Generation and Stability Analysis of Self Similar Pulses Through Specialty Microstructured Optical Fibers in Mid Infrared Regime
Piyali Biswas, Pratik Adhikary, Abhijit Biswas, Somnath Ghosh
aa r X i v : . [ phy s i c s . op ti c s ] J a n Generation and Stability Analysis of Self Similar Pulses Through Specialty Microstructured OpticalFibers in Mid Infrared Regime
Piyali Biswas, Pratik Adhikary, Abhijit Biswas and Somnath Ghosh ∗ Institute of Radiophysics and ElectronicsUniversity of Calcutta92, A.P.C. Road, Kolkata 700009, India (Dated: August 27, 2018)We report a numerical study on generation and stability of parabolic pulses during their propagation throughhighly nonlinear specialty optical fibers. Here, we have generated a parabolic pulse at 2.1 µ m wavelength froma Gaussian input pulse with 1.9 ps FWHM and 75 W peak power after travelling through only 20 cm length of achalcogenide glass based microstructured optical fiber (MOF). Dependence on input pulse shapes towards mostefficient conversion into self similar states is reported. The stability in terms of any deviation from dissipativeself-similar nature of such pulses has been analyzed by introducing a variable longitudinal loss profile within thespectral loss window of the MOF, and detailed pulse shapes are captured. Moreover, three different dispersionregimes of propagation have been considered to study the suitability to support most stable propagation of thepulse. I. INTRODUCTION
In mid infrared regime ranging from 2 to 12 µ m high power optical pulses have found their applications in near field mi-croscopy or spectroscopy, mid infrared fiber sources, chemical sensing, biomedical surgeries, imaging and so on [1–3]. In thisregime, chalcogenide glass based optical fibers are found to be highly efficient in generating high power ultra short optical pulsesdue to their transperancy to mid IR radiations and extraordinary linear and nonlinear properties [4–6]. Parabolic pulses (PP) area special kind of high power optical pulses that can withstand high nonlinearity of optical fiber being freed from optical wavebreaking [7] and also maintains their parabolic temporal profile throughout the propagation length with their characteristic linearchirp across the pulse width when operated in normal group velocity dispersion (GVD) regime [8]. Lately, the generation of highpower PPs in fiber amplifiers, fiber bragg gratings and passive fibers have already been demonstrated [9–12]. However, most ofthese studies deliberately ignore any detailed analysis of the self-similar states during propagation through real waveguides. Thispaper presents a numerical study of generation of a high power parabolic pulse through a chalcogenide glass based optical fiberunder various input conditions and analyzes its stability during its propagation through the fiber with high nonlinearity, tailoreddispersion and suitably customized losses.Most of the developments in the generation of parabolic pulses have been done at telecommunication range of wavelengths( ∼ µ m) and in active media such as fiber amplifiers [13, 14]. Parabolic pulses have also been generated in a millimeterlong tapered silicon photonic nanowire (Si-PhNW) at ∼ µ m wavelength which is found to be more stable than that generatedat 1.55 µ m [15]. Using a chalcogenide glass based microstructured optical fiber (MOF), PP with ∼ ∼
46 W peak power has been generated at 2.04 µ m wavelength [12]. Although these works demonstrateefficient generation of PPs with quite high power and short temporal width at mid IR, less attention has been paid to investigatethe stability of such high power pulses. As these optical pulses are extremely short (in the range of picosecond/sub-picosecond),they are highly susceptible to various fiber nonlinearities as well as dispersion behavior which lead them to break after a fewcentimeters of propagation. Fiber losses with their limited and nonuniform bandwidth are also an important design issue forconsideration, as in practice fiber deformations during its fabrication cause losses which turn out to be fatal for an optical pulse.In our work, we have generated numerically a parabolic pulse at 2.1 µ m wavelength from an input Gaussian pulse with 75 Wpeak power and 1.9 ps FWHM after its travel through a 20 cm long arsenic sulphide ( As S ) matrix based up-tapered MOF.Moreover parabolic pulses generated from different input pulse shapes other than Gaussian have been studied. Accordingly, tostudy the stability of the generated PP, a variable longitudinal loss profile along with its frequency dependence is incorporatedat 2.1 µ m wavelength and the corresponding changes in output pulse characteristics have been reported. Further propagationof the generated PP through different dispersion regimes have been investigated and compared for obtaining the most stablepropagation dynamics in such geometries. ∗ Electronic address: [email protected]
II. GENERATION OF PARABOLIC PULSEA. Numerical Modelling for PP Generation
The study of most nonlinear effects in optical fibers involves the short pulses with widths ranging from a few picoseconds(ps) to a few femtoseconds (fs). Propagation of such short pulses within the optical fiber is accompanied by dispersion andnonlinearity which influence their shapes and spectra. The pulse evolution along the tapered dispersion decreasing MOF hasbeen modeled by solving the following nonlinear Schr¨odinger equation (NLSE). Considering a slowly varying pulse envelope A ( z, T ) , NLSE for propagation of short optical pulses takes the form [16], ∂A∂z + β ∂A∂T + i β ∂ A∂T − β ∂ A∂T + α A = iγ ( ω ) | A | A, (1)where nonlinear parameter γ is defined as, γ ( ω ) = n ( ω ) ω cA e . (2) α is the loss parameter, β and β are the first and second order dispersions respectively, β is the third order dispersion (TOD), A e is the effective mode area of the fiber and n is the nonlinear coefficient of the medium. The pulse amplitude is assumed tobe normalized such that | A | represents the optical power. In an ideal loss-less optical fiber with normal GVD i.e., when valueof β is positive and a hyperbolic dispersion decreasing profile along length of the fiber, the asymptotic solution of NLSE yieldsa parabolic intensity profile. Under this condition, the propagation of optical pulses is governed by the NLSE of the form [8], i ∂A∂z − β D ( z ) ∂ A∂T − i β ∂ A∂T + γ ( z ) | A | A = 0 , (3)where D ( z ) is length dependent dispersion profile along the tapered length, β ( nd order GVD parameter) > , β is theTOD value and γ ( z ) is longitudinally varying nonlinear (NL) coefficient. By making use of the coordinate transformation, ξ = R z D ( z ′ ) dz ′ and defining a new amplitude U ( ξ, T ) = A ( ξ,T ) √ D ( ξ ) , eq. (3) transforms to, i ∂U∂ξ − β ∂ U∂T − i β D ( ξ ) ∂ U∂T + γ ( z ) | U | U = i Γ( ξ )2 U, (4)where Γ( ξ ) = − D dDdξ = − D dDdz (5)As D ( z ) is a decreasing function of z , Γ in eq.(5) is positive since D is a decreasing function with increasing z ; and hence itmimics as a gain term in eq.(3). In the chosen dispersion decreasing fiber (DDF), the varying dispersion term is equivalent tothe varying gain term of a fiber amplifier with normal GVD. Specifically, with the choice of D ( z ) = z the gain coefficientbecomes constant, i.e., Γ = Γ . The NLS equation in a fiber with normal GVD and a constant gain coefficient permits self-similarpropagation of a linearly chirped parabolic pulse as an asymptotic solution.To study the pulse propagation in nonlinear dispersive media Split-step Fourier method (SSFM) has been extensively efficient,which is much faster than any other numerical approach to achieve the same accuracy. In general, dispersion and nonlinearityact together along the length of the fiber. SSFM obtains an approximate solution by assuming that in propagating the opticalfield over a small distance h , the dispersion and nonlinear effects can be considered to act independently. More specifically,propagation from z to z + h is carried out in two steps. In the first step, nonlinearity acts alone while in the second stepdispersion acts alone. In this method eq.(4) can be written in the form [16] ∂A∂z = ( ˆ D + ˆ N ) A, (6)where ˆ D is the differential operator that accounts for the dispersion and losses within the medium and ˆ N is the nonlinear operatorthat governs the effect of fiber nonlinearities on pulse propagation. B. Pulse Evolution
We aim to generate a high power parabolic pulse in the mid infrared regime. A parabolic pulse has been efficiently gener-ated through numerical simulation at 2.1 µ m wavelength in arsenic sulphide ( As S ) based MOF geometry with a solid core,surrounded by a holey cladding consisted of 4 hexagonally arranged rings of air holes embedded in the As S matrix [2, 12]. As S possesses lowest transmission loss ( α T ∼ µ m) among chalcogenide glasses and very high nonlinearity ( n ∼ . × − m /W at 2 µ m). A meter long up-tapered MOF with suitably tailored dispersion and nonlinearity is shown infigure 1. A Gaussian pulse of peak power 75 W and initial full-width-at-half-maximum (FWHM) 1.9 ps was fed at the inputend of the fiber and after propagating only 20 cm of the fiber length, the shape of the pulse in time domain is transformed intoparabolic. The pulse evolution is shown in figure 2. The Gaussian pulse is reshaped to parabolic under the combined influence ofself phase modulation (SPM) and normal GVD. In figure 3(b), the top-hat nature of the output pulse in logarithmic scale carriesthe hallmark that the generated pulse is essentially parabolic. With the inclusion of third order dispersion (TOD), the parabolicprofile of the pulse is still maintained. The output pulse is broadened up to 4.65 ps (figure 3(a)) and a linear chirp is generatedacross the entire pulse width as shown in the inset of figure 3(a). d0 h0 d1 h1Input cross section Output cross sesction UP Tapered MOF
L = 1 m
FIG. 1: (color online) A schematic of the linearly up-tapered MOF. The length of the MOF is considered to be 1 m. At the input end d0 is theindividual air hole diameter and d1 the same at the output end, h0 and h1 are the air hole separation at the input and output end, respectively.The chosen taper ratio is 1.05FIG. 2: (color online) Pulse evolution from Gaussian input pulse (blue curve) to parabolic pulse (black curve) at only 20 cm length of the fiber.
C. Effect of Input Pulse Shapes
In this section we will discuss how various input pulse shapes affect the parabolic pulse evolution through the MOF. In order toinvestigate this we employ different pulse shapes at the input, such as a hyperbolic secant pulse with A (0 , T ) ∝ sech ( T /T ) , atriangular pulse with A (0 , T ) ∝ tripuls ( T /T ) and a supergaussian pulse with A (0 , T ) ∝ exp( − ( T /T ) m ) . The input pulsesare unchirped with the same peak power and energy as shown in figure 4(a) and the medium of propagation is assumed to belossless. First, we consider a secant hyperbolic pulse with 159 pJ of energy and 1.685 ps FWHM which after propagating through15 cm of the fiber length is converted to a parabolic intensity profile having a temporal FWHM of 3.48 ps. The output spectral −2 0 200.51 T (ps) N o r m a li ze d P o w e r −2 0 2−4−2024 T (ps) C h i r p ( T H z ) −2 0 210 −3 −2 −1 T (ps) N o r m a li ze d P o w e r −2 0 2−4−2024 T (ps) C h i r p ( T H z ) (a) (b) FIG. 3: (color online) (a) Time domain plot of the input Gaussian (black) and the output parabolic pulse (red). The linear chirp across theparabolic pulse width is shown in the inset, (b) the logarithmic plot of the input (black) and the output (red) pulses with linear chirp profileshown in the inset. TABLE I: Comparison of output pulses generated from various input pulse shapes.
Input Input Input Optimum fiber Fiber length Output Output Energy Spectralpulse energy FWHM length for PP before wave FWHM energy conversion broadeningshapes (pJ) (ps) generation (cm) breaking (cm) (ps) (pJ) efficiency (%) (nm)Gaussian 159 1.90 15 25 4.05 159 100 122Hyperbolic 159 1.52 15 22 3.48 159 100 125secantTriangular 159 1.70 17 19 3.72 159 100 110Supergaussian 159 2.21 8 13 2.38 159 100 120 broadening is estimated to 125 nm. Relative to the PP evolved from a Gaussian input, corresponding PP for the counterpartsecant pulse in both temporal and spectral domain is less broadened.With a triangular pulse of energy 159 pJ and FWHM of 1.70 ps, a nearly parabolic pulse is generated after 17 cm propagationthrough the fiber. Spectral output is broadened by 110 nm.Further, a supergaussian pulse of the same energy and FWHM of 2.21 ps is fed at the input end of the fiber. After propagatingonly 8 cm, a very weak parabolic intensity profile with FWHM 2.38 ps and a nearly linear chirp across is observed. Furtherpropagation of the pulse results in a triangular shaped temporal profile. The output pulse shapes obtained at optimum length ofthe fiber are depicted in figure 4(b). A comparative study for the characteristic parameters of the output pulses for different inputpulse shapes are presented in Table I. −2 −1 0 1 20204060
T (ps) P ea k P o w e r ( W ) −3 −2 −1 0 1 2 300.51 T (ps) N o r m a li ze d P o w e r (a) (b) FIG. 4: (color online) Plot of (a) four different input pulse shapes - Gaussian (dashed black), hyperbolic secant (solid black), triangular (red)and supergaussian (blue), all having the same energy 159 pJ; and (b) output pulses obtained from various input pulses.
III. STABILITY OF SIMILARITON PROPAGATIONA. Loss Window
We start our study with the generation of parabolic pulses in a lossless, suitably dispersion and nonlinearity tailored fiber.Here we examine the stability of the generated PP under the influence of a lossy medium.For this purpose we considered the specific material loss window for the As S chalcogenide MOF around 2.1 µ m wave-lengths, following [17], as shown in figure 5(a). The material loss of As S glass is around 0.3 dB/m [17] and the confinementloss of the MOF is taken as 1.0 dB/m [12]. So a total loss of 1.30 dB/m has been considered as the mean value and certainamount of deliberate fluctuations is introduced. Additionally, our investigation include two different mean values of the over allloss exceeding the previous value. λ ( µ m) L o ss ( d B / m ) L (m) L o ss ( d B / m ) (b)(a) FIG. 5: (a) Loss window for As S matrix based chalcogenide MOF and (b) loss variation of the fiber along the fiber length. B. Loss Fluctuations
In order to check the stability of the output pulse spectrum, we introduce a variable loss along the fiber length with certainamount of randomness as shown in figure 5(b), corresponding to three different loss values. To address the tolerance issue ofthe state-of-the-art fabrication process in terms of loss variation along the fiber length, loss fluctuations as high as 10% and 20%around the mean values have been considered. Figure 6(a) illustrates the spectral power reduction due to 10% fluctuations ofvarious loss values as compared to the lossless spectrum. Accordingly, spectral power change due to 20% fluctuations of lossvalues are shown in figure 6(b). Exact quantifications of the spectral modifications in terms of 3dB bandwidth are shown inTable II.
TABLE II: Comparison of PP with variable loss effect.
Loss Loss 3dB Maximum Output(dB/m) fluctuation Bandwidth spectral energy (pJ)(%) change (%) rippling (dB)0 0.0 0.0 2.5 1591.30 10 5.5 2.9 12720 6.9 3.01.80 10 8.3 3.4 11620 9.8 3.52.30 10 8.9 3.5 10520 9.6 3.6 λ ( µ m) S p ec t r a l po w e r ( d B ) λ ( µ m) S p ec t r a l po w e r ( d B ) (a) (b) FIG. 6: (color online) Various output spectra obtained at different loss values with (a) 10% and (b) 20% fluctuations of the longitudinal lossprofiles respectively.
C. Different Dispersion Regimes
For the stability analysis of the generated parabolic pulse, we use a two stage propagation of the pulse. Once a parabolic pulseis generated, it has been shown in various experiments that it retains its shape throughout the propagation length and followsself-similar propagation. To investigate the self similar propagation of the PP through a passive medium, we consider MOFswith three different configurations in which the parabolic pulse will be generated in first few centimeters of the fiber length.Hence the PP will be propagating through rest of the fiber length engineered with three distinct dipersion profiles, respectively.Chosen relevant fiber geometries are shown in figure 7.
L (m) D ( p s / k m . n m ) L (m) D ( p s / k m . n m ) L (m) D ( p s / k m . n m ) (a)(b)(c) FIG. 7: Dispersion profiles of the (a) up-taper MOF structure, (b) up-down taper MOF and (c) up-no taper MOF.
Firstly, we consider a fully up tapered MOF in which, up to 20 cm from the input end of the MOF, evolution of the parabolicpulse from a Gaussian seed pulse has been observed. Through rest of the fiber length, the self similar characteristic of thegenerated PP has been studied. Before explaining the obtained results from this up-tapered MOF, we will reconsider this pulseevolution process in other two cases. The second kind of fiber geometry under consideration is an up-down tapered fiber. Here,the first 20 cm of the total fiber is a dispersion decreasing MOF to genearte the PP efficiently. Then this PP has been fed to adown tapered MOF of same material as the up tapered fiber with increasing dispersion profile along the fiber length. Finally, theparabolic pulse generation and propagation have been studied in an up-straight MOF, where the PP is evolved through 20 cmfiber length and its propagation is made through a untapered MOF with a constant dispersion profile. Results of these chosenthree different configurations are shown in figure 8. −4 −2 0 2 4 600.51
T (sec) N o r m a li ze d P o w e r configuration (a)configuration (b)configuration (c) −2 0 2−505 T (ps) C h i r p ( T H z ) λ ( µ m) S p ec t r a l P o w e r ( d B ) configuration (a)configuration (b)configuration (c) (a) (b) FIG. 8: (color online) (a) Plot of temporal profiles of the output pulses obtained from three different MOF configurations at the end of 40 cmlength and (b) spectral variation of the corresponding output pulses.
The output pulses and spectra after propagation through 40 cm fiber length respectively, look almost identical irrespective oftheir fiber geometries. Notably the striking feature is that the pulses are no longer parabolic in shape as we could expect fromthe self similar propagation characteristics of parabolic pulses. Rather, we have obtained a parabolic pulse unlike a similariton.The input pulse has evolved to a parabolic shape at 20 cm of the fiber length which is just a transient state of the input pulseevolution in the passive media. On further propagation it essentially transformed into a nearly trapezoidal shape with linearchirp across most of the pulse. As our chosen MOF is a highly nonlinear fiber (HNLF), the large γ / β ratio makes SPM todominate over dispersion. The combined effect of SPM and GVD has resulted in broadening of the pulse but failed to maintainits shape. If we examine the chirp variation of the parabolic pulse in figure 3, it could be seen that its linear nature extendsalmost over the entire pulse width with steepened transitions at the leading and trailing edges. However in all three chosen fiberconfigurations (as shown in figure 7) the chirp evolution of the propagating pulse carries an interesting signature of flippingthe both edges in temporal domain. In addition as anticipated, the parabolic profile is gradually transformed into a nearlytrapezoidal profile with increasing propagation length. The nonmonotonic nature of the chirp is somewhat responsible for there-reshaping of the propagating pulse [18]. The output spectra carry signature of spectral broadening up to 140 nm and a nearlyflat top (fluctuations falls within 3 dB). The unwanted side side-lobes appears as a result of interference between newly generatedfrequency components. IV. CONCLUSION
In conclusion, numerically generated parabolic pulses from various input optical pulse shapes such as Gaussian, hyperbolicsecant, triangular and super-gaussian keeping initial energy constant has been demonstrated. A in-depth qualitative and quanti-tative analysis of these PPs has established that the PP obtained by reshaping of the input Gaussian pulse has turned out to bethe most efficient irrespective of the choice of chalcogenide glass based MOF designs. Moreover, the PP generation in differentlength dependent dispersion regimes such as up-taper, up-down taper and up-no taper geometries has been studied. From adirect comparison of PPs obtained from different structures, its evident that PPs look almost similar in every aspect for differentcases. Hence, from practical point of view we may propose the up-tapered MOF geometry as preferable fiber structure forgenerating PP owing to its fabrication friendly geometry. In addition, the stability of the PP has been investigated by introducinglongitudinally variable and customized fluctuating loss profiles within the specific loss window of the chosen MOFs. From thepulse dynamics through these dissipative structures, though no significantly adverse effect particularly on the shape of the outputspectra was observed, however a reduction in the spectral power along with lesser 3dB bandwidth has been noticed. Moreover,from the propagation characteristics through the dispersion tailored MOFs, it has been settled that the generated parabolic shapeis a transient state which merely capable of retaining its shape unless an optimized fiber design/scheme is proposed for self-consistent solution. Our findings would be of key interest for design and fabrication of self-consistent and stable PP sources formid infrared spectroscopy, fiber-based biomedical surgeries, chemical sensing etc.
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