PRL 111, 033902 (2013)
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PHYSICAL REVIEW LETTERS
PHz-wide Supercontinua of Nondispersing Subcycle Pulses Generated by Extreme Modulational Instability F. Tani,1,* J. C. Travers,1 and P. St. J. Russell1,2 1
Max Planck Institute for the Science of Light, Gu¨nther-Scharowsky-Strasse 1, 91058 Erlangen, Germany 2 Department of Physics, University of Erlangen-Nuremberg, 91058 Erlangen, Germany (Received 28 February 2013; published 19 July 2013) Modulational instability (MI) of 500 fs, 5 J pulses, propagating in gas-filled hollow-core kagome photonic crystal fiber, is studied numerically and experimentally. By tuning the pressure and launched energy, we control the duration of the pulses emerging as a consequence of MI and hence are able to study two regimes: the classical MI case leading to few-cycle solitons of the nonlinear Schro¨dinger equation; and an extreme case leading to the formation of nondispersing subcycle pulses (0.5 to 2 fs) with peak intensities of order 1014 W cm2 . Insight into the two regimes is obtained using a novel statistical analysis of the soliton parameters. Numerical simulations and experimental measurements show that, when a train of these pulses is generated, strong ionization of the gas occurs. This extreme MI is used to experimentally generate a high energy (> 1 J) and spectrally broad supercontinuum extending from the deep ultraviolet (320 nm) to the infrared (1300 nm). PACS numbers: 42.81.Dp, 32.80.Fb, 42.65.Ky, 42.65.Re
Modulational instability (MI) is a universal phenomenon occurring during the propagation of nonlinear waves. Amplification of noise causes the wave to breakup, leading to the formation of new structures. This instability can occur in both space and time. In both cases it leads to the spontaneous growth of sidebands in the spectrum and to the formation of pulses shorter than the mother wave. MI has been studied and observed for over fifty years in many different fields including water hydrodynamics, plasma physics and nonlinear optics to name just a few [1], but is nevertheless still of interest. Besides being a very universal phenomenon MI has many practical applications: in fiber optics it can be used for the formation of ultrashort pulses [2,3], and supercontinuum (SC) generation [4–8]. Generally, MI occurs for the same conditions as solitons— another universal phenomenon, and it is widely observed that MI leads to fundamental soliton formation as part of the process of supercontinuum evolution. MI dynamics can be described using Akhmediev breathers (AB) [9], valid for the same conditions as the nonlinear Schro¨dinger equation (NLSE), and the subsequent emergence of fundamental NLSE solitons has been found to arise through perturbations of the AB solutions by higher order nonlinearities [10]. In this Letter we explore what happens to the temporal MI dynamics when the modulation period becomes comparable to a single fundamental cycle of the pump carrier wave. We first show novel MI dynamics in this regime through numerical simulations and a statistical approach to studying the soliton parameters. We then show corresponding experimental results, including a high energy supercontinuum extending into deep ultraviolet, and compare them with the simulations. We studied extreme MI in gas filled hollow-core kagome style photonic-crystal fiber (PCF). This fiber has very low 0031-9007=13=111(3)=033902(5)
dispersion and can support high intensity laser pulses [11,12]. The fiber dispersion is accurately described by a model due to Marcatili [13,14]. For the 9 m core radius used in the experiments, the calculated dispersion is shown in Fig. 1 for Xe filling pressures of 3 and 10 bar. In both cases there is weak anomalous dispersion around 800 nm. Although the corresponding nonlinearity is rather low at 105 W1 m1 , this is compensated for by the extremely high intensities achievable in the core. For example, over 1014 W=cm2 was demonstrated by Ho¨lzer et al. [15], and we show similar levels in this work. Since a noble gas is used, and numerical calculations indicate that the fractional light-glass overlap can be as low as 0:01% [11,16], the system is also almost completely Raman free.
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DOI: 10.1103/PhysRevLett.111.033902
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FIG. 1 (color online). Group velocity dispersion of a 9 m core radius kagome-PCF filled with 3 and 10 bar Xe.
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orders of magnitude, exceeding 1018 cm3 , and remains above 1017 cm3 along the rest of the fiber. The origins of this strong ionization become clear by looking at the temporal power distribution at a position where the MI has fully developed. Figure 2(d) shows the squared temporal envelope (red line) and electric field (blue line) at z ¼ 6 cm. Temporal structures shorter than 1 fs with peak powers exceeding tens of MW are generated by the MI process. This corresponds to intensities of the order of 1014 W cm2 , well above the ionization threshold of Xe. Note that the high free electron density reported here is created by the high intensity MI induced pulses; this is distinct from the recently reported process where the MI itself is driven by ionization [23]. Figure 2(e) shows the time-frequency spectrograms of the pulse at four different positions. After 3.6 cm the pulse breaks up into many short subpulses, causing the spectrum to broaden. From the spectrograms we observe that the 1 fs pulses to not disperse upon further cm-scale propagation through the fiber—remarkable given their mm-scale dispersion lengths (LD ¼ T02 =j2 j)—and they have the appearance of solitons (spectral and temporal shape). This is followed ultimately by dispersive wave emission and soliton recoil, leading to further spectral broadening. Note that in the absence of Raman effects the soliton redshift should be much weaker than observed in Fig. 2, a conclusion that we confirmed by simulating the propagation of an isolated few-femtosecond soliton in the fiber. Careful analysis of the spectrograms (not shown) revealed that four-wave mixing between the solitons and dispersive waves strongly enhances the observed redshift and
In the experiments laser pulses of duration 500 fs and central wavelength 800 nm were launched into the fiber. Using the definition of NLSE soliton order qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N ¼ Pp T02 =j2 j, where Pp and T0 are the peak pulse power and duration and 2 the group velocity dispersion, an energy of a few J is required to achieve the high soliton orders (N > 100) needed to access MI dynamics [6–8] (usually observed for N 15). The system was numerically modeled by solving the unidirectional full field equation for the fundamental mode [17–20], using a standard split-step Fourier method (with adaptive axial step-size and 50 attosecond time mesh), including fiber loss and radially dependent ionization. The nonadiabatic Yudin Ivanov model [21], which takes into account both tunnel and multiphoton ionization, was used. Quantum noise was added to the input optical field in all the simulations through the addition of one photon per mode with random phase [22]. Figures 2(a) and 2(b) show the temporal and spectral evolution of the field along a fiber filled with 10 bar of Xe, pumped by a 500 fs pulse with an energy of 5 J. The MI dynamics are clear: the pulse propagates for a few cm before it breaks up into many short pulses, while the spectrum broadens considerably first into sidebands and then into a continuum. Note that since the system is seeded by noise, each laser shot produces a slightly different spectral distribution, which on average converges to a smooth continuum [5]. Figure 2(c) shows that, at the point (z 3 cm) where the onset of MI causes pulse breakup, the free electron density increases by over 2 20
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FIG. 2 (color). MI dynamics of a 500 fs pulse at 800 nm with 5 J energy, launched through a 9 m core radius kagome-PCF filled with 10 bar Xe. (a) Temporal intensity and (b) spectral intensity (log scale). (c) Corresponding free electron density. (d) Temporal profile after 6 cm; the red line is the squared envelope and the blue line the squared electric field; inset shows a subcycle pulse structure. (e) Spectrograms at different positions individually normalized. The dashed lines represent the zero dispersion wavelength and A and N indicate anomalous and normal dispersion regions.
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Note that this approach is different to that in Ref. [10], which found that equating the soliton and AB peak powers broke energy conservation; here we conserve energy. Although individual solitons may be shorter than Eq. (2) suggests, its neighboring solitons must consequently by longer to maintain energy conservation, and Eq. (2) provides the expected center of this distribution. We further justify this approach a posteriori based on the excellent agreement we achieve for the few-cycle soliton case, shown later [Fig. 3(b)]. For the subcycle case, using the same parameters as for the simulations in Fig. 2, we predict 0 0:67 fs, corresponding to a full-width at half-maximum (FWHM) duration of 1:2 fs, which agrees remarkably well with the structures seen in Fig. 2(d). Note that a single optical cycle at 800 nm lasts 2:6 fs; even though the central frequency of the solitons may shift, it is clear from Fig. 2(d) that the pulses are subcycle. Furthermore, the subplots in Fig. 2(e) clearly shows individual solitons exceeding the bandwidth of a single-cycle pulse. The simple analysis above is, however, arguably incorrect for the subcycle case because it is based on the NLSE, which is invalid when self-steepening and higher order dispersion become important. Two questions arise: (i) does MI behave differently in this regime and (ii) are these subcycle pulses actually solitons? Subcycle solitons have been analytically and numerically studied in the context of temporal compression, showing evidence of sub-single-cycle solitons under similar conditions to ours [26–28]. Until now, to the best of our knowledge, there have been no studies of MI leading to the formation of subcycle structures, nor any exploring whether these represent a new kind of soliton. In order to address these issues, we have numerically studied the MI behavior using a statistical approach, comparing the extreme case of
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subcycle pulse generation to the conventional case of MIinduced few-cycle soliton formation [29]. This involved developing an accurate and robust numerical method to extract the temporal location, FWHM duration, and peak power of the discrete peaks in the intensity envelope at a fixed fiber position. Finding that neither fiber loss nor ionization significantly affected the MI behavior, we neglected them both in what follows. Figure 3 shows the distribution of the retrieved peaks as a function of peak power and FWHM duration at two positions along the fiber (left and right columns) for parameters corresponding to MI induced few-cycle pulses (top row) and subcycle pulses (bottom row). The solid lines represent the expected behavior for fundamental NLSE solitons. The dashed lines mark the expected distribution peaks based on Eq. (2). Figure 3(a) shows the few-cycle MI case for 2 ps, 2:5 J pump pulses at a Xe pressure of 3 bar (pump pulse soliton order N ¼ 120), calculated from a set of 1000 simulations. The distribution of pulses
Frequency
concomitant dispersive wave blueshift [24]; indeed in the absence of Raman scattering this becomes the dominant cause of the redshift. The numerical observation of these very short pulses agrees well with a simple physical analysis based on assuming that all of the energy in a single modulation period of the Akhmediev breather solution—describing the initial MI dynamics—evolves into a single soliton upon perturbation. Explicitly, we equate the average energy Esol of an MI induced fundamental soliton to the energy EMI contained p within one MI period TMI ¼ 2=!MI (!MI ¼ 2N=T0 is the modulation frequency [2,25], which is identical to the AB modulation frequency [9])
0 ¼ TMI =2 :
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PRL 111, 033902 (2013)
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FIG. 3 (color online). Distribution of MI induced pulses as a function of peak power and FWHM duration in different MI regimes in a 9 m radius Xe-filled kagome PCF. Top row: 2 ps, 2:5 J pump pulse at 3 bar pressure; the expected MI induced soliton duration is 7.6 fs (few-cycle pulse); (a) initial stage of MI; (b) distribution after propagation over 4LD . Bottom row: 0.5 ps, 5 J pulse at 10 bar pressure; expected MI induced pulse duration is 1 fs (subcycle pulses); (c) initial stages of extreme MI; (d) converged distribution after propagation over 20LD . The solid lines represent the expected behavior for fundamental NLSE solitons. The histograms are normalized to the peak value within each row. Red dashed lines indicate the expected distribution peaks based on Eq. (2).
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PHYSICAL REVIEW LETTERS (a)
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Input energy (µ J)
clearly follows the solid fundamental soliton curve, some additional dispersive waves appearing below it. Even after these pulses have propagated for approximately four dispersion lengths the distribution does not change significantly [Fig. 3(b)], indicating solitonlike behavior. Equation (2) predicts an average FWHM soliton duration of 7:6 fs (two cycles), which is in excellent agreement with the peak of the upper distribution in Fig. 3(b), confirming the analysis leading to Eq. (2). Additionally, at the initial stages of MI, the temporal peaks should correspond to AB solutions, which have a FWHM of 0:148TMI ; i.e., they are shorter than the NLSE solitons which eventually emerge (FWHM 0:179TMI ). Indeed the peak in Fig. 3(a) is centered at a shorter duration of 7 fs. Therefore, Figs. 3(a) and 3(b) explicitly illustrate that the ABs arising from MI evolve into NLSE solitons upon propagation. This scenario changes for parameters leading to subcycle structures, with expected FWHM durations of 1 fs. To realize this configuration we modeled a 500 fs, 5 J pump pulse at a Xe gas pressure of 10 bar (pump pulse soliton order N ¼ 210). The distribution shown in Fig. 3(c) is much narrower than the two-cycle case, and lies well below the fundamental soliton line, being centered at a FWHM duration of 1.5 fs and 7 MW peak power. Two far less populated distributions also appear with durations of approximately half a cycle and one cycle. Upon further propagation the distribution slowly evolves and the initial peaks change in amplitude and width, eventually after 20LD converging to the distribution in Fig. 3(d), peaked at 1 fs and 6.5 MW: although there is a long tail extending above 50 MW. After that the distribution remains unchanged over the remaining fiber length (> 50LD ). The number of peaks between Figs. 3(c) and 3(d) drastically increases through further MI evolution. Despite their short duration, these pulses do not significantly broaden along the fiber—much longer pulses would be expected after 5 cm propagation for a dispersion length LD 1 mm. Therefore, even though these pulses are clearly not NLSE solitons, they do show solitonic behavior. This is additionally supported by the solitonic pulse trajectories and collisions shown in Fig. 2(a), which are very similar to those observed in a conventional MI driven SC. All the parameters used in the simulations discussed above are experimentally accessible. In the experiments we used a 20 cm long kagome PCF with an 18 m core diameter filled with 10 bar Xe. Each end of the fiber was placed in a gas cell, equipped with an MgF2 window for launching and extracting light. Pulses at 800 nm with 500 fs duration where launched into the fundamental mode of the fiber at energies up to 8 J without inducing any damage. Recorded output spectra are shown in Fig. 4. MI leads to the generation of a flat and high energy SC extending from 320 to 1300 nm. Figure 4(a) shows the experimental output spectra for different input energies and Fig. 4(b) shows the results of numerical simulations; the overall agreement is extremely good.
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PRL 111, 033902 (2013)
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FIG. 4 (color). (a) Energy dependent experimental spectra for 500 fs pump pulses at 800 nm at the output of a 20 cm long 9 m radius kagome-PCF filled with 10 bar Xe; (b) corresponding numerical results. (c) Comparison between simulated (blue line) and experimental (black line) spectra at 5 J pump energy. (d) Output energy scaling with pump energy: experiment (red dots); simulation including fiber loss (i, blue dashed line); and simulations including fiber and photoionization-induced losses (ii, green line).
Note that the bandwidth of the SC is mainly limited by energy loss due to photoionization of the gas. The linear transmission loss of the kagome PCF also reduces the bandwidth, but only marginally. Additionally, a wider SC and a modified MI dynamics may be possible when making use of a Raman active gas; however, this significantly complicates the analysis and is beyond the scope of this work. Conventional MI sidebands appear at low energies, and for higher energies a SC is formed, reaching its maximum spectral width at 3 J. Figure 4(b) shows the output spectrum for 5 J of pump energy. The black line is the experimental spectrum, and the blue line is the result of an average over 20 noise-seeded numerical simulations for the experimental parameters. The agreement between experiment and simulation is remarkably good in the low-loss transmission window of the fiber and only slightly worse in the high loss band around 400 nm (caused by resonances in the cladding). Figure 4(d) shows the output versus input energy. The dots are the experimental measurements, showing that the output energy saturates at 1 J. The dashed line represents the simulated behavior including the loss of the fiber, while the solid line is the expected behavior when additionally taking into account loss due to photoionization of the gas. It is interesting to note that the input energy at which saturation is seen coincides with the appearance of MI sidebands, indicating that the onset of ionization is linked to MI-induced pulse breakup. The accurate experimental confirmation of the numerical simulations, for parameters corresponding to the creation of
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PHYSICAL REVIEW LETTERS
subcycle pulses through MI, lends strong support to the correctness of our analysis. In conclusion, gas-filled hollow core kagome-style PCF permits the realization of an extreme form of MI, resulting in generation of a supercontinuum consisting of a shower of high intensity subcycle solitonlike pulses whose behavior differs from that of NLSE solitons. Numerical simulations predict that the high intensities obtained in this way result in strong and sustained photoionization of the gas. These results are confirmed in experiments, only a few J of energy being needed to generate a broad and flat SC extending from the UV to the near IR. Although this pump energy is remarkably low in gas-based nonlinear optics, the resulting supercontinuum has extremely high spectral energy density compared to the SC generated in solid-core PCFs.
*[email protected] [1] V. E. Zakharov and L. A. Ostrovsky, Physica (Amsterdam) 238D, 540 (2009). [2] A. Hasegawa and W. Brinkman, IEEE J. Quantum Electron. 16, 694 (1980). [3] K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986). [4] A. S. Gouveia-Neto, A. S. L. Gomes, and J. R. Taylor, IEEE J. Quantum Electron. 24, 332 (1988). [5] M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, J. Opt. Soc. Am. B 6, 1149 (1989). [6] E. M. Dianov, A. B. Grudini, A. M. Prokhorov, and V. N. Serkin, in Optical Solitons: Theory and Experiment (Cambridge University Press, Cambridge, England, 1992). [7] J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006). [8] J. C. Travers, J. Opt. 12, 113001 (2010). [9] J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, Opt. Express 17, 21497 (2009).
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[10] C. Mahnke and F. Mitschke, Phys. Rev. A 85, 033808 (2012). [11] J. C. Travers, W. Chang, J. Nold, N. Y. Joly, and P. St. J. Russell, J. Opt. Soc. Am. B 28, A11 (2011). [12] P. St. J. Russell, Science 299, 358 (2003). [13] E. A. J. Marcatili and R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964). [14] J. Nold, P. Ho¨lzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St. J. Russell, Opt. Lett. 35, 2922 (2010). [15] P. Ho¨lzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St. J. Russell, Phys. Rev. Lett. 107, 203901 (2011). [16] N. Y. Joly, J. Nold, W. Chang, P. Ho¨lzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, Phys. Rev. Lett. 106, 203901 (2011). [17] A. V. Husakou and J. Herrmann, Phys. Rev. Lett. 87, 203901 (2001). [18] M. Kolesik, J. V. Moloney, and M. Mlejnek, Phys. Rev. Lett. 89, 283902 (2002). [19] P. Kinsler, Phys. Rev. A 81, 013819 (2010). [20] W. Chang, A. Nazarkin, J. C. Travers, J. Nold, P. Ho¨lzer, N. Y. Joly, and P. St. J. Russell, Opt. Express 19, 21 018 (2011). [21] G. L. Yudin and M. Y. Ivanov, Phys. Rev. A 64, 013409 (2001). [22] P. D. Drummond and J. F. Corney, J. Opt. Soc. Am. B 18, 139 (2001). [23] M. Saleh, W. Chang, J. C. Travers, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 109, 113902 (2012). [24] D. V. Skryabin and A. V. Yulin, Phys. Rev. E 72, 016619 (2005). [25] J. N. Kutz, C. Lynga˚, and B. Eggleton, Opt. Express 13, 3989 (2005). [26] A. Nazarkin and G. Korn, Phys. Rev. Lett. 83, 4748 (1999). [27] D. V. Kartashov, A. V. Kim, and S. A. Skobelev, JETP Lett. 78, 276 (2003). [28] S. Amiranashvili, A. G. Vladimirov, and U. Bandelow, Eur. Phys. J. D 58, 219 (2010). [29] A. Schwache and F. MItschke, Phys. Rev. E 55, 7720 (1997).
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PHYSICAL REVIEW LETTERS
PHz-wide Supercontinua of Nondispersing Subcycle Pulses Generated by Extreme Modulational Instability F. Tani,1,* J. C. Travers,1 and P. St. J. Russell1,2 1
Max Planck Institute for the Science of Light, Gu¨nther-Scharowsky-Strasse 1, 91058 Erlangen, Germany 2 Department of Physics, University of Erlangen-Nuremberg, 91058 Erlangen, Germany (Received 28 February 2013; published 19 July 2013) Modulational instability (MI) of 500 fs, 5 J pulses, propagating in gas-filled hollow-core kagome photonic crystal fiber, is studied numerically and experimentally. By tuning the pressure and launched energy, we control the duration of the pulses emerging as a consequence of MI and hence are able to study two regimes: the classical MI case leading to few-cycle solitons of the nonlinear Schro¨dinger equation; and an extreme case leading to the formation of nondispersing subcycle pulses (0.5 to 2 fs) with peak intensities of order 1014 W cm2 . Insight into the two regimes is obtained using a novel statistical analysis of the soliton parameters. Numerical simulations and experimental measurements show that, when a train of these pulses is generated, strong ionization of the gas occurs. This extreme MI is used to experimentally generate a high energy (> 1 J) and spectrally broad supercontinuum extending from the deep ultraviolet (320 nm) to the infrared (1300 nm). PACS numbers: 42.81.Dp, 32.80.Fb, 42.65.Ky, 42.65.Re
Modulational instability (MI) is a universal phenomenon occurring during the propagation of nonlinear waves. Amplification of noise causes the wave to breakup, leading to the formation of new structures. This instability can occur in both space and time. In both cases it leads to the spontaneous growth of sidebands in the spectrum and to the formation of pulses shorter than the mother wave. MI has been studied and observed for over fifty years in many different fields including water hydrodynamics, plasma physics and nonlinear optics to name just a few [1], but is nevertheless still of interest. Besides being a very universal phenomenon MI has many practical applications: in fiber optics it can be used for the formation of ultrashort pulses [2,3], and supercontinuum (SC) generation [4–8]. Generally, MI occurs for the same conditions as solitons— another universal phenomenon, and it is widely observed that MI leads to fundamental soliton formation as part of the process of supercontinuum evolution. MI dynamics can be described using Akhmediev breathers (AB) [9], valid for the same conditions as the nonlinear Schro¨dinger equation (NLSE), and the subsequent emergence of fundamental NLSE solitons has been found to arise through perturbations of the AB solutions by higher order nonlinearities [10]. In this Letter we explore what happens to the temporal MI dynamics when the modulation period becomes comparable to a single fundamental cycle of the pump carrier wave. We first show novel MI dynamics in this regime through numerical simulations and a statistical approach to studying the soliton parameters. We then show corresponding experimental results, including a high energy supercontinuum extending into deep ultraviolet, and compare them with the simulations. We studied extreme MI in gas filled hollow-core kagome style photonic-crystal fiber (PCF). This fiber has very low 0031-9007=13=111(3)=033902(5)
dispersion and can support high intensity laser pulses [11,12]. The fiber dispersion is accurately described by a model due to Marcatili [13,14]. For the 9 m core radius used in the experiments, the calculated dispersion is shown in Fig. 1 for Xe filling pressures of 3 and 10 bar. In both cases there is weak anomalous dispersion around 800 nm. Although the corresponding nonlinearity is rather low at 105 W1 m1 , this is compensated for by the extremely high intensities achievable in the core. For example, over 1014 W=cm2 was demonstrated by Ho¨lzer et al. [15], and we show similar levels in this work. Since a noble gas is used, and numerical calculations indicate that the fractional light-glass overlap can be as low as 0:01% [11,16], the system is also almost completely Raman free.
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Dispersion (fs2/cm)
DOI: 10.1103/PhysRevLett.111.033902
10 0 3 bar
-10
10 bar
-20 -30 0.4
0.6
0.8
1.0
Wavelength (µ m)
FIG. 1 (color online). Group velocity dispersion of a 9 m core radius kagome-PCF filled with 3 and 10 bar Xe.
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Ó 2013 American Physical Society
orders of magnitude, exceeding 1018 cm3 , and remains above 1017 cm3 along the rest of the fiber. The origins of this strong ionization become clear by looking at the temporal power distribution at a position where the MI has fully developed. Figure 2(d) shows the squared temporal envelope (red line) and electric field (blue line) at z ¼ 6 cm. Temporal structures shorter than 1 fs with peak powers exceeding tens of MW are generated by the MI process. This corresponds to intensities of the order of 1014 W cm2 , well above the ionization threshold of Xe. Note that the high free electron density reported here is created by the high intensity MI induced pulses; this is distinct from the recently reported process where the MI itself is driven by ionization [23]. Figure 2(e) shows the time-frequency spectrograms of the pulse at four different positions. After 3.6 cm the pulse breaks up into many short subpulses, causing the spectrum to broaden. From the spectrograms we observe that the 1 fs pulses to not disperse upon further cm-scale propagation through the fiber—remarkable given their mm-scale dispersion lengths (LD ¼ T02 =j2 j)—and they have the appearance of solitons (spectral and temporal shape). This is followed ultimately by dispersive wave emission and soliton recoil, leading to further spectral broadening. Note that in the absence of Raman effects the soliton redshift should be much weaker than observed in Fig. 2, a conclusion that we confirmed by simulating the propagation of an isolated few-femtosecond soliton in the fiber. Careful analysis of the spectrograms (not shown) revealed that four-wave mixing between the solitons and dispersive waves strongly enhances the observed redshift and
In the experiments laser pulses of duration 500 fs and central wavelength 800 nm were launched into the fiber. Using the definition of NLSE soliton order qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N ¼ Pp T02 =j2 j, where Pp and T0 are the peak pulse power and duration and 2 the group velocity dispersion, an energy of a few J is required to achieve the high soliton orders (N > 100) needed to access MI dynamics [6–8] (usually observed for N 15). The system was numerically modeled by solving the unidirectional full field equation for the fundamental mode [17–20], using a standard split-step Fourier method (with adaptive axial step-size and 50 attosecond time mesh), including fiber loss and radially dependent ionization. The nonadiabatic Yudin Ivanov model [21], which takes into account both tunnel and multiphoton ionization, was used. Quantum noise was added to the input optical field in all the simulations through the addition of one photon per mode with random phase [22]. Figures 2(a) and 2(b) show the temporal and spectral evolution of the field along a fiber filled with 10 bar of Xe, pumped by a 500 fs pulse with an energy of 5 J. The MI dynamics are clear: the pulse propagates for a few cm before it breaks up into many short pulses, while the spectrum broadens considerably first into sidebands and then into a continuum. Note that since the system is seeded by noise, each laser shot produces a slightly different spectral distribution, which on average converges to a smooth continuum [5]. Figure 2(c) shows that, at the point (z 3 cm) where the onset of MI causes pulse breakup, the free electron density increases by over 2 20
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eq 0.4 ue nc y
0.0 0.0 s)
N 0.6 Hz )
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1017 1018 1019 Free electrons (cm-3)
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Fr
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(e) z = 0 cm
0.0
(d )
30
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0
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0.5
(P
1.0
lay
De
(p
eq 0.4 ue nc y
0.0 N 0.6 Hz )
130
140 150 Delay (fs)
z = 5.5 cm
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Fr
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0.5
0.0 s) (p y a el
D
(P
z = 1 5 cm
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Fr
eq 0.4 ue nc y
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Fr
eq 0.4 ue nc y
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De
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FIG. 2 (color). MI dynamics of a 500 fs pulse at 800 nm with 5 J energy, launched through a 9 m core radius kagome-PCF filled with 10 bar Xe. (a) Temporal intensity and (b) spectral intensity (log scale). (c) Corresponding free electron density. (d) Temporal profile after 6 cm; the red line is the squared envelope and the blue line the squared electric field; inset shows a subcycle pulse structure. (e) Spectrograms at different positions individually normalized. The dashed lines represent the zero dispersion wavelength and A and N indicate anomalous and normal dispersion regions.
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(b) Frequency
1 6.9 fs
0 15
N=1
Peak power (MW)
Note that this approach is different to that in Ref. [10], which found that equating the soliton and AB peak powers broke energy conservation; here we conserve energy. Although individual solitons may be shorter than Eq. (2) suggests, its neighboring solitons must consequently by longer to maintain energy conservation, and Eq. (2) provides the expected center of this distribution. We further justify this approach a posteriori based on the excellent agreement we achieve for the few-cycle soliton case, shown later [Fig. 3(b)]. For the subcycle case, using the same parameters as for the simulations in Fig. 2, we predict 0 0:67 fs, corresponding to a full-width at half-maximum (FWHM) duration of 1:2 fs, which agrees remarkably well with the structures seen in Fig. 2(d). Note that a single optical cycle at 800 nm lasts 2:6 fs; even though the central frequency of the solitons may shift, it is clear from Fig. 2(d) that the pulses are subcycle. Furthermore, the subplots in Fig. 2(e) clearly shows individual solitons exceeding the bandwidth of a single-cycle pulse. The simple analysis above is, however, arguably incorrect for the subcycle case because it is based on the NLSE, which is invalid when self-steepening and higher order dispersion become important. Two questions arise: (i) does MI behave differently in this regime and (ii) are these subcycle pulses actually solitons? Subcycle solitons have been analytically and numerically studied in the context of temporal compression, showing evidence of sub-single-cycle solitons under similar conditions to ours [26–28]. Until now, to the best of our knowledge, there have been no studies of MI leading to the formation of subcycle structures, nor any exploring whether these represent a new kind of soliton. In order to address these issues, we have numerically studied the MI behavior using a statistical approach, comparing the extreme case of
(a)
10 5
5
(c)
10 15 20 FWHM (fs)
0 1 Frequency
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1
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(d) Frequency
(2)
Peak power (MW)
where Pp is the pump peak power. Using the definition of soliton order N together with Esol ¼ 2P0 0 , where P0 and 0 are the soliton peak power and temporal width, we can extract the expected soliton duration 0 [6,8]:
Frequency
(1)
Peak power (MW)
Esol ¼ EMI ¼ Pp TMI ;
subcycle pulse generation to the conventional case of MIinduced few-cycle soliton formation [29]. This involved developing an accurate and robust numerical method to extract the temporal location, FWHM duration, and peak power of the discrete peaks in the intensity envelope at a fixed fiber position. Finding that neither fiber loss nor ionization significantly affected the MI behavior, we neglected them both in what follows. Figure 3 shows the distribution of the retrieved peaks as a function of peak power and FWHM duration at two positions along the fiber (left and right columns) for parameters corresponding to MI induced few-cycle pulses (top row) and subcycle pulses (bottom row). The solid lines represent the expected behavior for fundamental NLSE solitons. The dashed lines mark the expected distribution peaks based on Eq. (2). Figure 3(a) shows the few-cycle MI case for 2 ps, 2:5 J pump pulses at a Xe pressure of 3 bar (pump pulse soliton order N ¼ 120), calculated from a set of 1000 simulations. The distribution of pulses
Frequency
concomitant dispersive wave blueshift [24]; indeed in the absence of Raman scattering this becomes the dominant cause of the redshift. The numerical observation of these very short pulses agrees well with a simple physical analysis based on assuming that all of the energy in a single modulation period of the Akhmediev breather solution—describing the initial MI dynamics—evolves into a single soliton upon perturbation. Explicitly, we equate the average energy Esol of an MI induced fundamental soliton to the energy EMI contained p within one MI period TMI ¼ 2=!MI (!MI ¼ 2N=T0 is the modulation frequency [2,25], which is identical to the AB modulation frequency [9])
0 ¼ TMI =2 :
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Peak power (MW)
PRL 111, 033902 (2013)
10 15 20 FWHM (fs)
0 1 Frequency
1 0.95 fs
0 60
N=1
40
20
1
2 3 FWHM (fs)
0 1 Frequency
FIG. 3 (color online). Distribution of MI induced pulses as a function of peak power and FWHM duration in different MI regimes in a 9 m radius Xe-filled kagome PCF. Top row: 2 ps, 2:5 J pump pulse at 3 bar pressure; the expected MI induced soliton duration is 7.6 fs (few-cycle pulse); (a) initial stage of MI; (b) distribution after propagation over 4LD . Bottom row: 0.5 ps, 5 J pulse at 10 bar pressure; expected MI induced pulse duration is 1 fs (subcycle pulses); (c) initial stages of extreme MI; (d) converged distribution after propagation over 20LD . The solid lines represent the expected behavior for fundamental NLSE solitons. The histograms are normalized to the peak value within each row. Red dashed lines indicate the expected distribution peaks based on Eq. (2).
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(c) 0 5
Power (-10 dB)
Input energy (µ J)
clearly follows the solid fundamental soliton curve, some additional dispersive waves appearing below it. Even after these pulses have propagated for approximately four dispersion lengths the distribution does not change significantly [Fig. 3(b)], indicating solitonlike behavior. Equation (2) predicts an average FWHM soliton duration of 7:6 fs (two cycles), which is in excellent agreement with the peak of the upper distribution in Fig. 3(b), confirming the analysis leading to Eq. (2). Additionally, at the initial stages of MI, the temporal peaks should correspond to AB solutions, which have a FWHM of 0:148TMI ; i.e., they are shorter than the NLSE solitons which eventually emerge (FWHM 0:179TMI ). Indeed the peak in Fig. 3(a) is centered at a shorter duration of 7 fs. Therefore, Figs. 3(a) and 3(b) explicitly illustrate that the ABs arising from MI evolve into NLSE solitons upon propagation. This scenario changes for parameters leading to subcycle structures, with expected FWHM durations of 1 fs. To realize this configuration we modeled a 500 fs, 5 J pump pulse at a Xe gas pressure of 10 bar (pump pulse soliton order N ¼ 210). The distribution shown in Fig. 3(c) is much narrower than the two-cycle case, and lies well below the fundamental soliton line, being centered at a FWHM duration of 1.5 fs and 7 MW peak power. Two far less populated distributions also appear with durations of approximately half a cycle and one cycle. Upon further propagation the distribution slowly evolves and the initial peaks change in amplitude and width, eventually after 20LD converging to the distribution in Fig. 3(d), peaked at 1 fs and 6.5 MW: although there is a long tail extending above 50 MW. After that the distribution remains unchanged over the remaining fiber length (> 50LD ). The number of peaks between Figs. 3(c) and 3(d) drastically increases through further MI evolution. Despite their short duration, these pulses do not significantly broaden along the fiber—much longer pulses would be expected after 5 cm propagation for a dispersion length LD 1 mm. Therefore, even though these pulses are clearly not NLSE solitons, they do show solitonic behavior. This is additionally supported by the solitonic pulse trajectories and collisions shown in Fig. 2(a), which are very similar to those observed in a conventional MI driven SC. All the parameters used in the simulations discussed above are experimentally accessible. In the experiments we used a 20 cm long kagome PCF with an 18 m core diameter filled with 10 bar Xe. Each end of the fiber was placed in a gas cell, equipped with an MgF2 window for launching and extracting light. Pulses at 800 nm with 500 fs duration where launched into the fundamental mode of the fiber at energies up to 8 J without inducing any damage. Recorded output spectra are shown in Fig. 4. MI leads to the generation of a flat and high energy SC extending from 320 to 1300 nm. Figure 4(a) shows the experimental output spectra for different input energies and Fig. 4(b) shows the results of numerical simulations; the overall agreement is extremely good.
3
exp 1 2 sim 3
1 0.4
0.4
0.8 1.2 Wavelength (µ m)
0.8 Wavelength (µ m)
1.2
(d)
(b) Input energy (µ J)
PRL 111, 033902 (2013)
5
5
3
3
1
exp
ii i
1 0.4
0.8 1.2 Wavelength (µ m)
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0.8 1.2 Output energy ( µ J)
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FIG. 4 (color). (a) Energy dependent experimental spectra for 500 fs pump pulses at 800 nm at the output of a 20 cm long 9 m radius kagome-PCF filled with 10 bar Xe; (b) corresponding numerical results. (c) Comparison between simulated (blue line) and experimental (black line) spectra at 5 J pump energy. (d) Output energy scaling with pump energy: experiment (red dots); simulation including fiber loss (i, blue dashed line); and simulations including fiber and photoionization-induced losses (ii, green line).
Note that the bandwidth of the SC is mainly limited by energy loss due to photoionization of the gas. The linear transmission loss of the kagome PCF also reduces the bandwidth, but only marginally. Additionally, a wider SC and a modified MI dynamics may be possible when making use of a Raman active gas; however, this significantly complicates the analysis and is beyond the scope of this work. Conventional MI sidebands appear at low energies, and for higher energies a SC is formed, reaching its maximum spectral width at 3 J. Figure 4(b) shows the output spectrum for 5 J of pump energy. The black line is the experimental spectrum, and the blue line is the result of an average over 20 noise-seeded numerical simulations for the experimental parameters. The agreement between experiment and simulation is remarkably good in the low-loss transmission window of the fiber and only slightly worse in the high loss band around 400 nm (caused by resonances in the cladding). Figure 4(d) shows the output versus input energy. The dots are the experimental measurements, showing that the output energy saturates at 1 J. The dashed line represents the simulated behavior including the loss of the fiber, while the solid line is the expected behavior when additionally taking into account loss due to photoionization of the gas. It is interesting to note that the input energy at which saturation is seen coincides with the appearance of MI sidebands, indicating that the onset of ionization is linked to MI-induced pulse breakup. The accurate experimental confirmation of the numerical simulations, for parameters corresponding to the creation of
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subcycle pulses through MI, lends strong support to the correctness of our analysis. In conclusion, gas-filled hollow core kagome-style PCF permits the realization of an extreme form of MI, resulting in generation of a supercontinuum consisting of a shower of high intensity subcycle solitonlike pulses whose behavior differs from that of NLSE solitons. Numerical simulations predict that the high intensities obtained in this way result in strong and sustained photoionization of the gas. These results are confirmed in experiments, only a few J of energy being needed to generate a broad and flat SC extending from the UV to the near IR. Although this pump energy is remarkably low in gas-based nonlinear optics, the resulting supercontinuum has extremely high spectral energy density compared to the SC generated in solid-core PCFs.
*[email protected] [1] V. E. Zakharov and L. A. Ostrovsky, Physica (Amsterdam) 238D, 540 (2009). [2] A. Hasegawa and W. Brinkman, IEEE J. Quantum Electron. 16, 694 (1980). [3] K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986). [4] A. S. Gouveia-Neto, A. S. L. Gomes, and J. R. Taylor, IEEE J. Quantum Electron. 24, 332 (1988). [5] M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, J. Opt. Soc. Am. B 6, 1149 (1989). [6] E. M. Dianov, A. B. Grudini, A. M. Prokhorov, and V. N. Serkin, in Optical Solitons: Theory and Experiment (Cambridge University Press, Cambridge, England, 1992). [7] J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006). [8] J. C. Travers, J. Opt. 12, 113001 (2010). [9] J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, Opt. Express 17, 21497 (2009).
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[10] C. Mahnke and F. Mitschke, Phys. Rev. A 85, 033808 (2012). [11] J. C. Travers, W. Chang, J. Nold, N. Y. Joly, and P. St. J. Russell, J. Opt. Soc. Am. B 28, A11 (2011). [12] P. St. J. Russell, Science 299, 358 (2003). [13] E. A. J. Marcatili and R. A. Schmeltzer, Bell Syst. Tech. J. 43, 1783 (1964). [14] J. Nold, P. Ho¨lzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St. J. Russell, Opt. Lett. 35, 2922 (2010). [15] P. Ho¨lzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St. J. Russell, Phys. Rev. Lett. 107, 203901 (2011). [16] N. Y. Joly, J. Nold, W. Chang, P. Ho¨lzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, Phys. Rev. Lett. 106, 203901 (2011). [17] A. V. Husakou and J. Herrmann, Phys. Rev. Lett. 87, 203901 (2001). [18] M. Kolesik, J. V. Moloney, and M. Mlejnek, Phys. Rev. Lett. 89, 283902 (2002). [19] P. Kinsler, Phys. Rev. A 81, 013819 (2010). [20] W. Chang, A. Nazarkin, J. C. Travers, J. Nold, P. Ho¨lzer, N. Y. Joly, and P. St. J. Russell, Opt. Express 19, 21 018 (2011). [21] G. L. Yudin and M. Y. Ivanov, Phys. Rev. A 64, 013409 (2001). [22] P. D. Drummond and J. F. Corney, J. Opt. Soc. Am. B 18, 139 (2001). [23] M. Saleh, W. Chang, J. C. Travers, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 109, 113902 (2012). [24] D. V. Skryabin and A. V. Yulin, Phys. Rev. E 72, 016619 (2005). [25] J. N. Kutz, C. Lynga˚, and B. Eggleton, Opt. Express 13, 3989 (2005). [26] A. Nazarkin and G. Korn, Phys. Rev. Lett. 83, 4748 (1999). [27] D. V. Kartashov, A. V. Kim, and S. A. Skobelev, JETP Lett. 78, 276 (2003). [28] S. Amiranashvili, A. G. Vladimirov, and U. Bandelow, Eur. Phys. J. D 58, 219 (2010). [29] A. Schwache and F. MItschke, Phys. Rev. E 55, 7720 (1997).
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