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Supercontinuum generation through cascaded four-wave mixing in photonic-crystal fibers: When picoseconds do it better
Optics Communications 274 (2007) 433–440 www.elsevier.com/locate/optcom
Supercontinuum generation through cascaded four-wave mixing in photonic-crystal fibers: When picoseconds do it better E.E. Serebryannikov, A.M. Zheltikov
*
Physics Department, International Laser Center, M.V. Lomonosov Moscow State University, Vorob’evy gory, Moscow 119992, Russia Received 21 September 2006; received in revised form 30 November 2006; accepted 1 December 2006
Abstract By numerically solving the generalized nonlinear Schro¨dinger equation, we provide a comparative analysis of spectral and temporal transformations of pico and femtosecond laser pulses in a guided mode with a generic dispersion profile typical of a photonic-crystal fiber. We demonstrate that with an appropriate choice of the input laser pulse parameters, picosecond pulses are efficiently transformed into a supercontinuum emission through a cascade of phase-matched four-wave mixing (FWM) processes. For femtosecond pulses, soliton frequency-shifting phenomena tend to arrest such FWM cascades. In this regime, picosecond pulses can provide much higher efficiencies of supercontinuum generation, as well as much broader and much smoother spectra of output radiation compared to femtosecond pulses with the same peak power. Ó 2007 Published by Elsevier B.V. PACS: 42.65.Wi; 42.81.Qb
1. Introduction Supercontinuum (SC) generation is an interesting and practically significant physical phenomenon involving a dramatic spectral broadening of a laser pulse propagating through a nonlinear medium and resulting in the generation of white-light emission with spectrum often spanning over octaves. This phenomenon, first observed by Alfano and Shapiro [1,2], remains a focus of interest over more than three decades due to numerous applications of SC light sources [3], as well as due to an interesting physics behind this effect, involving a complex interplay of nonlinear-optical processes [4]. The early work on SC generation was largely aimed at explaining the origin of this broadband emission in the solid phase [5], organic and inorganic liquids [6–9], as well as in gas media [10,11]. Early experiments on SC generation in silica fibers, performed with pico and nanosecond light sources [12,13], suggested the ways toward a substantial reduction of laser *
Corresponding author. Tel.: +7 9393959; fax: +7 9393113. E-mail address: [email protected] (A.M. Zheltikov).
0030-4018/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.optcom.2007.01.080
peak power required for white-light transformation and provided important insights into the role of stimulated Raman scattering (SRS) and four-wave mixing (FWM) processes in SC generation. SC generation has been also examined as a promising strategy for the development of multiwavelength optical sources for WDM applications [14–18]. In these studies, SC was produced through solion compression in the regime of anomalous dispersion in a dispersion-decreasing fiber [18] and through self-phase modulation (SPM) in a dispersion-flattened fiber, as well as by cascading these two approaches [17]. Photonic-crystal fibers (PCFs) [19,20] have offered a variety of new possibilities for efficient SC generation. These waveguides allow dispersion profile engineering through fiber structure modifications [21] and provide high optical nonlinearities due to the strong field confinement in small-size fiber cores [22,23]. As a result, the nonlinear-optical effects contributing to SC generation are radically enhanced, leading to highly efficient spectral broadening of laser pulses [24,25]. In fact, the basic physical phenomena resulting in SC generation in PCFs have been identified and understood [3] long before the advent of PCFs. In
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particular, it has been known since the early days of the laser era that the intensity dependence of the refractive index leads to spectral broadening through self-phase modulation (SPM) [26,27]. Shock waves have been identified [28] as one of the mechanism leading to the asymmetry of SPM-broadened spectra. Early experiments on SC generation in silica fibers using pico and nanosecond sources [12,13] have demonstrated an important role of FWM and SRS in enriching the spectral contents of the light field with new spectral components. Golovchenko et al. [29] have highlighted the importance of mutual influence of Raman and parametric processes, showing that the SRS threshold can be increased because of parametric processes. Beaud et al. [30] have applied the generalized nonlinear Schro¨dinger equation (GNSE) [31,32], including high-order dispersion and the Raman effect, to provide an in-depth study of the dynamics of subpicosecond laser pulses in singlemode fibers, discussing pulse breakup and soliton self-frequency shift. Among other interesting findings, Schu¨tz et al. [33] have analyzed pulse propagation at the zero dispersion wavelength of a standard optical fiber and revealed the physics behind pulse distortions in this regime. In PCFs, all the above-outlined mechanisms of pulse transformation are radically enhanced [25]. This often leads to a complicated interplay of several nonlinear-optical processes [34,35], which can be controlled to some extent by tailoring an appropriate fiber dispersion profile. A detailed analysis of SC in PCFs can be found in the extensive literature (see e.g. [4,36] for a review). SC-generating PCFs are currently extensively used in many areas of optical science [4], giving an excess to the carrier–envelope phase [37,38], allowing the generation of stabilized frequency combs for optical metrology and femtosecond clockwork [39,40,37,41], permitting creation of novel optical coherence tomographs [42], nonlinear spectrometers [43,44] and microscopes [45–47], and generators of correlated photons [48], and simplifying pump–seed synchronization in few-cycle optical parametric chirped-pulse amplifiers [49]. In view of rapidly growing applications of PCF white-light sources, SC generation remains one of the hottest topics in optical science. The main focus of research in the past few years was placed on understanding the ways to generate SC radiation with spectral properties best suited for specific applications, optimization of nonlinear-optical pulse propagation dynamics in PCFs aimed at loosening the requirements on input pulse parameters, and integration of PCF-based SC generators [50,51], pulse compressors [52–54], frequency shifters [49], and components for dispersion compensation [54] into advanced laser systems based on fiber-laser technologies [50–52,54,55] and thin-disk lasers [53]. The input pulse width is an important parameter that in many ways influences the regime of nonlinear-optical pulse transformation in a PCF. While femtosecond pulses can efficiently generate SC through solitonic mechanisms in the region of anomalous dispersion [25,56], picosecond pulses, as shown by Coen et al. [57], can be transformed
into supercontinuum radiation through stimulated Raman scattering and four-wave mixing (FWM). Interesting insights into supercontinuum generation in PCFs by nanosecond laser pulses have been provided by Dudley et al. [58]. Comparison of the physical mechanisms and optimal regimes of SC generation by femto and picosecond pulses is motivated by the rapid progress in compact laser systems, including advanced fiber-laser sources [59–61], capable of generating pulses with pulse widths ranging from 10s of femtoseconds to a few picoseconds. Recent experiments [50,51] have shown that due to an advantageous combination of high optical nonlinearity and dispersion flexibility, PCFs are ideally suited for SC transformation of picosecond fiber-laser output. On the other hand, such an analysis raises interesting physical issues related to the interplay of nonlinear-optical processes in guided-wave short-pulse dynamics. In this paper, we show that cascaded FWM processes in the region of anomalous dispersion in PCFs offer interesting possibilities for efficient SC transformation of picosecond laser pulses. We numerically solve the generalized nonlinear Schro¨dinger equation and provide a comparative analysis of spectral and temporal transformations of pico and femtosecond laser pulses in a guided mode with a generic dispersion profile typical of a PCF. Based on this analysis, we demonstrate that, with an appropriate choice of the input laser pulse parameters, picosecond pulses can be efficiently transformed into a supercontinuum emission through a cascade of phase-matched FWM interactions. For femtosecond pulses, on the other hand, soliton frequency-shifting phenomena tend to arrest such FWM cascades. In this regime, picosecond pulses can outperform femtosecond pulses in supercontinuum generation, providing much higher efficiencies of supercontinuum generation, as well as much broader and much smoother spectra of output radiation compared to femtosecond pulses with the same peak power. 2. Numerical procedure Our numerical procedure is based on the solution of the generalized nonlinear Schro¨dinger equation [45,46] for the field envelope A ¼ Aðz; tÞ: 6 X oA ðiÞ ðkÞ ok A ¼i b þ P nl ðn; sÞ; on k! osk k¼2 k
ð1Þ
where z is the propagation coordinate, t is the time variable, s is the retarded time, and bðkÞ ¼ ok b=oxk are the coefficients in the Taylor-series expansion of the propagation constant b. The nonlinear polarization P nl ðn; sÞ in Eq. (1) is defined in such a way as to include the retarded nonlinearity of the fiber material and the wavelength dependence of the effective mode area: n2 x 1 b ~pnl ðn; x0 xÞ ; P nl ðn; sÞ ¼ i F ð2Þ cS eff ðxÞ
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where n2 is the nonlinear refractive index of the fiber material, x is the current frequency, x0 is the central frequency of theR input field, c is R theR speed of light, 1 R1 1 1 2 2 4 S eff ¼ ½ 1 1 j F ðx; yÞj dxdy = 1 1 j F ðx; yÞj dxdy is the effective mode area, F ðx,yÞ is the field profile in a waveguide mode, and the operator Fb 1 ðÞ denotes the inverse Fourier transform. The frequency-domain nonlinear polarization in Eq. (2) is defined through the direct Fourier transform, Z 1 2 ~ pnl ðn; x x0 Þ ¼ Fb Aðn; sÞ RðtÞjAðn; s tÞj dt ; ð3Þ 1
including both the instantaneous, Kerr nonlinearity and the retarded, Raman contribution via the nonlinear response function s2 þ s2 t t RðtÞ ¼ ð1 fR ÞdðtÞ þ fR HðtÞ 1 2 2 e s2 sin ; ð4Þ s1 s1 s2 where fR is the fractional contribution of the Raman response; d(t) and HðtÞ are the delta and the Heaviside step functions, respectively; s1 and s2 are the characteristic times of the Raman response of the fiber material. For fused silica, fR ¼ 0:18, s1 ¼ 12:5 fs, and s2 ¼ 32 fs. The nonlinear polarization P nl ðn; sÞ defined in the form of Eq. (2) not only helps to include the influence of the frequency-dependent effective mode area S eff on the nonlinear coefficient c ¼ ðn2 xÞ=ðcS eff Þ, but also gives a correct definition of the local field intensity, which also depends on S eff ðxÞ. We consider spectral and temporal transformations of pico and femtosecond pulses with parameters typical of fiber laser output propagating in a waveguide mode with a wavelength dependence of group-velocity dispersion (GVD) shown by line 1 in Fig. 1. This dispersion profile is typical of a generic type of PCF, including several types of commercial fibers, referred to as highly nonlinear PCFs. Parameters bðkÞ in Eq. (1) with k taking the values from 2 to 6 were defined by fitting the GVD profile shown in Fig. 1 with a polynomial. This procedure yields the following set
Fig. 1. Group-velocity dispersion of the fiber mode as a function of the wavelength (1) and the input spectra of laser pulses centered at 1070 nm (2, 3) and 800 nm (4) with an initial pulse width of 2 ps (2, 4) and 200 fs (3).
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of dispersion coefficients: bð2Þ 0:012 ps2/m, bð3Þ 7:9 105 ps3/m, bð4Þ 1:4 107 ps4/m, bð5Þ 2:0 1010 ps5/m, and bð6Þ 2:9 1012 ps6/m. For input laser pulses, the initial pulse widths are taken equal to 2 ps and 200 fs. The input pulse energy is set equal to 20 nJ for 2-ps pulses and 2 nJ for 200-fs pulses. The input peak power is thus the same in both cases and is equal to 10 kW. To identify the differences between the dynamics of laser pulses propagating in guided modes of PCFs with normal and anomalous dispersion, we will consider laser pulses with input radiation wavelengths of 1070 nm (curves 2 and 3 in Fig. 1) and 800 nm (curve 4 in Fig. 1). Numerical simulations were performed by solving the GNSE with the use of the split-step Fourier-transform technique [32]. In these simulations, a 30-ps temporal window was discretized by 217 points. The discretization step Dz for the propagation length was chosen in such a way as to eliminate artifacts originating from numerical instabilities induced by high-order dispersion terms in the GNSE. With sufficiently small Dz (Dz 10 lm in our simulations), such instabilities were shifted outside the spectral window, leading to no interference with the spectrum of the laser field within the considered spectral range. 3. Results and discussion 3.1. Picosecond pulses We start our analysis with spectral and temporal transformations of 2-ps 800-nm laser pulses in the region of normal dispersion of the considered PCF. In this regime, the evolution of laser pulses is dominated by self-phase modulation (SPM). The spectrum of the laser pulse remains centered around 800 nm, with the nonlinear phase shift manifesting itself in a symmetric spectral broadening (Fig. 2a). A slight asymmetry in the laser pulse spectrum becomes noticeable for fiber lengths exceeding 35 cm (Fig. 2a), when high-order dispersion and shock terms come into play. These effects also distort the temporal envelope of the field intensity, which can deviate quite noticeably from the input Gaussian pulse shape for large fiber lengths (50 cm in Fig. 2b). To analyze the evolution of picosecond pulses in the regime of anomalous dispersion, we consider the propagation of 2-ps 1070-nm laser pulses through the PCF with the GVD profile shown by curve 1 in Fig. 1. The initial stage of evolution of such pulses in the PCF (Fig. 3a and b) is also dominated by self-phase modulation and has much in common with the evolution of picosecond pulses in the region of normal dispersion. Although the pulse now senses negative dispersion of the fiber, soliton phenomena do not play a noticeable role at this stage, as the dispersion length Ld ¼ s2 =jb2 j (s is the pulse width) for the chosen input pulse width is Ld 400 m, remaining much larger than the maximum fiber length in the considered series of computations. As the pulse propagates further on along the fiber, pronounced sidebands start to build up in the spectrum of the
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Fig. 2. Spectral (a) and temporal (b) transformation of a laser pulse with an initial pulse width of 2 ps, the input central wavelength of 800 nm, and an input energy of 20 nJ in the fiber with the GVD profile shown by curve 1 in Fig. 1. The fiber length increases from top to bottom, taking the values of 5, 20, 35, and 50 cm.
Fig. 3. Spectral (a) and temporal (b) transformation of a laser pulse with an initial pulse width of 2 ps, the input central wavelength of 1070 nm, and an input energy of 20 nJ in the fiber with the GVD profile shown by curve 1 in Fig. 1. The fiber length increases from top to bottom, taking the values of 0.5, 1, 5, and 10 cm.
laser field (Fig. 4a) at frequencies x0 X as a result of parametric four-wave mixing 2x0 ¼ ðx0 þ XÞ þ ðx0 XÞ, where x0 is the frequency of the pump field. Lines 1 and 2 in Fig. 5 show the phase mismatch db for the considered FWM process calculated with pump wavelengths of 1070 and 1050 nm. For a pulse width of 2 ps and a pulse energy of 20 nJ, the wavelengths corresponding to zero db agree very well with the wavelengths where the sidebands build up in numerical simulations in Fig. 4a. Lines 3 and 4 in Fig. 5 present the FWM gain G calculated as a function of the frequency shift X with pump wavelengths of 1070 and 1050 nm using the standard formula [32] GðXÞ ¼ 1=2 jb2 jXð4cP =jb2 j X2 Þ , where P is the peak power. Analysis of FWM phase matching and gain shows that the spectrally broadened laser field (Fig. 4a) provides frequency components that serve as a pump for phase-matched
FWM, giving rise to intense sidebands in the spectrum of the light field in the fiber. With a pump wavelength of 1050 nm, the maximum gain, as can be seen from Fig. 5, is achieved for the sidebands centered at 930 and 1230 nm, i.e., exactly where the sidebands build up in numerical simulations in Fig. 4a. In the time domain, sideband generation through parametric FWM translates into an oscillating field structure (see the inset in Fig. 4b), which becomes noticeable already for a fiber length L 10 cm (the lowest plot in Fig. 3b) and which becomes more and more pronounced as the amplitude of the sidebands increases in the frequency domain (Fig. 4b). An instructive way to understand this tendency in the temporal field evolution is to consider the parametric FWM process 2x0 ¼ ðx0 þ XÞ þ ðx0 XÞ as a modulation instability (MI) [32] of the temporal field envelope with
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Fig. 5. The propagation-constant mismatch db (1,2) and the gain G (3, 4) for parametric FWM 2x0 ¼ ðx0 þ XÞ þ ðx0 XÞ, resulting in the generation of sidebands at x0 X calculated with pump wavelengths of (1, 3) 1050 and (2, 4) 1070 nm. The inset shows the propagation-constant mismatch for the xp1 þ xp2 ¼ xa þ xs process with a two-color pump involving the spectral components at 1070 and 930 nm.
Fig. 4. Spectral (a) and temporal (b) transformation of a laser pulse with an initial pulse width of 2 ps, the input central wavelength of 1070 nm, and an input energy of 20 nJ in the fiber with the GVD profile shown by curve 1 in Fig. 1. The fiber length increases from top to bottom, taking the values of 11, 12, 13, and 14 cm. The inset displays the close-up of the fragment of the field intensity envelope shown by a circle in the panel corresponding to a fiber length of 12 cm, providing a closer view of oscillations building up on top of the laser pulse.
respect to perturbations whose spectrum is controlled by SPM-induced phase matching, as illustrated in Fig. 5. For larger fiber lengths (L ¼ 12 cm in Fig. 4a, b), an intense spectral component centered at 830 nm shows up in the spectrum of the laser pulse. Generation of this spectral component is attributed to a parametric FWM process xp1 þ xp2 ¼ xa þ xs , where the 1220-nm sideband plays
the role of the Stokes field amplified by the pump fields xp1 and xp2 , provided by the 1070-nm pump and 930-nm sideband. The phase matching for this process is illustrated in the inset to Fig. 5, which shows that the propagationconstant mismatch db for the xp1 þ xp2 ¼ xa þ xs process with a two-color pump involving the spectral components at 1070 and 930 nm vanishes at Stokes and anti-Stokes wavelengths equal to 1220 and 830 nm, respectively. Efficient generation of the 830-nm spectral component thus becomes possible through a cascade of two phase-matched FWM processes. The first stage involves sideband generation through modulation instability of the pump field with SPM-induced phase matching. At the second stage, the Stokes sideband is parametrically amplified in the field of the two-color pump, provided by the central part of the field spectrum and the anti-Stokes sideband, leading to efficient generation of the second anti-Stokes component centered at 830 nm. As the fiber length increases (Figs. 4a, 6a), multiple sidebands generated through cascaded FWM are broadened through self- and cross-phase modulation, merging together and eventually giving rise to a broadband (supercontinuum) emission with a spectrum stretching (for L 30 cm) from 650 to 2200 nm. The spectrum of the light field in this regime is broad enough to access the Ramanactive modes of the fiber material. In the frequency domain, the Raman effect gives rise to an intense longwavelength wing of the spectrum, whose maximum shifts from approximately 1.7 to 2.1 lm as the fiber length is increased from 20 to 30 cm (Fig. 6a). Broad red-shifted
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Fig. 7. Spectral (a) and temporal (b) transformation of a laser pulse with an initial pulse width of 200 fs, the input central wavelength of 1070 nm, and an input energy of 2 nJ in the fiber with the GVD profile shown by curve 1 in Fig. 1. The fiber length increases from top to bottom, taking the values of 5, 20, 35, and 50 cm.
Fig. 6. Spectral (a) and temporal (b) transformation of a laser pulse with an initial pulse width of 2 ps, the input central wavelength of 1070 nm, and an input energy of 20 nJ in the fiber with the GVD profile shown by curve 1 in Fig. 1. The fiber length increases from top to bottom, taking the values of 15, 20, 25, and 30 cm. The inset presents a close-up of the solitonic feature shown by a circle in the panel corresponding to a fiber length of 30 cm.
bands observed in the spectra presented in Fig. 6a correspond to well-localized femtosecond solitonic features in the time domain (Fig. 6b). Anomalous fiber dispersion translates the red shift of these spectral bands into a positive delay time. As a result, in the time domain, these solitonic features are noticeably delayed with respect to the
remaining part of the light field. Numerical analysis of pulse dynamics performed with and without the retarded part of optical nonlinearity shows that the 1.7–2.1-lm region of supercontinuum emission in Fig. 6a is fully controlled by the Raman effect and red-shifted solitons. 3.2. Femtosecond pulses We now turn to the analysis of the dynamics of femtosecond laser pulses in the region of anomalous dispersion of PCF. Fig. 7a and b illustrate the main tendencies of spectral and temporal transformations of 200-fs 1070-nm pulses in a PCF with the dispersion profile shown by curve 1 in Fig. 1. Pulse evolution in this case radically differs from the dynamics of picosecond pulses discussed in Section 3.1. For femtosecond laser pulses, the spectrum becomes asymmetric and
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quite complicated within the first few centimeters of the fiber. A distinct gap around the zero-GVD wavelength kz 970 nm is one of the prominent features observed in the spectrum of the laser field. This gap, as shown in earlier work, originates from parametric FWM processes xp1 þ xp2 ¼ xa þ xs , which are easily phase-matched when the pump-field components xp1 and xp2 lie close to the zeroGVD point, depleting the spectral range around kz . The low-frequency part of the field spectrum falls within the region of anomalous dispersion and tends to generate solitons, which are readily observed in the time-domain structure of the field in Fig. 7b. These solitons undergo a continuous red shift due to the Raman effect as the pulse propagates through the fiber, giving rise to an intense long-wavelength wing of the field spectrum. The blueshifted part of PCF output spectra is dominated by dispersive-wave (Cherenkov) emission of solitons. As a result, 2-ps pulses with the same input peak power and the same initial central wavelength generate in the case under consideration much broader and much smoother spectra at the output of a 50-cm PCF, offering a promising solution for the creation of PCF supercontinuum sources pumped by picosecond fiber lasers. 4. Conclusion We have shown that cascaded FWM processes in the region of anomalous dispersion in PCFs offer interesting possibilities for efficient SC transformation of picosecond laser pulses. By numerically solving the generalized nonlinear Schro¨dinger equation we provided a comparative analysis of spectral and temporal transformations of pico and femtosecond laser pulses in a guided mode with a generic dispersion profile typical of a PCF. Based on this analysis, we have demonstrated that, with an appropriate choice of the input laser pulse parameters, picosecond pulses can be efficiently transformed into a supercontinuum emission through a cascade of phase-matched FWM interactions. For femtosecond pulses, on the other hand, soliton frequency-shifting phenomena tend to arrest such FWM cascades. In this regime, picosecond pulses can outperform femtosecond pulses in supercontinuum generation, providing much higher efficiencies of supercontinuum generation, as well as much broader and much smoother spectra of output radiation compared to femtosecond pulses with the same peak power. Acknowledgements This study was supported in part by the Russian Foundation for Basic Research (Projects 06-02-16880 and 05-0290566-NNS) and INTAS (Projects Nos. 03-51-5037 and 03-51-5288). The research described in this publication was made possible in part by Award No. RUP2-2695 of the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF).
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Supercontinuum generation through cascaded four-wave mixing in photonic-crystal fibers: When picoseconds do it better E.E. Serebryannikov, A.M. Zheltikov
*
Physics Department, International Laser Center, M.V. Lomonosov Moscow State University, Vorob’evy gory, Moscow 119992, Russia Received 21 September 2006; received in revised form 30 November 2006; accepted 1 December 2006
Abstract By numerically solving the generalized nonlinear Schro¨dinger equation, we provide a comparative analysis of spectral and temporal transformations of pico and femtosecond laser pulses in a guided mode with a generic dispersion profile typical of a photonic-crystal fiber. We demonstrate that with an appropriate choice of the input laser pulse parameters, picosecond pulses are efficiently transformed into a supercontinuum emission through a cascade of phase-matched four-wave mixing (FWM) processes. For femtosecond pulses, soliton frequency-shifting phenomena tend to arrest such FWM cascades. In this regime, picosecond pulses can provide much higher efficiencies of supercontinuum generation, as well as much broader and much smoother spectra of output radiation compared to femtosecond pulses with the same peak power. Ó 2007 Published by Elsevier B.V. PACS: 42.65.Wi; 42.81.Qb
1. Introduction Supercontinuum (SC) generation is an interesting and practically significant physical phenomenon involving a dramatic spectral broadening of a laser pulse propagating through a nonlinear medium and resulting in the generation of white-light emission with spectrum often spanning over octaves. This phenomenon, first observed by Alfano and Shapiro [1,2], remains a focus of interest over more than three decades due to numerous applications of SC light sources [3], as well as due to an interesting physics behind this effect, involving a complex interplay of nonlinear-optical processes [4]. The early work on SC generation was largely aimed at explaining the origin of this broadband emission in the solid phase [5], organic and inorganic liquids [6–9], as well as in gas media [10,11]. Early experiments on SC generation in silica fibers, performed with pico and nanosecond light sources [12,13], suggested the ways toward a substantial reduction of laser *
Corresponding author. Tel.: +7 9393959; fax: +7 9393113. E-mail address: [email protected] (A.M. Zheltikov).
0030-4018/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.optcom.2007.01.080
peak power required for white-light transformation and provided important insights into the role of stimulated Raman scattering (SRS) and four-wave mixing (FWM) processes in SC generation. SC generation has been also examined as a promising strategy for the development of multiwavelength optical sources for WDM applications [14–18]. In these studies, SC was produced through solion compression in the regime of anomalous dispersion in a dispersion-decreasing fiber [18] and through self-phase modulation (SPM) in a dispersion-flattened fiber, as well as by cascading these two approaches [17]. Photonic-crystal fibers (PCFs) [19,20] have offered a variety of new possibilities for efficient SC generation. These waveguides allow dispersion profile engineering through fiber structure modifications [21] and provide high optical nonlinearities due to the strong field confinement in small-size fiber cores [22,23]. As a result, the nonlinear-optical effects contributing to SC generation are radically enhanced, leading to highly efficient spectral broadening of laser pulses [24,25]. In fact, the basic physical phenomena resulting in SC generation in PCFs have been identified and understood [3] long before the advent of PCFs. In
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particular, it has been known since the early days of the laser era that the intensity dependence of the refractive index leads to spectral broadening through self-phase modulation (SPM) [26,27]. Shock waves have been identified [28] as one of the mechanism leading to the asymmetry of SPM-broadened spectra. Early experiments on SC generation in silica fibers using pico and nanosecond sources [12,13] have demonstrated an important role of FWM and SRS in enriching the spectral contents of the light field with new spectral components. Golovchenko et al. [29] have highlighted the importance of mutual influence of Raman and parametric processes, showing that the SRS threshold can be increased because of parametric processes. Beaud et al. [30] have applied the generalized nonlinear Schro¨dinger equation (GNSE) [31,32], including high-order dispersion and the Raman effect, to provide an in-depth study of the dynamics of subpicosecond laser pulses in singlemode fibers, discussing pulse breakup and soliton self-frequency shift. Among other interesting findings, Schu¨tz et al. [33] have analyzed pulse propagation at the zero dispersion wavelength of a standard optical fiber and revealed the physics behind pulse distortions in this regime. In PCFs, all the above-outlined mechanisms of pulse transformation are radically enhanced [25]. This often leads to a complicated interplay of several nonlinear-optical processes [34,35], which can be controlled to some extent by tailoring an appropriate fiber dispersion profile. A detailed analysis of SC in PCFs can be found in the extensive literature (see e.g. [4,36] for a review). SC-generating PCFs are currently extensively used in many areas of optical science [4], giving an excess to the carrier–envelope phase [37,38], allowing the generation of stabilized frequency combs for optical metrology and femtosecond clockwork [39,40,37,41], permitting creation of novel optical coherence tomographs [42], nonlinear spectrometers [43,44] and microscopes [45–47], and generators of correlated photons [48], and simplifying pump–seed synchronization in few-cycle optical parametric chirped-pulse amplifiers [49]. In view of rapidly growing applications of PCF white-light sources, SC generation remains one of the hottest topics in optical science. The main focus of research in the past few years was placed on understanding the ways to generate SC radiation with spectral properties best suited for specific applications, optimization of nonlinear-optical pulse propagation dynamics in PCFs aimed at loosening the requirements on input pulse parameters, and integration of PCF-based SC generators [50,51], pulse compressors [52–54], frequency shifters [49], and components for dispersion compensation [54] into advanced laser systems based on fiber-laser technologies [50–52,54,55] and thin-disk lasers [53]. The input pulse width is an important parameter that in many ways influences the regime of nonlinear-optical pulse transformation in a PCF. While femtosecond pulses can efficiently generate SC through solitonic mechanisms in the region of anomalous dispersion [25,56], picosecond pulses, as shown by Coen et al. [57], can be transformed
into supercontinuum radiation through stimulated Raman scattering and four-wave mixing (FWM). Interesting insights into supercontinuum generation in PCFs by nanosecond laser pulses have been provided by Dudley et al. [58]. Comparison of the physical mechanisms and optimal regimes of SC generation by femto and picosecond pulses is motivated by the rapid progress in compact laser systems, including advanced fiber-laser sources [59–61], capable of generating pulses with pulse widths ranging from 10s of femtoseconds to a few picoseconds. Recent experiments [50,51] have shown that due to an advantageous combination of high optical nonlinearity and dispersion flexibility, PCFs are ideally suited for SC transformation of picosecond fiber-laser output. On the other hand, such an analysis raises interesting physical issues related to the interplay of nonlinear-optical processes in guided-wave short-pulse dynamics. In this paper, we show that cascaded FWM processes in the region of anomalous dispersion in PCFs offer interesting possibilities for efficient SC transformation of picosecond laser pulses. We numerically solve the generalized nonlinear Schro¨dinger equation and provide a comparative analysis of spectral and temporal transformations of pico and femtosecond laser pulses in a guided mode with a generic dispersion profile typical of a PCF. Based on this analysis, we demonstrate that, with an appropriate choice of the input laser pulse parameters, picosecond pulses can be efficiently transformed into a supercontinuum emission through a cascade of phase-matched FWM interactions. For femtosecond pulses, on the other hand, soliton frequency-shifting phenomena tend to arrest such FWM cascades. In this regime, picosecond pulses can outperform femtosecond pulses in supercontinuum generation, providing much higher efficiencies of supercontinuum generation, as well as much broader and much smoother spectra of output radiation compared to femtosecond pulses with the same peak power. 2. Numerical procedure Our numerical procedure is based on the solution of the generalized nonlinear Schro¨dinger equation [45,46] for the field envelope A ¼ Aðz; tÞ: 6 X oA ðiÞ ðkÞ ok A ¼i b þ P nl ðn; sÞ; on k! osk k¼2 k
ð1Þ
where z is the propagation coordinate, t is the time variable, s is the retarded time, and bðkÞ ¼ ok b=oxk are the coefficients in the Taylor-series expansion of the propagation constant b. The nonlinear polarization P nl ðn; sÞ in Eq. (1) is defined in such a way as to include the retarded nonlinearity of the fiber material and the wavelength dependence of the effective mode area: n2 x 1 b ~pnl ðn; x0 xÞ ; P nl ðn; sÞ ¼ i F ð2Þ cS eff ðxÞ
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where n2 is the nonlinear refractive index of the fiber material, x is the current frequency, x0 is the central frequency of theR input field, c is R theR speed of light, 1 R1 1 1 2 2 4 S eff ¼ ½ 1 1 j F ðx; yÞj dxdy = 1 1 j F ðx; yÞj dxdy is the effective mode area, F ðx,yÞ is the field profile in a waveguide mode, and the operator Fb 1 ðÞ denotes the inverse Fourier transform. The frequency-domain nonlinear polarization in Eq. (2) is defined through the direct Fourier transform, Z 1 2 ~ pnl ðn; x x0 Þ ¼ Fb Aðn; sÞ RðtÞjAðn; s tÞj dt ; ð3Þ 1
including both the instantaneous, Kerr nonlinearity and the retarded, Raman contribution via the nonlinear response function s2 þ s2 t t RðtÞ ¼ ð1 fR ÞdðtÞ þ fR HðtÞ 1 2 2 e s2 sin ; ð4Þ s1 s1 s2 where fR is the fractional contribution of the Raman response; d(t) and HðtÞ are the delta and the Heaviside step functions, respectively; s1 and s2 are the characteristic times of the Raman response of the fiber material. For fused silica, fR ¼ 0:18, s1 ¼ 12:5 fs, and s2 ¼ 32 fs. The nonlinear polarization P nl ðn; sÞ defined in the form of Eq. (2) not only helps to include the influence of the frequency-dependent effective mode area S eff on the nonlinear coefficient c ¼ ðn2 xÞ=ðcS eff Þ, but also gives a correct definition of the local field intensity, which also depends on S eff ðxÞ. We consider spectral and temporal transformations of pico and femtosecond pulses with parameters typical of fiber laser output propagating in a waveguide mode with a wavelength dependence of group-velocity dispersion (GVD) shown by line 1 in Fig. 1. This dispersion profile is typical of a generic type of PCF, including several types of commercial fibers, referred to as highly nonlinear PCFs. Parameters bðkÞ in Eq. (1) with k taking the values from 2 to 6 were defined by fitting the GVD profile shown in Fig. 1 with a polynomial. This procedure yields the following set
Fig. 1. Group-velocity dispersion of the fiber mode as a function of the wavelength (1) and the input spectra of laser pulses centered at 1070 nm (2, 3) and 800 nm (4) with an initial pulse width of 2 ps (2, 4) and 200 fs (3).
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of dispersion coefficients: bð2Þ 0:012 ps2/m, bð3Þ 7:9 105 ps3/m, bð4Þ 1:4 107 ps4/m, bð5Þ 2:0 1010 ps5/m, and bð6Þ 2:9 1012 ps6/m. For input laser pulses, the initial pulse widths are taken equal to 2 ps and 200 fs. The input pulse energy is set equal to 20 nJ for 2-ps pulses and 2 nJ for 200-fs pulses. The input peak power is thus the same in both cases and is equal to 10 kW. To identify the differences between the dynamics of laser pulses propagating in guided modes of PCFs with normal and anomalous dispersion, we will consider laser pulses with input radiation wavelengths of 1070 nm (curves 2 and 3 in Fig. 1) and 800 nm (curve 4 in Fig. 1). Numerical simulations were performed by solving the GNSE with the use of the split-step Fourier-transform technique [32]. In these simulations, a 30-ps temporal window was discretized by 217 points. The discretization step Dz for the propagation length was chosen in such a way as to eliminate artifacts originating from numerical instabilities induced by high-order dispersion terms in the GNSE. With sufficiently small Dz (Dz 10 lm in our simulations), such instabilities were shifted outside the spectral window, leading to no interference with the spectrum of the laser field within the considered spectral range. 3. Results and discussion 3.1. Picosecond pulses We start our analysis with spectral and temporal transformations of 2-ps 800-nm laser pulses in the region of normal dispersion of the considered PCF. In this regime, the evolution of laser pulses is dominated by self-phase modulation (SPM). The spectrum of the laser pulse remains centered around 800 nm, with the nonlinear phase shift manifesting itself in a symmetric spectral broadening (Fig. 2a). A slight asymmetry in the laser pulse spectrum becomes noticeable for fiber lengths exceeding 35 cm (Fig. 2a), when high-order dispersion and shock terms come into play. These effects also distort the temporal envelope of the field intensity, which can deviate quite noticeably from the input Gaussian pulse shape for large fiber lengths (50 cm in Fig. 2b). To analyze the evolution of picosecond pulses in the regime of anomalous dispersion, we consider the propagation of 2-ps 1070-nm laser pulses through the PCF with the GVD profile shown by curve 1 in Fig. 1. The initial stage of evolution of such pulses in the PCF (Fig. 3a and b) is also dominated by self-phase modulation and has much in common with the evolution of picosecond pulses in the region of normal dispersion. Although the pulse now senses negative dispersion of the fiber, soliton phenomena do not play a noticeable role at this stage, as the dispersion length Ld ¼ s2 =jb2 j (s is the pulse width) for the chosen input pulse width is Ld 400 m, remaining much larger than the maximum fiber length in the considered series of computations. As the pulse propagates further on along the fiber, pronounced sidebands start to build up in the spectrum of the
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Fig. 2. Spectral (a) and temporal (b) transformation of a laser pulse with an initial pulse width of 2 ps, the input central wavelength of 800 nm, and an input energy of 20 nJ in the fiber with the GVD profile shown by curve 1 in Fig. 1. The fiber length increases from top to bottom, taking the values of 5, 20, 35, and 50 cm.
Fig. 3. Spectral (a) and temporal (b) transformation of a laser pulse with an initial pulse width of 2 ps, the input central wavelength of 1070 nm, and an input energy of 20 nJ in the fiber with the GVD profile shown by curve 1 in Fig. 1. The fiber length increases from top to bottom, taking the values of 0.5, 1, 5, and 10 cm.
laser field (Fig. 4a) at frequencies x0 X as a result of parametric four-wave mixing 2x0 ¼ ðx0 þ XÞ þ ðx0 XÞ, where x0 is the frequency of the pump field. Lines 1 and 2 in Fig. 5 show the phase mismatch db for the considered FWM process calculated with pump wavelengths of 1070 and 1050 nm. For a pulse width of 2 ps and a pulse energy of 20 nJ, the wavelengths corresponding to zero db agree very well with the wavelengths where the sidebands build up in numerical simulations in Fig. 4a. Lines 3 and 4 in Fig. 5 present the FWM gain G calculated as a function of the frequency shift X with pump wavelengths of 1070 and 1050 nm using the standard formula [32] GðXÞ ¼ 1=2 jb2 jXð4cP =jb2 j X2 Þ , where P is the peak power. Analysis of FWM phase matching and gain shows that the spectrally broadened laser field (Fig. 4a) provides frequency components that serve as a pump for phase-matched
FWM, giving rise to intense sidebands in the spectrum of the light field in the fiber. With a pump wavelength of 1050 nm, the maximum gain, as can be seen from Fig. 5, is achieved for the sidebands centered at 930 and 1230 nm, i.e., exactly where the sidebands build up in numerical simulations in Fig. 4a. In the time domain, sideband generation through parametric FWM translates into an oscillating field structure (see the inset in Fig. 4b), which becomes noticeable already for a fiber length L 10 cm (the lowest plot in Fig. 3b) and which becomes more and more pronounced as the amplitude of the sidebands increases in the frequency domain (Fig. 4b). An instructive way to understand this tendency in the temporal field evolution is to consider the parametric FWM process 2x0 ¼ ðx0 þ XÞ þ ðx0 XÞ as a modulation instability (MI) [32] of the temporal field envelope with
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Fig. 5. The propagation-constant mismatch db (1,2) and the gain G (3, 4) for parametric FWM 2x0 ¼ ðx0 þ XÞ þ ðx0 XÞ, resulting in the generation of sidebands at x0 X calculated with pump wavelengths of (1, 3) 1050 and (2, 4) 1070 nm. The inset shows the propagation-constant mismatch for the xp1 þ xp2 ¼ xa þ xs process with a two-color pump involving the spectral components at 1070 and 930 nm.
Fig. 4. Spectral (a) and temporal (b) transformation of a laser pulse with an initial pulse width of 2 ps, the input central wavelength of 1070 nm, and an input energy of 20 nJ in the fiber with the GVD profile shown by curve 1 in Fig. 1. The fiber length increases from top to bottom, taking the values of 11, 12, 13, and 14 cm. The inset displays the close-up of the fragment of the field intensity envelope shown by a circle in the panel corresponding to a fiber length of 12 cm, providing a closer view of oscillations building up on top of the laser pulse.
respect to perturbations whose spectrum is controlled by SPM-induced phase matching, as illustrated in Fig. 5. For larger fiber lengths (L ¼ 12 cm in Fig. 4a, b), an intense spectral component centered at 830 nm shows up in the spectrum of the laser pulse. Generation of this spectral component is attributed to a parametric FWM process xp1 þ xp2 ¼ xa þ xs , where the 1220-nm sideband plays
the role of the Stokes field amplified by the pump fields xp1 and xp2 , provided by the 1070-nm pump and 930-nm sideband. The phase matching for this process is illustrated in the inset to Fig. 5, which shows that the propagationconstant mismatch db for the xp1 þ xp2 ¼ xa þ xs process with a two-color pump involving the spectral components at 1070 and 930 nm vanishes at Stokes and anti-Stokes wavelengths equal to 1220 and 830 nm, respectively. Efficient generation of the 830-nm spectral component thus becomes possible through a cascade of two phase-matched FWM processes. The first stage involves sideband generation through modulation instability of the pump field with SPM-induced phase matching. At the second stage, the Stokes sideband is parametrically amplified in the field of the two-color pump, provided by the central part of the field spectrum and the anti-Stokes sideband, leading to efficient generation of the second anti-Stokes component centered at 830 nm. As the fiber length increases (Figs. 4a, 6a), multiple sidebands generated through cascaded FWM are broadened through self- and cross-phase modulation, merging together and eventually giving rise to a broadband (supercontinuum) emission with a spectrum stretching (for L 30 cm) from 650 to 2200 nm. The spectrum of the light field in this regime is broad enough to access the Ramanactive modes of the fiber material. In the frequency domain, the Raman effect gives rise to an intense longwavelength wing of the spectrum, whose maximum shifts from approximately 1.7 to 2.1 lm as the fiber length is increased from 20 to 30 cm (Fig. 6a). Broad red-shifted
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Fig. 7. Spectral (a) and temporal (b) transformation of a laser pulse with an initial pulse width of 200 fs, the input central wavelength of 1070 nm, and an input energy of 2 nJ in the fiber with the GVD profile shown by curve 1 in Fig. 1. The fiber length increases from top to bottom, taking the values of 5, 20, 35, and 50 cm.
Fig. 6. Spectral (a) and temporal (b) transformation of a laser pulse with an initial pulse width of 2 ps, the input central wavelength of 1070 nm, and an input energy of 20 nJ in the fiber with the GVD profile shown by curve 1 in Fig. 1. The fiber length increases from top to bottom, taking the values of 15, 20, 25, and 30 cm. The inset presents a close-up of the solitonic feature shown by a circle in the panel corresponding to a fiber length of 30 cm.
bands observed in the spectra presented in Fig. 6a correspond to well-localized femtosecond solitonic features in the time domain (Fig. 6b). Anomalous fiber dispersion translates the red shift of these spectral bands into a positive delay time. As a result, in the time domain, these solitonic features are noticeably delayed with respect to the
remaining part of the light field. Numerical analysis of pulse dynamics performed with and without the retarded part of optical nonlinearity shows that the 1.7–2.1-lm region of supercontinuum emission in Fig. 6a is fully controlled by the Raman effect and red-shifted solitons. 3.2. Femtosecond pulses We now turn to the analysis of the dynamics of femtosecond laser pulses in the region of anomalous dispersion of PCF. Fig. 7a and b illustrate the main tendencies of spectral and temporal transformations of 200-fs 1070-nm pulses in a PCF with the dispersion profile shown by curve 1 in Fig. 1. Pulse evolution in this case radically differs from the dynamics of picosecond pulses discussed in Section 3.1. For femtosecond laser pulses, the spectrum becomes asymmetric and
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quite complicated within the first few centimeters of the fiber. A distinct gap around the zero-GVD wavelength kz 970 nm is one of the prominent features observed in the spectrum of the laser field. This gap, as shown in earlier work, originates from parametric FWM processes xp1 þ xp2 ¼ xa þ xs , which are easily phase-matched when the pump-field components xp1 and xp2 lie close to the zeroGVD point, depleting the spectral range around kz . The low-frequency part of the field spectrum falls within the region of anomalous dispersion and tends to generate solitons, which are readily observed in the time-domain structure of the field in Fig. 7b. These solitons undergo a continuous red shift due to the Raman effect as the pulse propagates through the fiber, giving rise to an intense long-wavelength wing of the field spectrum. The blueshifted part of PCF output spectra is dominated by dispersive-wave (Cherenkov) emission of solitons. As a result, 2-ps pulses with the same input peak power and the same initial central wavelength generate in the case under consideration much broader and much smoother spectra at the output of a 50-cm PCF, offering a promising solution for the creation of PCF supercontinuum sources pumped by picosecond fiber lasers. 4. Conclusion We have shown that cascaded FWM processes in the region of anomalous dispersion in PCFs offer interesting possibilities for efficient SC transformation of picosecond laser pulses. By numerically solving the generalized nonlinear Schro¨dinger equation we provided a comparative analysis of spectral and temporal transformations of pico and femtosecond laser pulses in a guided mode with a generic dispersion profile typical of a PCF. Based on this analysis, we have demonstrated that, with an appropriate choice of the input laser pulse parameters, picosecond pulses can be efficiently transformed into a supercontinuum emission through a cascade of phase-matched FWM interactions. For femtosecond pulses, on the other hand, soliton frequency-shifting phenomena tend to arrest such FWM cascades. In this regime, picosecond pulses can outperform femtosecond pulses in supercontinuum generation, providing much higher efficiencies of supercontinuum generation, as well as much broader and much smoother spectra of output radiation compared to femtosecond pulses with the same peak power. Acknowledgements This study was supported in part by the Russian Foundation for Basic Research (Projects 06-02-16880 and 05-0290566-NNS) and INTAS (Projects Nos. 03-51-5037 and 03-51-5288). The research described in this publication was made possible in part by Award No. RUP2-2695 of the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF).
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