot noise is semiclassically modeled by adding a stochastic variation in the magnitude of the input electrical field whose standard deviation equals to the square root of the number of photons in small temporal steps . The spectral coherence |g (1) 12(λ)| is calculated from an ensemble average on the results of 20 simulations from input pulses that have different random quantum noise. The result for a 1.0 nJ input pulse energy is plotted in Fig. 4(b). The continuum possesses nearly perfect spectral coherence over the whole continuum spectral range.
Clearly, within the current concept the wavelength to which the generated continuum can be extended is rather limited by the efficiency of Raman self-frequency shift and fiber loss in the mid-infrared. Using higher pump pulse energies and thus larger soliton numbers will result in faster Raman shifts of solitons and DW generation at even longer wavelengths. An example is shown in Fig. 4(a), where the continuum spectrum obtained for a 2.0 nJ input pulse (corresponding to a soliton number of N = 3.3) is plotted. The overall shape of the resulting spectrum is, however, determined mostly by the tapering design, rather than by pump pulse energy. This observation suggests stability of the continuum spectrum with respect to fluctuations in pump pulse parameters, however in high-power regime multiple solitons will form, each generating its own DW rendering the continuum incompressible.
We present and numerically demonstrate a new concept for coherent continuum generation in a non-uniformly tapered fiber structure based on phase-matched dispersive wave generation by a soliton stabilized in the vicinity of a second zero dispersion wavelength, which progressively shifts to shorter wavelengths along the fiber. A ZBLAN fiber taper with exponentially-decreasing core size and a fixed Δn is numerically modeled by solving a generalized nonlinear Schrödinger equation where the dispersion, nonlinear coefficient and Raman contribution are included to all orders. One-octave coherent, compressible and stable continuum is generated asymmetrically with near 100% efficient energy transfer from the pump pulse to the desired spectral region. Importantly, compact femtosecond fiber lasers delivering ~1 nJ, 100 fs pulses can be used to generate such broad spectra in the mid-infrared extending to ~ 3 μm and possibly beyond with further optimized taper structures.
This work was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. Department of Energy, Office of Basic Energy Sciences user facility. Los Alamos National Laboratory, an affirmative action equal opportunity employer, is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under contract DE-AC52-06NA25396.
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