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Dispersive shock mediated resonant radiations in defocused nonlinear medium
Optics Communications 412 (2018) 226–229
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Dispersive shock mediated resonant radiations in defocused nonlinear medium Surajit Bose a,b, *, Rik Chattopadhyay c , Shyamal Kumar Bhadra c a b c
Fiber Optics and Photonics Division, CSIR-Central Glass and Ceramic Research Institute, 196 Raja S. C. Mullick Road, Kolkata-700032, India Academy of Scientific and Innovative Research (AcSIR), CGCRI, India RCAMOS, Indian Association for the Cultivation of Science, 2A & 2B Raja S. C. Mullick Road, Kolkata-700032, India
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Keywords: Dispersive shock wave Resonant radiation Soliton Defocused Kerr nonlinear medium Silver doped PCF
a b s t r a c t We report the evolution of resonant radiation (RR) in a self-defocused nonlinear medium with two zero dispersion wavelengths. RR is generated from dispersive shock wave (DSW) front when the pump pulse is in non-solitonic regime close to first zero dispersion wavelength (ZDW). DSW is responsible for pulse splitting resulting in the generation of blue solitons when leading edge of the pump pulse hits the first ZDW. DSW also generates a red shifted dispersive wave (DW) in the presence of higher order dispersion coefficients. Further, DSW through cross-phase modulation with red shifted dispersive wave (DW) excites a localized radiation. The presence of zero nonlinearity point in the system restricts red-shift of RR and enhances the red shifting of DW. It also helps in the formation of DSW at shorter distance and squeezes the solitonic region beyond second zero dispersion point. Predicted results indicate that the spectral evolution depends on the product of Kerr nonlinearity and group velocity dispersion. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Nonlinear guided media manifest an interesting source for exhibiting broadband supercontinuum light for various applications. In particular, geometry of silica based photonic crystal fiber (PCF) provides good tunability of dispersion profile with more than one zero dispersion points [1,2]. Dispersive wave (DW) forms when a soliton, in presence of higher order dispersions, radiates its energy across the zero dispersion wavelength (ZDW) [3–6]. The number of zero dispersion points and the dispersion profile of a fiber control the frequency and energy of DWs [7,8]. The phase matching between the dispersive and nonlinear phase dictates the efficiency of DW generation from solitonic pulse. A DW can also be excited efficiently at lower power if the pump wavelength is close to ZDW. The wavelength of generated DW is either red-shifted or blue-shifted based on the phase-matching condition [8]. This study attracts much attention owing to the richness of physical processes involved mainly Raman-induced DW trapping [9], soliton self-frequency shift cancellation [10], formation of negative resonant radiation [11]. Soliton trapping of DWs is an important phenomenon in the blue part of the supercontinuum spectrum [12], which involves new physics beyond elementary nonlinear processes like cross-phase modulation (XPM) [13]. Different works related to the soliton trapping
reported- e.g. in tapered PCF [14] where changing the group velocity in a non-uniform fiber leads to the same trapping mechanism as for decelerating Raman soliton in a uniform fiber [15], in soliton trapping of DWs in a PCF having two ZDW [16], and in a recent work on the effect of self-steepening coefficient on soliton trapping [17]. We have reported in a previous study that soliton pulse is not always necessary for the evolution of DWs [18]. Even a pump pulse in the normal dispersion domain and focused (positive) nonlinear medium can generate a DW [18,19]. The DW is typically emanated from dispersive shock wave (DSW) front [20,21]. Recently, nonlinear dynamics in silver nanoparticle doped PCF is investigated which acts as natural [22,23] Kerr self-defocused (negative) nonlinear medium. The silver nanoparticles doped in the PCF core provide negative nonlinearity over a certain wavelength range that may be tuned by changing the filling factor of the metal particles. DSWs are expanding part of pulse envelope that contain fast oscillations originating from the dispersive homogenization of classical shock waves [21,24]. Perturbed DSWs can emit resonant radiation (RR) at frequencies given by phase matching condition, where the velocity of shock front plays an important role [20,21]. DSW generation in defocused nonlinear medium has never been studied before. Here we present the effects of negative nonlinearity and dispersion on DSW
* Corresponding author at: Fiber Optics and Photonics Division, CSIR-Central Glass and Ceramic Research Institute, 196 Raja S. C. Mullick Road, Kolkata 700032, India.
E-mail addresses: [email protected] (S. Bose), [email protected] (S.K. Bhadra). https://doi.org/10.1016/j.optcom.2017.12.016 Received 13 September 2017; Received in revised form 5 December 2017; Accepted 7 December 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
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Optics Communications 412 (2018) 226–229
generation and its evolution along propagation length. It is shown that DSWs can be generated by pumping in anomalous dispersive (AD) regime in a defocused Kerr nonlinear medium. This in turn facilitates the generation of RR. We have chosen a typical configuration of a PCF which contains two ZDWs and one zero nonlinear wavelength (ZNW) in the wavelength region of interest. It is observed that both blue and red solitons can be excited from DSW. DSW traps one of the red-shifted radiations. We observed the evolution of RR when ZNW is absent. However, incorporation of ZNW restricts such evolution and enhances the amplitude of radiation of red DWs. 2. Numerical analysis 2.1. Waveguide description In order to quantify the amount of metal nanoparticle inclusions in silica glass, a parameter called filling factor (𝑓 ) is defined. Fig. 1(a) shows the microstructure of a heterogeneous medium comprising two phases — A (metallic inclusions) and B (host silica glass). This composite structure in Fig. 1(a) can be represented by a random unit cell embedded in the effective medium (Fig. 1(b)). In other words, the extinction of the random unit cell is same as if it was replaced with a material having the effective dielectric permittivity. It is calculated using Maxwell– Garnett theory [25]. This effective medium is considered as the core of the metal nanoparticle doped PCF. The volume ratio of the concentric spheres shown in Fig. 1(b) resembles with 𝑓 of metallic inclusions. Parameters like pitch (𝛬, distance between two adjacent air holes), diameter of the air holes, core diameter and volume fraction (𝑓 ) of metal nanoparticles are 1.7 μm, 0.663 μm, 2.5 μm and 1.3×10−2 respectively for the modeled silver nanoparticle doped solid core photonic crystal fiber (SNPCF). The geometry as well as the dispersion and nonlinearity characteristics of the SNPCF are shown in Fig. 1(c). Earlier we have demonstrated the generation of shock-front mediated bright solitons in pure silica core PCF [18] and silver nanoparticle doped single mode optical fiber [26] producing fluorescence. Here we have adopted the concept of these two fibers and designed a SNPCF which exhibits two zero dispersion points and one zero nonlinearity point. The existence of zero dispersion and nonlinear points divide the entire spectral region into the solitonic (𝛽 2 𝛾𝐾𝑒𝑟𝑟 < 0) and non-solitonic regimes (𝛽 2 𝛾𝐾𝑒𝑟𝑟 > 0) which in turn offer exciting nonlinear pulse dynamics. Dispersion and nonlinear profiles of such a SNPCF can always be tuned by changing the geometrical parameters (𝑑, 𝛬) and 𝑓 . In Fig. 1(d), we show a specific case where nonlinear profile of the fiber is tailored by changing the doping concentration of metal nanoparticles i.e. by changing 𝑓 . In general, due to doping of silver nanoparticles in silica glass main absorption band occurs around 400–450 nm, however, it is expected that there would be little increase in the background loss but it would not affect the operational wavelength range considered in the present work.
Fig. 1. (Color Online) (a) Microstructure for heterogeneous two-phase media. The shaded circles indicate the silver nano-particles. (b) Random unit cell. (c) Dispersion and nonlinear profile for the proposed SNPCF. The solitonic (𝛽2 𝛾𝐾𝑒𝑟𝑟 < 0) and nonsolitonic (𝛽2 𝛾𝐾𝑒𝑟𝑟 > 0) regimes are indicated by 1–4. Inset shows the schematic crosssection of the fiber. (d) Variation of Kerr nonlinearity for different volume fraction of metal nanoparticles. Inset shows the fundamental mode field distribution at 1060 nm (operating wavelength).
nonlinear parameter for the undoped PCF in silica matrix. The other nonlinear coefficients considered in this study are as follows: 𝛾0𝑒𝑓 𝑓 = 𝛾1 =
𝜆0 𝐴𝑒𝑓 𝑓
, 𝛾1𝑒𝑓 𝑓 =
𝑑𝛾𝑒𝑓 𝑓 | 2𝜋Re(𝑛2 ) , | , 𝛾 = 𝑑𝜔 |𝜔0 0 𝜆0 𝐴𝑒𝑓 𝑓
(3) 3𝜒𝑒𝑓 3𝜒ℎ(3) 𝛾0 𝑓 , 𝑛2𝑒𝑓 𝑓 = and 𝑛2 = . 𝜔0 4𝜀0 𝑐𝜀𝑒𝑓 𝑓 4𝜀0 𝑐
Here 𝑛2 is the nonlinear-index coefficient of the host silica, 𝑛2𝑒𝑓 𝑓 is the nonlinear-index coefficient of the composite material. 𝜒ℎ is the third order susceptibility of host silica glass. 𝜀𝑒𝑓 𝑓 is the dielectric constant of the composite and 𝜒𝑒𝑓 𝑓 is the effective third order susceptibility which are calculated from effective medium theory and theory of composite nonlinear materials [22,23,26,27] respectively. 2.3. Phase matching equation It is already established that steep front formation acts as a seed to form RR, which is phase-matched to the shock in its moving frame at velocity 𝑉𝑠 [20]. Linear waves in such a frame have wave-number 𝐾(𝜔) = 𝑘(𝜔) − 𝜔∕𝑉𝑠 and the pump has 𝐾𝑝 (𝜔) = 𝑘(𝜔𝑝 ) − 𝜔𝑝 ∕𝑉𝑠 + 𝑘𝑛𝑙 . 𝑘𝑛𝑙 = 𝛾𝑒𝑓 𝑓 𝑃0 ∕2 is the nonlinear contribution at the pump with peak power 𝑃0 . Using Taylor series expansion of 𝐾(𝜔) around 𝜔𝑝 and taking 𝛺 = 𝜔−𝜔𝑝 , we find the phase-matching equation [20] for 𝐾(𝜔) = 𝐾𝑝 (𝜔) as ( ) ∑ 𝛽2 (𝛾0𝑒𝑓 𝑓 + 𝛾1𝑒𝑓 𝑓 𝛺)𝑃0 1 1 − 𝛺= . (2) (𝛺)𝑛 + 𝑛! 𝑉 𝑉 2 𝑔 𝑠 𝑛=2 The frequency shift 𝛺 = 𝛺RR gives the location of RR, 𝑉𝑔 is the group velocity of the pump pulse and 𝑉𝑠 is the velocity of the shock wave.
2.2. Simulation Pulse propagation in SNPCF can be described by modifying the generalized nonlinear Schrödinger equation for which the Kerr nonlinearity changes rapidly with wavelength [4,23] 𝛽𝑛 ( 𝜕 )𝑛 𝜕𝐴 −𝑖 𝑖 𝐴 = 𝑖(1 − 𝑓𝑅 )𝛾𝑒𝑓 𝑓 |𝐴(𝑧, 𝑡)|2 𝐴(𝑧, 𝑡) 𝜕𝑧 𝑛! 𝜕𝑡 𝑛=1 ∞ { } ′ | ′ |2 ′ ℎ𝑅 (𝑡 )|𝐴(𝑧, 𝑡 − 𝑡 )| 𝑑𝑡 . + 𝑖𝑓𝑅 𝛾0 + 𝛾1 (𝜔 − 𝜔0 ) 𝐴(𝑧, 𝑡) | | ∫0
2𝜋Re(𝑛2𝑒𝑓 𝑓 )
3. Results and discussion The dynamics of an optical pulse in the proposed SNPCF can be better described if we start with a simpler case where nonlinear parameter is assumed to be constant and negative. The dispersion and nonlinear profiles are shown in Fig. 2(a). A 50 fs sech-pulse with a peak power 3 kW is launched at 1060 nm, close to the first ZDW 1006 nm. The launching regime is non-solitonic (𝛽 2 𝛾𝐾𝑒𝑟𝑟 > 0) in nature and hence the, pulse spectrum initially broadens due to SPM followed by optical wave-breaking. As the ZDW (1006 nm) is close to pump it leads to an asymmetric temporal growth in the leading edge of the pulse. This forms
∞ ∑
(1)
Here, 𝐴 is the pulse envelope and 𝛽𝑛 represents the 𝑛th order dispersion coefficient. 𝛾𝑒𝑓 𝑓 is frequency dependent nonlinear parameter for SNPCF and it can be approximated as, 𝛾𝑒𝑓 𝑓 ≈ 𝛾0𝑒𝑓 𝑓 + 𝛾1𝑒𝑓 𝑓 (𝜔 − 𝜔0 ), 𝛾0 is the 227
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Fig. 3. (Color online) XFROG at (a) 50 cm (b) 70 cm (c) 100 cm and (d) 250 cm. The black dashed line indicates the first and second ZDW at 1006 nm and 1330 nm respectively. The red box indicates the trapped radiation and the black ellipse indicates radiations at same group velocity.
soliton (red) beyond second ZDW have same group delay as indicated by the black dotted ellipse, whereas the part of localized radiation around 1530 nm shown by the red box are at same group delay with the DSW (Fig. 3(b)–(c)) which is unusual. We infer that this part develops from cross-phase modulation between an intense DSW and red-shifted DW. Spectrogram at 100 cm length of PCF (Fig. 3(c)) confirms that this DSW in defocused Kerr nonlinear medium is able to trap the radiation which was not known before. Spectrogram at 250 cm (Fig. 3(d)) shows a spectral attraction between the blue soliton and red spectral components of the RR which overlap in time domain. Cross-phase modulation (XPM) between these two waves causes chirping which shifts the solitonic spectrum towards red and non-solitonic part (i.e. RR) towards blue side. This spectral attraction is also reported in [28]. Here, the entire phenomenon is reconstructed considering self-defocusing Kerr nonlinearity where the blue soliton, RR and the premature solitons are propagating with same group velocity [16]. Fascinating part of this work is that pump in non-solitonic domain can produce simultaneous red and blue shifted solitons and they possess same delay characteristics. We also observed a radiation that may be trapped by DSW in such medium. Next we extend our study to a more realistic case where the value of Kerr nonlinearity changes with wavelength and even changes its sign. We considered the doped fiber as a waveguide that has been already described in Section 2.1. In this case, the pump pulse (parameters are same as described in Fig. 2) encounters four different spectral regions as shown in Fig. 1(c). Regions 1 and 3 are solitonic (𝛽2 𝛾𝐾𝑒𝑟𝑟 < 0) and regions 2 and 4 are non-solitonic (𝛽2 𝛾𝐾𝑒𝑟𝑟 > 0) in nature. The spectral and temporal evolutions of the pulse are shown in Fig. 4(a) and (b) respectively. Here the input pulse is launched in region 2 and we observe two very close radiations around 890 nm and 940 nm. The presence of zero nonlinearity point (nonlinear dispersion) allows the generation of the shock front at a relatively smaller distance and hence radiations appear earlier. The velocity of the shock wave changes in such a manner that RR develops from the DSW at 1200 nm. The effect of nonlinear dispersion is incorporated in the general phase matching condition given by Eq. (2) and it correctly locates the position of blue DWs, RR and red DW as shown in Fig. 4(c). It is evident that the red DW changes its location slightly from 1680 nm to 1700 nm. The presence of ZNW at 1403 nm acts as a barrier and restricts the red shift of RR [23]. Since region 3 (solitonic region) is now narrowed down, the RR does not convert to a soliton. Hence most of the energy is transformed to the red DW and it becomes more intense. It is observed that in presence of
Fig. 2. (Color online) (a) Dispersion-nonlinear curve. Two black vertical lines indicate ZDWs. (b) Spectral output at 20 cm showing DSW, blue DW, weak RR. Black vertical dotted line indicates first ZDW at 1006 nm bifurcating normal dispersion (ND) and anomalous dispersion (AD) regime. (c) Temporal output showing shock waves and optical wave breaking phenomenon at 20 cm. (d) Spectral density plots. Two black vertical lines indicate two ZDWs. Soliton like radiation and localized radiation are enclosed in blue and red rectangle respectively. Red elliptical region reveals RR. (e) Temporal density plot indicating soliton like radiation by a circular region. (f) Phase matching curve locating all the radiation. Black vertical dotted line indicates the pumping wavelength.
the shock wave front observed in Fig. 2(c) indicated by the red ellipse. GVD tries to regularize the shock waves by producing fast oscillating wave-trains (DSWs) which is spontaneously developed from leading pulse edge. Spectral signature of DSW is seen at 1180 nm (Fig. 2(b)). This is accompanied by the formation of modulated nonlinear periodic wave that forms the soliton like wave at higher propagation length. This solitary wave is interpreted as shock wave mediated soliton. We try to capture the entire event through Fig. 2(b)–(e). As shown in Fig. 2(b) and (d), an asymmetric spectral broadening is developed towards the blue side due to shock wave formation. A fraction of this energy is identified as blue DW at 980 nm and rest is converted to a soliton-like radiation (around 890 nm) at higher propagation length of the fiber indicated by a blue rectangle in Fig. 2(d). Perturbed DSWs tend to emit resonant radiation (RR) shown by an ellipse in Fig. 2(d) and red shifted DW at 1680 nm in the presence of higher order dispersion. The radiations are well predicted by the graphical solution of Eq. (2) as shown in Fig. 2(f). A localized radiation is observed around 1500 nm as shown by the large red rectangle in Fig. 2(d). Fraction of this radiation is coming from the RR which gets red-shifted and forces its spectrum to cross the second ZDW during propagation. Gradually it evolves into a solitonic radiation after propagating for quite a long length of the fiber. The evolution is captured in the XFROG-spectrogram shown in Fig. 3(a)–(d). The spectrogram for 50 cm length of PCF shows that all the radiations RR, red DW, blue soliton and the localized radiation have the same group delay indicated by the red dot-dashed line in Fig. 3(a). After propagating 70 cm, it becomes clear that the RR, blue soliton and the 228
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Scheme-21(1017)/15/EMR-II. Authors are indebted to the Director, CSIR-CGCRI, Kolkata for his support and encouragement and the Director, IACS, Kolkata for his unstinted cooperation. The authors are also thankful to Govind P. Agrawal, Samudra Roy for valuable discussions and Debashri Ghosh for editing the manuscript. References [1] P.S.J. Russell, Photonic crystal fibers, Science 299 (2003) 358–362. [2] K.M. Hilligsøe, T. Andersen, H. Paulsen, C. Nielsen, K. Mølmer, S. Keiding, R. Kristiansen, K. Hansen, J. Larsen, Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths, Opt. Express 12 (2004) 1045. [3] N. Akhmediev, M. Karlsson, Cherenkov radiation emitted by solitons in optical fibers, Phys. Rev. A. 51 (1995) 2602–2607. [4] G.P. Agrawal, Nonlinear Fiber Optics, fifth ed., Academic Press, New York, 2013. [5] I. Cristiani, R. Tediosi, L. Tartara, V. Degiorgio, Dispersive wave generation by solitons in microstructured optical fibers, Opt. Express 12 (2004) 124. [6] J.M. Dudley, G. Genty, S. Coen, Supercontinuum generation in photonic crystal fiber, Rev. Modern Phys. 78 (2006) 1135–1184. [7] S. Roy, D. Ghosh, S.K. Bhadra, G.P. Agrawal, Role of dispersion profile in controlling emission of dispersive waves by solitons in supercontinuum generation, Opt. Comm. 283 (2010) 3081–3088. [8] S. Roy, S.K. Bhadra, G.P. Agrawal, Effects of higher-order dispersion on resonant dispersive waves emitted by solitons, Opt. Lett. 34 (2009) 2072–2074. [9] A.V. Gorbach, D.V. Skryabin, Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres, Nat. Photonics 1 (2007) 653– 657. [10] D.V. Skryabin, Soliton self-frequency shift cancellation in photonic crystal fibers, Science 301 (2003) 1705–1708. [11] E. Rubino, J. McLenaghan, S.C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C.E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, Negative-frequency resonant radiation, Phys. Rev. Lett. 108 (2012) 1–5. [12] D.V. Skryabin, A.V. Gorbach, Colloquium: Looking at a soliton through the prism of optical supercontinuum, Rev. Modern Phys. 82 (2010) 1287. [13] A.V. Gorbach, D.V. Skryabin, Theory of radiation trapping by the accelerating solitons in optical fibers, Phys. Rev. A 76 (2007) 1–10. [14] J.C. Travers, J.R. Taylor, Soliton trapping of dispersive waves in tapered optical fibers, Opt. Lett. 34 (2009) 115–117. [15] A.C. Judge, O. Bang, C.M. de Sterke, Theory of dispersive wave frequency shift via trapping by a soliton in an axially nonuniform optical fiber, J. Opt. Soc. Am. B. 27 (2010) 2195–2202. [16] W.B. Wang, H. Yang, P.H. Tang, F. Han, Soliton trapping of dispersive waves during supercontinuum generation in photonic crystal fiber, Acta Phys. Sin. 62 (2013) 184202. [17] H. Yang, B. Wang, N. Chen, X. Tong, S. Zhao, The impact of self-steepening effect on soliton trapping in photonic crystal fibers, Opt. Commun. 359 (2016) 20–25. [18] S. Bose, S. Roy, R. Chattopadhyay, M. Pal, S.K. Bhadra, Experimental and theoretical study of red-shifted solitonic resonant radiation in photonic crystal fibers and generation of radiation seeded Raman soliton, J. Opt. 17 (2015) 105506. [19] K.E. Webb, Y.Q. Xu, M. Erkintalo, S.G. Murdoch, Generalized dispersive wave emission in nonlinear fiber optics, Opt. Lett. 38 (2013) 151. [20] M. Conforti, S. Trillo, Dispersive wave emission from wave breaking, Opt. Lett. 38 (2013) 3815–3818. [21] M. Conforti, F. Baronio, S. Trillo, Resonant radiation shed by dispersive shock waves, Phys. Rev. A 89 (2014) 1–8. [22] S. Bose, R. Chattopadhyay, S. Roy, S.K. Bhadra, Study of nonlinear dynamics in silver- nanoparticle-doped photonic crystal fiber, J. Opt. Soc. Amer. B 33 (2016) 1014–1021. [23] S. Bose, A. Sahoo, R. Chattopadhyay, S. Roy, S.K. Bhadra, G.P. Agrawal, Implications of a zero-nonlinearity wavelength in photonic crystal fibers doped with silver nanoparticles, Phys. Rev. A 94 (2016) 043835. [24] S. Malaguti, M. Conforti, S. Trillo, Dispersive radiation induced by shock waves in passive resonators, Opt. Lett. 39 (2014) 5626–5629. [25] J.C.M. Garnett, Colours in metal glasses and in metallic films, Phil. Trans. R. Soc. A 203 (1904) 385–420. [26] A. Halder, R. Chattopadhyay, S. Majumder, S. Bysakh, M.C. Paul, S. Das, S.K. Bhadra, M. Unnikrishnan, Highly fluorescent silver nanoclusters in alumina silicate composite optical fiber, Appl. Phys. Lett. 106 (2015) 011101. [27] J.E. Sipe, R.W. Boyd, Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model, Phys. Rev. A46 (1992) 1614. [28] S. Roy, S.K. Bhadra, K. Saitoh, M. Koshiba, G.P. Agrawal, Dynamics of Raman soliton during supercontinuum generation near the zero-dispersion wavelength of optical fibers, Opt. Express 19 (2011). [29] S. Zhao, H. Yang, N. Chen, X. Fu, C. Zhao, Soliton trapping of dispersive waves in photonic crystal fiber with three zero-dispersive wavelengths, IEEE Photon. J. 7 (2015).
Fig. 4. (Color online) (a) Spectral density plots. Two black vertical lines indicate two ZDW and red vertical line indicates ZNW. Blue DWs and RR are shown in a circular region (b) Temporal density plot. (c) Phase matching curve, (d) Spectrogram at 15 cm clearly visualizing RR, blue and red DWs.
the ZNW the intensity of the radiation trapped by the DSW is diminished. It may be inferred that the ZNW affects excitation of any frequency component in its vicinity. The XFROG at 15 cm length of propagation is shown in Fig. 4(d). The main difference between Figs. 2(d) and 4(a) is the boundary of the generated spectrum. In the former case we observed that the spectrum is bound in the blue side by a soliton like radiation and on the red side by a red DW. On the other hand, in the latter case (Fig. 4(a)) we observed that the spectrum is no more bound by soliton like radiation in the blue part though the region is still solitonic. This is exactly a similar kind of situation as shown in [29] where three zero dispersion points are present in a focused Kerr medium. In the present case we have only two zero dispersion points and one zero nonlinearity point. Hence we may infer that the pulse dynamics is actually governed by the product 𝛽2 𝛾𝐾𝑒𝑟𝑟 rather than the GVD profile (𝛽2 ) or Kerr nonlinearity (𝛾𝐾𝑒𝑟𝑟 ) itself. The results also indicate that the efficiency of blue soliton is indeed dependent on the nonlinear dispersion. Since the ZNW degrades the quality of RR hence the blue soliton also disappears from the spectrum. 4. Conclusion We have studied the dispersive shock generated RRs for defocused nonlinear medium. A femtosecond pulse is launched in the non-solitonic region close to ZDW which produces a DSW and shock mediated blue soliton. The DSW under higher order dispersion gives RR and red DW. Since the negative Kerr nonlinearity produces a solitonic region beyond the second ZDW, hence the RR gradually evolves into a red shifted soliton. Next we observed the trapping of radiation by the DSW in such a defocused medium which was never observed before. On the other hand if the nonlinearity becomes dispersive in nature such evolution of RR is restricted due to narrow solitonic region beyond the second ZDW. This affects the excitation of blue soliton and hence reveals that origin of such blue soliton is strictly governed by the DSW. ZNW also weakens the trapped radiation and enhances the amplitude of red DW. The present study reveals that soliton dynamics is dictated by the product of GVD and Kerr nonlinearity term. Acknowledgments The author SB is thankful to CSIR for providing CSIR-SRF fellowship. Part of the work of RC & SKB is supported by CSIR Emeritus Scientist 229
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Dispersive shock mediated resonant radiations in defocused nonlinear medium Surajit Bose a,b, *, Rik Chattopadhyay c , Shyamal Kumar Bhadra c a b c
Fiber Optics and Photonics Division, CSIR-Central Glass and Ceramic Research Institute, 196 Raja S. C. Mullick Road, Kolkata-700032, India Academy of Scientific and Innovative Research (AcSIR), CGCRI, India RCAMOS, Indian Association for the Cultivation of Science, 2A & 2B Raja S. C. Mullick Road, Kolkata-700032, India
a r t i c l e
i n f o
Keywords: Dispersive shock wave Resonant radiation Soliton Defocused Kerr nonlinear medium Silver doped PCF
a b s t r a c t We report the evolution of resonant radiation (RR) in a self-defocused nonlinear medium with two zero dispersion wavelengths. RR is generated from dispersive shock wave (DSW) front when the pump pulse is in non-solitonic regime close to first zero dispersion wavelength (ZDW). DSW is responsible for pulse splitting resulting in the generation of blue solitons when leading edge of the pump pulse hits the first ZDW. DSW also generates a red shifted dispersive wave (DW) in the presence of higher order dispersion coefficients. Further, DSW through cross-phase modulation with red shifted dispersive wave (DW) excites a localized radiation. The presence of zero nonlinearity point in the system restricts red-shift of RR and enhances the red shifting of DW. It also helps in the formation of DSW at shorter distance and squeezes the solitonic region beyond second zero dispersion point. Predicted results indicate that the spectral evolution depends on the product of Kerr nonlinearity and group velocity dispersion. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Nonlinear guided media manifest an interesting source for exhibiting broadband supercontinuum light for various applications. In particular, geometry of silica based photonic crystal fiber (PCF) provides good tunability of dispersion profile with more than one zero dispersion points [1,2]. Dispersive wave (DW) forms when a soliton, in presence of higher order dispersions, radiates its energy across the zero dispersion wavelength (ZDW) [3–6]. The number of zero dispersion points and the dispersion profile of a fiber control the frequency and energy of DWs [7,8]. The phase matching between the dispersive and nonlinear phase dictates the efficiency of DW generation from solitonic pulse. A DW can also be excited efficiently at lower power if the pump wavelength is close to ZDW. The wavelength of generated DW is either red-shifted or blue-shifted based on the phase-matching condition [8]. This study attracts much attention owing to the richness of physical processes involved mainly Raman-induced DW trapping [9], soliton self-frequency shift cancellation [10], formation of negative resonant radiation [11]. Soliton trapping of DWs is an important phenomenon in the blue part of the supercontinuum spectrum [12], which involves new physics beyond elementary nonlinear processes like cross-phase modulation (XPM) [13]. Different works related to the soliton trapping
reported- e.g. in tapered PCF [14] where changing the group velocity in a non-uniform fiber leads to the same trapping mechanism as for decelerating Raman soliton in a uniform fiber [15], in soliton trapping of DWs in a PCF having two ZDW [16], and in a recent work on the effect of self-steepening coefficient on soliton trapping [17]. We have reported in a previous study that soliton pulse is not always necessary for the evolution of DWs [18]. Even a pump pulse in the normal dispersion domain and focused (positive) nonlinear medium can generate a DW [18,19]. The DW is typically emanated from dispersive shock wave (DSW) front [20,21]. Recently, nonlinear dynamics in silver nanoparticle doped PCF is investigated which acts as natural [22,23] Kerr self-defocused (negative) nonlinear medium. The silver nanoparticles doped in the PCF core provide negative nonlinearity over a certain wavelength range that may be tuned by changing the filling factor of the metal particles. DSWs are expanding part of pulse envelope that contain fast oscillations originating from the dispersive homogenization of classical shock waves [21,24]. Perturbed DSWs can emit resonant radiation (RR) at frequencies given by phase matching condition, where the velocity of shock front plays an important role [20,21]. DSW generation in defocused nonlinear medium has never been studied before. Here we present the effects of negative nonlinearity and dispersion on DSW
* Corresponding author at: Fiber Optics and Photonics Division, CSIR-Central Glass and Ceramic Research Institute, 196 Raja S. C. Mullick Road, Kolkata 700032, India.
E-mail addresses: [email protected] (S. Bose), [email protected] (S.K. Bhadra). https://doi.org/10.1016/j.optcom.2017.12.016 Received 13 September 2017; Received in revised form 5 December 2017; Accepted 7 December 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
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generation and its evolution along propagation length. It is shown that DSWs can be generated by pumping in anomalous dispersive (AD) regime in a defocused Kerr nonlinear medium. This in turn facilitates the generation of RR. We have chosen a typical configuration of a PCF which contains two ZDWs and one zero nonlinear wavelength (ZNW) in the wavelength region of interest. It is observed that both blue and red solitons can be excited from DSW. DSW traps one of the red-shifted radiations. We observed the evolution of RR when ZNW is absent. However, incorporation of ZNW restricts such evolution and enhances the amplitude of radiation of red DWs. 2. Numerical analysis 2.1. Waveguide description In order to quantify the amount of metal nanoparticle inclusions in silica glass, a parameter called filling factor (𝑓 ) is defined. Fig. 1(a) shows the microstructure of a heterogeneous medium comprising two phases — A (metallic inclusions) and B (host silica glass). This composite structure in Fig. 1(a) can be represented by a random unit cell embedded in the effective medium (Fig. 1(b)). In other words, the extinction of the random unit cell is same as if it was replaced with a material having the effective dielectric permittivity. It is calculated using Maxwell– Garnett theory [25]. This effective medium is considered as the core of the metal nanoparticle doped PCF. The volume ratio of the concentric spheres shown in Fig. 1(b) resembles with 𝑓 of metallic inclusions. Parameters like pitch (𝛬, distance between two adjacent air holes), diameter of the air holes, core diameter and volume fraction (𝑓 ) of metal nanoparticles are 1.7 μm, 0.663 μm, 2.5 μm and 1.3×10−2 respectively for the modeled silver nanoparticle doped solid core photonic crystal fiber (SNPCF). The geometry as well as the dispersion and nonlinearity characteristics of the SNPCF are shown in Fig. 1(c). Earlier we have demonstrated the generation of shock-front mediated bright solitons in pure silica core PCF [18] and silver nanoparticle doped single mode optical fiber [26] producing fluorescence. Here we have adopted the concept of these two fibers and designed a SNPCF which exhibits two zero dispersion points and one zero nonlinearity point. The existence of zero dispersion and nonlinear points divide the entire spectral region into the solitonic (𝛽 2 𝛾𝐾𝑒𝑟𝑟 < 0) and non-solitonic regimes (𝛽 2 𝛾𝐾𝑒𝑟𝑟 > 0) which in turn offer exciting nonlinear pulse dynamics. Dispersion and nonlinear profiles of such a SNPCF can always be tuned by changing the geometrical parameters (𝑑, 𝛬) and 𝑓 . In Fig. 1(d), we show a specific case where nonlinear profile of the fiber is tailored by changing the doping concentration of metal nanoparticles i.e. by changing 𝑓 . In general, due to doping of silver nanoparticles in silica glass main absorption band occurs around 400–450 nm, however, it is expected that there would be little increase in the background loss but it would not affect the operational wavelength range considered in the present work.
Fig. 1. (Color Online) (a) Microstructure for heterogeneous two-phase media. The shaded circles indicate the silver nano-particles. (b) Random unit cell. (c) Dispersion and nonlinear profile for the proposed SNPCF. The solitonic (𝛽2 𝛾𝐾𝑒𝑟𝑟 < 0) and nonsolitonic (𝛽2 𝛾𝐾𝑒𝑟𝑟 > 0) regimes are indicated by 1–4. Inset shows the schematic crosssection of the fiber. (d) Variation of Kerr nonlinearity for different volume fraction of metal nanoparticles. Inset shows the fundamental mode field distribution at 1060 nm (operating wavelength).
nonlinear parameter for the undoped PCF in silica matrix. The other nonlinear coefficients considered in this study are as follows: 𝛾0𝑒𝑓 𝑓 = 𝛾1 =
𝜆0 𝐴𝑒𝑓 𝑓
, 𝛾1𝑒𝑓 𝑓 =
𝑑𝛾𝑒𝑓 𝑓 | 2𝜋Re(𝑛2 ) , | , 𝛾 = 𝑑𝜔 |𝜔0 0 𝜆0 𝐴𝑒𝑓 𝑓
(3) 3𝜒𝑒𝑓 3𝜒ℎ(3) 𝛾0 𝑓 , 𝑛2𝑒𝑓 𝑓 = and 𝑛2 = . 𝜔0 4𝜀0 𝑐𝜀𝑒𝑓 𝑓 4𝜀0 𝑐
Here 𝑛2 is the nonlinear-index coefficient of the host silica, 𝑛2𝑒𝑓 𝑓 is the nonlinear-index coefficient of the composite material. 𝜒ℎ is the third order susceptibility of host silica glass. 𝜀𝑒𝑓 𝑓 is the dielectric constant of the composite and 𝜒𝑒𝑓 𝑓 is the effective third order susceptibility which are calculated from effective medium theory and theory of composite nonlinear materials [22,23,26,27] respectively. 2.3. Phase matching equation It is already established that steep front formation acts as a seed to form RR, which is phase-matched to the shock in its moving frame at velocity 𝑉𝑠 [20]. Linear waves in such a frame have wave-number 𝐾(𝜔) = 𝑘(𝜔) − 𝜔∕𝑉𝑠 and the pump has 𝐾𝑝 (𝜔) = 𝑘(𝜔𝑝 ) − 𝜔𝑝 ∕𝑉𝑠 + 𝑘𝑛𝑙 . 𝑘𝑛𝑙 = 𝛾𝑒𝑓 𝑓 𝑃0 ∕2 is the nonlinear contribution at the pump with peak power 𝑃0 . Using Taylor series expansion of 𝐾(𝜔) around 𝜔𝑝 and taking 𝛺 = 𝜔−𝜔𝑝 , we find the phase-matching equation [20] for 𝐾(𝜔) = 𝐾𝑝 (𝜔) as ( ) ∑ 𝛽2 (𝛾0𝑒𝑓 𝑓 + 𝛾1𝑒𝑓 𝑓 𝛺)𝑃0 1 1 − 𝛺= . (2) (𝛺)𝑛 + 𝑛! 𝑉 𝑉 2 𝑔 𝑠 𝑛=2 The frequency shift 𝛺 = 𝛺RR gives the location of RR, 𝑉𝑔 is the group velocity of the pump pulse and 𝑉𝑠 is the velocity of the shock wave.
2.2. Simulation Pulse propagation in SNPCF can be described by modifying the generalized nonlinear Schrödinger equation for which the Kerr nonlinearity changes rapidly with wavelength [4,23] 𝛽𝑛 ( 𝜕 )𝑛 𝜕𝐴 −𝑖 𝑖 𝐴 = 𝑖(1 − 𝑓𝑅 )𝛾𝑒𝑓 𝑓 |𝐴(𝑧, 𝑡)|2 𝐴(𝑧, 𝑡) 𝜕𝑧 𝑛! 𝜕𝑡 𝑛=1 ∞ { } ′ | ′ |2 ′ ℎ𝑅 (𝑡 )|𝐴(𝑧, 𝑡 − 𝑡 )| 𝑑𝑡 . + 𝑖𝑓𝑅 𝛾0 + 𝛾1 (𝜔 − 𝜔0 ) 𝐴(𝑧, 𝑡) | | ∫0
2𝜋Re(𝑛2𝑒𝑓 𝑓 )
3. Results and discussion The dynamics of an optical pulse in the proposed SNPCF can be better described if we start with a simpler case where nonlinear parameter is assumed to be constant and negative. The dispersion and nonlinear profiles are shown in Fig. 2(a). A 50 fs sech-pulse with a peak power 3 kW is launched at 1060 nm, close to the first ZDW 1006 nm. The launching regime is non-solitonic (𝛽 2 𝛾𝐾𝑒𝑟𝑟 > 0) in nature and hence the, pulse spectrum initially broadens due to SPM followed by optical wave-breaking. As the ZDW (1006 nm) is close to pump it leads to an asymmetric temporal growth in the leading edge of the pulse. This forms
∞ ∑
(1)
Here, 𝐴 is the pulse envelope and 𝛽𝑛 represents the 𝑛th order dispersion coefficient. 𝛾𝑒𝑓 𝑓 is frequency dependent nonlinear parameter for SNPCF and it can be approximated as, 𝛾𝑒𝑓 𝑓 ≈ 𝛾0𝑒𝑓 𝑓 + 𝛾1𝑒𝑓 𝑓 (𝜔 − 𝜔0 ), 𝛾0 is the 227
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Fig. 3. (Color online) XFROG at (a) 50 cm (b) 70 cm (c) 100 cm and (d) 250 cm. The black dashed line indicates the first and second ZDW at 1006 nm and 1330 nm respectively. The red box indicates the trapped radiation and the black ellipse indicates radiations at same group velocity.
soliton (red) beyond second ZDW have same group delay as indicated by the black dotted ellipse, whereas the part of localized radiation around 1530 nm shown by the red box are at same group delay with the DSW (Fig. 3(b)–(c)) which is unusual. We infer that this part develops from cross-phase modulation between an intense DSW and red-shifted DW. Spectrogram at 100 cm length of PCF (Fig. 3(c)) confirms that this DSW in defocused Kerr nonlinear medium is able to trap the radiation which was not known before. Spectrogram at 250 cm (Fig. 3(d)) shows a spectral attraction between the blue soliton and red spectral components of the RR which overlap in time domain. Cross-phase modulation (XPM) between these two waves causes chirping which shifts the solitonic spectrum towards red and non-solitonic part (i.e. RR) towards blue side. This spectral attraction is also reported in [28]. Here, the entire phenomenon is reconstructed considering self-defocusing Kerr nonlinearity where the blue soliton, RR and the premature solitons are propagating with same group velocity [16]. Fascinating part of this work is that pump in non-solitonic domain can produce simultaneous red and blue shifted solitons and they possess same delay characteristics. We also observed a radiation that may be trapped by DSW in such medium. Next we extend our study to a more realistic case where the value of Kerr nonlinearity changes with wavelength and even changes its sign. We considered the doped fiber as a waveguide that has been already described in Section 2.1. In this case, the pump pulse (parameters are same as described in Fig. 2) encounters four different spectral regions as shown in Fig. 1(c). Regions 1 and 3 are solitonic (𝛽2 𝛾𝐾𝑒𝑟𝑟 < 0) and regions 2 and 4 are non-solitonic (𝛽2 𝛾𝐾𝑒𝑟𝑟 > 0) in nature. The spectral and temporal evolutions of the pulse are shown in Fig. 4(a) and (b) respectively. Here the input pulse is launched in region 2 and we observe two very close radiations around 890 nm and 940 nm. The presence of zero nonlinearity point (nonlinear dispersion) allows the generation of the shock front at a relatively smaller distance and hence radiations appear earlier. The velocity of the shock wave changes in such a manner that RR develops from the DSW at 1200 nm. The effect of nonlinear dispersion is incorporated in the general phase matching condition given by Eq. (2) and it correctly locates the position of blue DWs, RR and red DW as shown in Fig. 4(c). It is evident that the red DW changes its location slightly from 1680 nm to 1700 nm. The presence of ZNW at 1403 nm acts as a barrier and restricts the red shift of RR [23]. Since region 3 (solitonic region) is now narrowed down, the RR does not convert to a soliton. Hence most of the energy is transformed to the red DW and it becomes more intense. It is observed that in presence of
Fig. 2. (Color online) (a) Dispersion-nonlinear curve. Two black vertical lines indicate ZDWs. (b) Spectral output at 20 cm showing DSW, blue DW, weak RR. Black vertical dotted line indicates first ZDW at 1006 nm bifurcating normal dispersion (ND) and anomalous dispersion (AD) regime. (c) Temporal output showing shock waves and optical wave breaking phenomenon at 20 cm. (d) Spectral density plots. Two black vertical lines indicate two ZDWs. Soliton like radiation and localized radiation are enclosed in blue and red rectangle respectively. Red elliptical region reveals RR. (e) Temporal density plot indicating soliton like radiation by a circular region. (f) Phase matching curve locating all the radiation. Black vertical dotted line indicates the pumping wavelength.
the shock wave front observed in Fig. 2(c) indicated by the red ellipse. GVD tries to regularize the shock waves by producing fast oscillating wave-trains (DSWs) which is spontaneously developed from leading pulse edge. Spectral signature of DSW is seen at 1180 nm (Fig. 2(b)). This is accompanied by the formation of modulated nonlinear periodic wave that forms the soliton like wave at higher propagation length. This solitary wave is interpreted as shock wave mediated soliton. We try to capture the entire event through Fig. 2(b)–(e). As shown in Fig. 2(b) and (d), an asymmetric spectral broadening is developed towards the blue side due to shock wave formation. A fraction of this energy is identified as blue DW at 980 nm and rest is converted to a soliton-like radiation (around 890 nm) at higher propagation length of the fiber indicated by a blue rectangle in Fig. 2(d). Perturbed DSWs tend to emit resonant radiation (RR) shown by an ellipse in Fig. 2(d) and red shifted DW at 1680 nm in the presence of higher order dispersion. The radiations are well predicted by the graphical solution of Eq. (2) as shown in Fig. 2(f). A localized radiation is observed around 1500 nm as shown by the large red rectangle in Fig. 2(d). Fraction of this radiation is coming from the RR which gets red-shifted and forces its spectrum to cross the second ZDW during propagation. Gradually it evolves into a solitonic radiation after propagating for quite a long length of the fiber. The evolution is captured in the XFROG-spectrogram shown in Fig. 3(a)–(d). The spectrogram for 50 cm length of PCF shows that all the radiations RR, red DW, blue soliton and the localized radiation have the same group delay indicated by the red dot-dashed line in Fig. 3(a). After propagating 70 cm, it becomes clear that the RR, blue soliton and the 228
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Scheme-21(1017)/15/EMR-II. Authors are indebted to the Director, CSIR-CGCRI, Kolkata for his support and encouragement and the Director, IACS, Kolkata for his unstinted cooperation. The authors are also thankful to Govind P. Agrawal, Samudra Roy for valuable discussions and Debashri Ghosh for editing the manuscript. References [1] P.S.J. Russell, Photonic crystal fibers, Science 299 (2003) 358–362. [2] K.M. Hilligsøe, T. Andersen, H. Paulsen, C. Nielsen, K. Mølmer, S. Keiding, R. Kristiansen, K. Hansen, J. Larsen, Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths, Opt. Express 12 (2004) 1045. [3] N. Akhmediev, M. Karlsson, Cherenkov radiation emitted by solitons in optical fibers, Phys. Rev. A. 51 (1995) 2602–2607. [4] G.P. Agrawal, Nonlinear Fiber Optics, fifth ed., Academic Press, New York, 2013. [5] I. Cristiani, R. Tediosi, L. Tartara, V. Degiorgio, Dispersive wave generation by solitons in microstructured optical fibers, Opt. Express 12 (2004) 124. [6] J.M. Dudley, G. Genty, S. Coen, Supercontinuum generation in photonic crystal fiber, Rev. Modern Phys. 78 (2006) 1135–1184. [7] S. Roy, D. Ghosh, S.K. Bhadra, G.P. Agrawal, Role of dispersion profile in controlling emission of dispersive waves by solitons in supercontinuum generation, Opt. Comm. 283 (2010) 3081–3088. [8] S. Roy, S.K. Bhadra, G.P. Agrawal, Effects of higher-order dispersion on resonant dispersive waves emitted by solitons, Opt. Lett. 34 (2009) 2072–2074. [9] A.V. Gorbach, D.V. Skryabin, Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres, Nat. Photonics 1 (2007) 653– 657. [10] D.V. Skryabin, Soliton self-frequency shift cancellation in photonic crystal fibers, Science 301 (2003) 1705–1708. [11] E. Rubino, J. McLenaghan, S.C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C.E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, Negative-frequency resonant radiation, Phys. Rev. Lett. 108 (2012) 1–5. [12] D.V. Skryabin, A.V. Gorbach, Colloquium: Looking at a soliton through the prism of optical supercontinuum, Rev. Modern Phys. 82 (2010) 1287. [13] A.V. Gorbach, D.V. Skryabin, Theory of radiation trapping by the accelerating solitons in optical fibers, Phys. Rev. A 76 (2007) 1–10. [14] J.C. Travers, J.R. Taylor, Soliton trapping of dispersive waves in tapered optical fibers, Opt. Lett. 34 (2009) 115–117. [15] A.C. Judge, O. Bang, C.M. de Sterke, Theory of dispersive wave frequency shift via trapping by a soliton in an axially nonuniform optical fiber, J. Opt. Soc. Am. B. 27 (2010) 2195–2202. [16] W.B. Wang, H. Yang, P.H. Tang, F. Han, Soliton trapping of dispersive waves during supercontinuum generation in photonic crystal fiber, Acta Phys. Sin. 62 (2013) 184202. [17] H. Yang, B. Wang, N. Chen, X. Tong, S. Zhao, The impact of self-steepening effect on soliton trapping in photonic crystal fibers, Opt. Commun. 359 (2016) 20–25. [18] S. Bose, S. Roy, R. Chattopadhyay, M. Pal, S.K. Bhadra, Experimental and theoretical study of red-shifted solitonic resonant radiation in photonic crystal fibers and generation of radiation seeded Raman soliton, J. Opt. 17 (2015) 105506. [19] K.E. Webb, Y.Q. Xu, M. Erkintalo, S.G. Murdoch, Generalized dispersive wave emission in nonlinear fiber optics, Opt. Lett. 38 (2013) 151. [20] M. Conforti, S. Trillo, Dispersive wave emission from wave breaking, Opt. Lett. 38 (2013) 3815–3818. [21] M. Conforti, F. Baronio, S. Trillo, Resonant radiation shed by dispersive shock waves, Phys. Rev. A 89 (2014) 1–8. [22] S. Bose, R. Chattopadhyay, S. Roy, S.K. Bhadra, Study of nonlinear dynamics in silver- nanoparticle-doped photonic crystal fiber, J. Opt. Soc. Amer. B 33 (2016) 1014–1021. [23] S. Bose, A. Sahoo, R. Chattopadhyay, S. Roy, S.K. Bhadra, G.P. Agrawal, Implications of a zero-nonlinearity wavelength in photonic crystal fibers doped with silver nanoparticles, Phys. Rev. A 94 (2016) 043835. [24] S. Malaguti, M. Conforti, S. Trillo, Dispersive radiation induced by shock waves in passive resonators, Opt. Lett. 39 (2014) 5626–5629. [25] J.C.M. Garnett, Colours in metal glasses and in metallic films, Phil. Trans. R. Soc. A 203 (1904) 385–420. [26] A. Halder, R. Chattopadhyay, S. Majumder, S. Bysakh, M.C. Paul, S. Das, S.K. Bhadra, M. Unnikrishnan, Highly fluorescent silver nanoclusters in alumina silicate composite optical fiber, Appl. Phys. Lett. 106 (2015) 011101. [27] J.E. Sipe, R.W. Boyd, Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model, Phys. Rev. A46 (1992) 1614. [28] S. Roy, S.K. Bhadra, K. Saitoh, M. Koshiba, G.P. Agrawal, Dynamics of Raman soliton during supercontinuum generation near the zero-dispersion wavelength of optical fibers, Opt. Express 19 (2011). [29] S. Zhao, H. Yang, N. Chen, X. Fu, C. Zhao, Soliton trapping of dispersive waves in photonic crystal fiber with three zero-dispersive wavelengths, IEEE Photon. J. 7 (2015).
Fig. 4. (Color online) (a) Spectral density plots. Two black vertical lines indicate two ZDW and red vertical line indicates ZNW. Blue DWs and RR are shown in a circular region (b) Temporal density plot. (c) Phase matching curve, (d) Spectrogram at 15 cm clearly visualizing RR, blue and red DWs.
the ZNW the intensity of the radiation trapped by the DSW is diminished. It may be inferred that the ZNW affects excitation of any frequency component in its vicinity. The XFROG at 15 cm length of propagation is shown in Fig. 4(d). The main difference between Figs. 2(d) and 4(a) is the boundary of the generated spectrum. In the former case we observed that the spectrum is bound in the blue side by a soliton like radiation and on the red side by a red DW. On the other hand, in the latter case (Fig. 4(a)) we observed that the spectrum is no more bound by soliton like radiation in the blue part though the region is still solitonic. This is exactly a similar kind of situation as shown in [29] where three zero dispersion points are present in a focused Kerr medium. In the present case we have only two zero dispersion points and one zero nonlinearity point. Hence we may infer that the pulse dynamics is actually governed by the product 𝛽2 𝛾𝐾𝑒𝑟𝑟 rather than the GVD profile (𝛽2 ) or Kerr nonlinearity (𝛾𝐾𝑒𝑟𝑟 ) itself. The results also indicate that the efficiency of blue soliton is indeed dependent on the nonlinear dispersion. Since the ZNW degrades the quality of RR hence the blue soliton also disappears from the spectrum. 4. Conclusion We have studied the dispersive shock generated RRs for defocused nonlinear medium. A femtosecond pulse is launched in the non-solitonic region close to ZDW which produces a DSW and shock mediated blue soliton. The DSW under higher order dispersion gives RR and red DW. Since the negative Kerr nonlinearity produces a solitonic region beyond the second ZDW, hence the RR gradually evolves into a red shifted soliton. Next we observed the trapping of radiation by the DSW in such a defocused medium which was never observed before. On the other hand if the nonlinearity becomes dispersive in nature such evolution of RR is restricted due to narrow solitonic region beyond the second ZDW. This affects the excitation of blue soliton and hence reveals that origin of such blue soliton is strictly governed by the DSW. ZNW also weakens the trapped radiation and enhances the amplitude of red DW. The present study reveals that soliton dynamics is dictated by the product of GVD and Kerr nonlinearity term. Acknowledgments The author SB is thankful to CSIR for providing CSIR-SRF fellowship. Part of the work of RC & SKB is supported by CSIR Emeritus Scientist 229