Source: http://tops.osa.org/oe/abstract.cfm?uri=oe-18-19-20151
Timestamp: 2019-04-18 18:52:41+00:00

Document:
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.
S. Coen, A. H. L. Chau, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation via stimulated Raman scattering and parametric four-wave-mixing in photonic crystal fibers,” J. Opt. Soc. Am. B 19, 753–764 (2002).
J. Teipel, K. Franke, D. Türke, F. Warken, D. Meiser, M. Leuschner, and H. Giessen, “Characteristics of supercontinuum generation in tapered fibers using femtosecond laser pulses,” Appl. Phys. B 77, 245–251 (2003).
G. Genty, M. Lehtonen, M. Kaivola, and H. Ludvigsen, “Enhanced bandwidth of supercontinuum generated in microstructured fibers,” Opt. Express 12, 3471–3480 (2004).
J. N. Kutz, C. Lynga, and B. J. Eggleton, “Enhanced supercontinuum generation through dispersion-management,” Opt. Express 13, 3989–3998 (2005).
F. Hoos, S. Pricking, and H. Giessen, “Compact portable 20 MHz solid-state femtosecond whitelight-laser,” Opt. Express 14, 10913–10920 (2006).
D.-I. Yeom, J. A. Bolger, G. D. Marshall, D. R. Austin, B. T. Kuhlmey, M. J. Withford, C. Martijn de Sterke, and B. J. Eggleton, “Tunable spectral enhancement of fiber supercontinuum,” Opt. Lett. 32, 1644–1646 (2007).
G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, “Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides,” Opt. Express 15, 5382–5387 (2007).
D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998).
G. P. Agrawal, Nonlinear fiber optics (Academic Press, San Diego, 1995).
L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
G. Genty, S. Coen, and J. M. Dudley, “Fiber supercontinuum sources (Invited),” J. Opt. Soc. Am. B 24, 1771–1785 (2007).
P. St. J. Russell, “Photonic Crystal Fibers,” Science 17, 358–362 (2003).
S. Pricking and H. Giessen, “Tapering fibers with complex shape,” Opt. Express 18, 3426–3437 (2010).
R. Zhang, X. Zhang, D. Meiser, and H. Giessen, “Mode and group velocity dispersion evolution in the tapered region of a single-mode tapered fiber,” Opt. Express 12, 5840–5849 (2004).
S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse compression in dispersion-decreasing fiber,” Opt. Lett. 18, 476–478 (1993).
F. Lu, Y. Deng, and W. H. Knox, “Generation of broadband femtosecond visible pulses in dispersion-micromanaged holey fibers,” Opt. Lett. 30, 1566–1568 (2005).
A. Kudlinski, A. K. George, J. C. Knight, J. C. Travers, A. B. Rulkov, S. V. Popov, and J. R. Taylor “Zero-dispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,” Opt. Express 14, 5715–5722 (2006).
J. C. Travers, J. M. Stone, A. B. Rulkov, B. A. Cumberland, A. K. George, S. V. Popov, J. C. Knight, and J. R. Taylor “Optical pulse compression in dispersion decreasing photonic crystal fiber,” Opt. Express 15, 13203–13211 (2007).
A. Kudlinski and A. Mussot, “Visible cw-pumped supercontinuum,” Opt. Lett. 33, 2407–2409 (2008).
J. C. Travers, S. V. Popov, and J. R. Taylor, “Extended blue supercontinuum generation in cascaded holey fibers,” Opt. Lett. 30, 3132–3134 (2005).
E. C. Mӓgi, P. Steinvurzel, and B. J. Eggleton, “Tapered photonic crystal fibers,” Opt. Express 12, 776–784 (2004).
W. J. Wadsworth, A. Witkowska, S. G. Leon-Saval, and T. A. Birks, “Hole inflation and tapering of stock photonic crystal fibres,” Opt. Express 13, 6541–6549 (2005).
C.-M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: Scaling laws and numerical analysis,” J. Opt. Soc. Am. B 19, 1961–1967 (2002).
N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
A. V. Yulin, D. V. Skryabin, and P. St. J. Russell, “Four-wave mixing of linear waves and solitons in fibers with higher-order dispersion,” Opt. Lett. 29, 2411–2413 (2004).
J. R. Taylor, Optical solitons: Theory and Experiment (Cambridge University Press, Cambridge, 2005).
A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photonics 1, 653–657 (2007).
S. Linden, H. Giessen, and J. Kuhl, “XFROG - A New Method for Amplitude and Phase Characterization of Weak Ultrashort Pulses,” Phys. Status Solidi B 206, 119–124 (1999).
A. Konyukhov, L. Melnikov, and Y. Mazhirina, “Dispersive wave generation in microstructured fiber with periodically modulated diameter,” Proc. SPIE 6165, 616508 (2006).
Fig. 1. We investigate the propagation of ultrafast laser pulses in different non-homogeneous fiber waists. Our simulations are able to deliver the temporal and spectral evolution of the pulses along the fiber axis as well as XFROG traces  at desired propagation distances.
Fig. 2. (a) Propagation distance of maximal solitonic pulse compression z MPC relative to the soliton period zs in dependence of the soliton number. The dashed line shows the normalized fission length, calculated by the theoretical approximation L fis/zs = √2(πN)−1. (b) Nonlinear parameter γ in dependence of diameter and wavelength. The inset shows γ for a constant λ = 773 nm.
Fig. 3. (a), (c), (e): Increasing cone from 0.9 µm to 2.7 µm over a distance of 60 mm. (b), (d), (f): Decreasing cone from 2.7 µm to 0.9 µm over a distance of 60 mm. (a), (b): Resulting output spectra (logarithmic scale). (c), (d): Position of the maximal pulse compression z MPC and the corresponding fiber diameter in dependence of the input peak power P 0. (e), (f): Dispersion parameter D and nonlinear parameter γ at the longitudinal position of maximum pulse compression z MPC. The input pulse energy is kept constant at 1 nJ, the central wavelength is 773 nm, the chirp parameter C equals 0.
Fig. 4. (a) Temporal (logarithmic scale) and (b) spectral (logarithmic scale) evolution of a femtosecond pulse in a increasing waist (the same as in Fig. 3). The pulse energy is 1 nJ, P 0 = 5 kW, T 0 = 100 fs and λ 0 = 773 nm. (c) Output XFROG trace . (d) Group velocity relative to c in dependence of diameter and wavelength. The solid line shows the path of the dominant soliton, the dashed line the group-velocity-matched path on the short wavelength side. (e) Group velocity dispersion in dependence of diameter and wavelength. (f) Phasematching condition for the NSR. The wavelength of the decaying soliton is 773 nm.
Fig. 5. (a) Temporal (linear scale) and (b) spectral (logarithmic scale) evolution of a femtosecond pulse in a decreasing waist (the same as in Fig. 3). The pulse energy is 1 nJ, P 0 = 5 kW, T 0 = 100 fs and λ 0 = 773 nm. (c) Output XFROG trace . (d) Group velocity relative to c in dependence of diameter and wavelength. The solid line shows the path of the dominant soliton, the dashed line the group-velocity-matched path on the short wavelength side, here nearly matched to the wavelength of the NSR. Note the reversed diameter axis to match the fiber profile.
Fig. 6. Temporal (left) and spectral (middle) evolution of a second-order soliton in a fiber with a step-like diameter profile (d 1 = 2 µm, d 2 = 3 µm). Right: Simulated XFROG trace of the output pulse. The letters in the circles denote the pulse features belonging together: (a) NSR of the first decay, (b) fundamental soliton, later turning into a dispersive wave, (c) dominant soliton, changing its order as described in the text, (d) NSR of the second decay, (e) fundamental soliton generated by the second decay. All in a logarithmic scale.
Fig. 7. Normalized output spectra: (a) Sinusoidal waist, d min = 1.3 µm, d max = 2.7 µm, period 3.3 mm, 24 periods; P in = 375 mW, P out = 205 mW. (b) Increasing diameter profile from 1.7 µm to 3.0 µm over 20 mm, P in = 380 mW, P out = 250 mW. (c) Same as (b) but as decreasing profile; P in = 395 mW, P out = 220 mW. (d) Increasing diameter profile from 0.9 µm to 2.3 µm over 25 mm, P in = 216 mW, P out = 100 mW. (e) Same as (d) but as decreasing profile; P in = 191 mW, P out = 103 mW.
(1) ∂ A ( z , T ) ∂ z = D ̂ A ( z , T ) + i γ ( 1 + i ω 0 ∂ ∂ T ) ( A ( z , T ) ∫ − ∞ + ∞ R ( t ′ ) ∣ A ( z , T − t ′ ) ∣ 2 d t ′ ) .

References: V. 
 V. 
 V. 
 V. 
 V. 
 V. 
 V. 
 V.