Self-phase modulation dominated supercontinuum generation employing cosh-Gaussian pulses in photonic crystal fibers

Self-phase modulation dominatedsupercontinuum generationemploying cosh-Gaussian pulses inphotonic crystal fibers

Nitu BorgohainSwapan Konar

Self-phase modulation dominatedsupercontinuum generation employing

cosh-Gaussian pulses in photonic crystal fibers

Nitu Borgohain* and Swapan KonarBirla Institute of Technology, Department of Physics, Mesra, Ranchi 835215, India

Abstract. This paper presents an investigation of broadband supercontinuum (SC) generation inphotonic crystal fibers using cosh-Gaussian optical pulses. It is found that the SC spectra of thesecosh-Gaussian pulses are composed of several internal oscillations. The number of the oscilla-tions increases with an increase in the value of the cosh parameter Ω0. The internal structure ofthe SC spectra shows an asymmetric behavior, possessing fewer oscillations as we move fromthe lower to higher wavelength region. Our results indicate that the SC generation dynamics isdominated by self-phase modulation. © 2015 Society of Photo-Optical Instrumentation Engineers(SPIE) [DOI: 10.1117/1.JNP.9.093098]

Keywords: cosh-Gaussian pulse; self-phase modulation; photonic crystal fiber; supercontinuumgeneration.

Paper 14103 received Sep. 18, 2014; accepted for publication Dec. 9, 2014; published onlineJan. 13, 2015.

1 Introduction

Supercontinuum (SC) generation is characterized by the dramatic spectral broadening of an opti-cal field which occurs when an intense narrow-band light pulse propagates through nonlinearmedia.1–5 The spectral broadening is contributed by a host of nonlinear optical processes such asself-phase modulation (SPM), cross-phase modulation, modulation instability, soliton fission,Raman scattering, dispersive wave generation, four wave mixing, and self-steepening.1,3,5

These nonlinear phenomena in the spectral broadening are highly dependent on the dispersionof the medium and a cleaver dispersion design can significantly reduce power requirements3 andimprove the quality of the generated SC.6,7 The first SC generation was experimentally dem-onstrated by Alfano and Shaprio in 1970 using borosilicate glass pumped by a picosecondspulse.8 Although the SC generation has been experimentally achieved in different nonlinearmedia, photonic crystal fibers (PCFs) have emerged as the most popular nonlinear media forSC generation due to the feasibility of dispersion and nonlinearity engineering.1,5 The discoveryof highly nonlinear soft glass PCFs has further enhanced their popularity as a nonlinear mediumfor successful SC generation.9 The SC spectra in a highly nonlinear PCF is usually generated bypumping femtosecond or picosecond pulses close to the zero dispersion point in the anomalousdispersion regime of the fiber. However, the SC can also be generated in the normal dispersionregime where the spectral broadening is mainly dominated through SPM, Raman scattering, andfour wave mixing.10 Although the majority of available literature on the SC generation in PCFsinvolve only a single zero dispersion wavelength,11 a considerable effort has been made to studySC generation in PCFs with two zero dispersion wavelengths12–14 yielding enhanced bandwidthwith improved flatness. Multiwavelength pumping has also been successfully attempted toachieve enhanced bandwidth.15 Nonsilica fibers, characterized by large optical nonlinearity,have also been exploited for SC generation.9,16 Solid core photonic band gap fibers17 and liquidfilled PCFs18 have also been employed to investigate SC generation.

Most of the experimental and numerical investigations on SC generation, as elucidatedabove, have been performed using “Gaussian” and “sech” pulses. Most optical sources emit

*Address all correspondence to: Nitu Borgohain, E-mail: [email protected]

0091-3286/2015/$25.00 © 2015 SPIE

Journal of Nanophotonics 093098-1 Vol. 9, 2015

light pulses whose temporal profiles are very close to Gaussian or “sech” and the qualitativefeatures of SPM induced spectral broadening of Gaussian and “sech” pulses are identical.Although the pulse shape does not change on propagation, the SPM induced spectral broadeningdepends both on the shape as well as on the chirp of the input pulse. A couple of years ago, Konarand Jana19 predicted that sinh-Gaussian pulses could lead to, under certain conditions, moreefficient SPM-induced spectral broadening. Hence, it would be worth examining the SC gen-eration phenomenon using optical pulses whose temporal shape is different from “sech” orGaussian. Therefore, in this paper, we have studied SC generation employing cosh-Gaussianpulses (not considering the issue of the generation of these pulses) in a soft lead silicate(SF57) PCF.

2 PCF Properties

Although our primary objective is to study the SC generation in a lead silicate (SF57) PCF, wefirst prefer to design the fiber, to study its optical properties which are essential for SC gener-ation, and at the end, to study SC generation in this fiber using cosh-Gaussian pulses. Leadsilicate glass PCFs offer many advantages because of their large Kerr nonlinearity and goodthermal and crystallization stability. These advantages have motivated various researchers tomodify the structure of the PCF to achieve higher nonlinearity16 and exploit this nonlinearityto generate a broad SC. Therefore, in the present paper, we first design an SF57 PCF with a smallgroup velocity dispersion and large optical nonlinearity, and thereafter, study the SC generationin this fiber using cosh-Gaussian pulses. The cross section of the fiber has been depicted inFig. 1(a). It has 11 rings of air holes arranged in a hexagonal lattice formation. The hole diameteris d and the hole pitch is Λ. One missing air hole in the fiber acts as the center of the fiber whosebackbone is made of SF57 glass. The effective index of the fundamental optical mode can becalculated using neff ¼ ðλ∕2πÞβ, where β is the propagation constant. Chromatic dispersion ofthe fiber is composed of two factors, the first one is the waveguide dispersion DwðλÞ ¼−ðλ∕cÞðd2neff∕dλ2Þ, while the second contribution comes from the material and is known asthe material dispersion DmðλÞ ¼ −ðλ∕cÞðd2nSF57∕dλ2Þ, where c is the velocity of light in vac-uum and nSF57 is the refractive index of the material.20,21 The refractive index is given by thewell-known Sellmeier equation, n2SF57 ¼ 1þP

3j¼1½Bjλ

2∕ðλ2 − CjÞ�, where B1 ¼ 1.81651371,B2 ¼ 0.37375749, B3 ¼ 1.07186278, C1 ¼ 0.0143704198, C2 ¼ 0.0592801172 and C3 ¼121.419942. The contribution of material dispersion to the total dispersion of the periodic struc-ture can be evaluated using this expression for nSF57. Total chromatic dispersion DðλÞ ¼DwðλÞ þDmðλÞ. There are a number of methods to model the propagation dynamics as wellas the dispersion profiles of the PCFs such as full vector BPM modeling,22 full vector FEMmodeling,23,24 etc. In the present investigation, we have used a software based on the finite differ-ence time domain method that is commercially available in the market.25 For our present work,we have chosen fiber parameters as follows: d ¼ 0.4Λ and Λ ¼ 1.4 μm. The optical mode fieldof the fiber has been depicted in Fig. 1(b), which signifies that the optical field is completelylocalized in the central core of the fiber. The total chromatic dispersion of the fiber has beendisplayed in Fig. 2(a). In order to have an idea about the contribution of material dispersion, wehave demonstrated its variation with the wavelength with the dash-dot line in Fig. 2(a). The zerototal dispersion point is located at 1.65 μm. At other wavelengths, the total dispersion profile is

Fig. 1 Schematic of the designed photonic crystal fiber: (a) cross section of the fiber and (b) modefield of the fiber. d ¼ 0.4Λ, Λ ¼ 1.4 μm.

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virtually flat. The value of the anomalous dispersion at 1.55 μm is about 20 ps∕nm∕km which isquite low, therefore we choose the pump wavelength at this value. The values of higher-orderdispersion terms are evaluated by taking the higher-order differential of the propagation constantat the pump wavelength, which arises as a result of Taylor expansion of the frequency dependentpropagation constant β. Typical values of different higher-order betas are β2 ¼ −2.50×10−3 ps2 m−1, β3¼2.07×10−5 ps3m−1, β4¼−1.02×10−7 ps4m−1, β5 ¼ 9.47 × 10−11 ps5 m−1,β6 ¼ −1.35 × 10−12 ps6 m−1, β7 ¼ 3.13 × 10−14 ps7 m−1, β8 ¼ −4.23 × 10−16 ps8 m−1, β9 ¼1.20 × 10−17 ps9 m−1, and β10 ¼ −1.40 × 10−18 ps10 m−1.

The optical nonlinearity γ has been evaluated using the relation γ ¼ ð2πn2∕λAeffÞ, where n2is the nonlinear index of refraction of the material and Aeff is the effective mode area of the fiber.Usually, Aeff changes with the wavelength, hence γ is a function of frequency. Therefore, itwould be more appropriate to incorporate the frequency dependence of γ in the modelingfor more accurate prediction. However, since modeling with a constant γ does produce resultswith a reasonable accuracy, we use a constant γ without much loss of accuracy. The variation ofoptical nonlinearity with wavelength has been demonstrated in Fig. 2(b). The value of nonli-nearity gradually decreases with the increase in λ. This behavior is quite usual since with theincrease in λ, the spot size of the optical field increases and consequently the value of nonli-nearity decreases. The typical value of the effective nonlinearity has been found to be∼415 W−1 km−1 at the operating wavelength.

3 Numerical Model for SC Generation

The usual approach in the study of SC generation in fibers involves in the examination of tem-poral and spectral dynamics of the high power short pulse that is propagating through the fiber.The pulse propagation through the fiber is modeled by the generalized nonlinear Schrödingerequation5

∂∂z

Aðz; TÞ ¼ −αðωÞ2

Aðz; TÞ þXn≥2

βninþ1

n!∂n

∂Tn Aðz; TÞ

þ iγ

�1þ i

ω0

∂∂T

�Aðz; TÞ

Z∞

−∞

RðT 0ÞjAðz; T − T 0Þj2dT 0; (1)

where Aðz; TÞ is the envelope of the electric field of the pulse in a frame that is moving at thegroup velocity of the pulse along the z-direction. βn ¼ ðdnβÞ∕dωn is the usual dispersion coef-ficient at the central frequency ω0. RðTÞ is the nonlinear response function which includes boththe instantaneous electronic response and the contribution of the delayed Raman response. It canbe written as, RðTÞ ¼ ð1 − fRÞδðTÞ þ fRhRðTÞ, fR ¼ 0.1 for lead silicate PCF and hRðTÞ iscalculated using hRðTÞ ¼ ½ðτ21 þ τ22Þ∕τ1τ22�e½− ðT∕τ2Þ� sinðT∕τ1Þ, with τ1 ¼ 5.5 fs andτ2 ¼ 32 fs. The nonlinear Schrödinger equation is solved by using the “split-step Fouriermethod,” the details of which can be found elsewhere.4,5

Fig. 2 (a) Group velocity dispersion of the fiber and (b) variation of fiber nonlinearity with wave-length. Air hole diameter d ¼ 0.4Λ and hole pitch Λ ¼ 1.4 μm.

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4 Linear Properties of cosh-Gaussian Pulses and Nonlinear FrequencyShift

Before proceeding to study the SC generation, it would be worth presenting a brief discussion onthe temporal and spectral properties of Gaussian and cosh-Gaussian pulses. A slowly varyingenvelope Aðz; TÞ of the propagating cosh-Gaussian pulse can be expressed as Aðz; TÞ ¼A0 expf−½ð1þ iCÞT2�∕2T2

0g × coshðΩTÞ, where A0 is a constant that determines the powerof the pulse, C is the chirp, and Ω is a parameter known as the cosh factor. The abovepulse form is rewritten as Aðz; τÞ ¼ A0 expf−½ð1þ iCÞ∕2�τ2g coshðΩ0τÞ, where τ ¼ T∕T0,Ω0 ¼ ΩT0 and T0 is associated with the pulse duration. Ω0 ¼ 0 corresponds to theGaussian pulses. The value of Ω0 determines the deviation from the Gaussian pulse. The tem-poral and spectral properties of Gaussian and cosh-Gaussian pulses for different values of Ω0

have been demonstrated in Fig. 3. It is worth mentioning that the cosh-Gaussian pulse possess adip at the center if the value of Ω0 is greater than one. The Fourier transform of these pulses fordifferent values of Ω0 are displayed in the same Fig. 3(b). Since the SPM plays a very importantrole in SC generation, before proceeding further, we investigate the nonlinear frequency shiftcaused by SPM. In order to study the phase shift separately, we neglect the dispersion, fiber loss,and Raman response terms in Eq. (1) and rewrite the equation as

i∂A∂z

þ γjAj2A ¼ 0: (2)

The solution to the above equation reads as Uðz; τÞ ¼ Uð0; τÞ exp½iϕnlðz; τÞ�, where ϕnlðz; τÞ ¼jUð0; τÞj2δ is the nonlinear phase shift due to the intensity dependent change in the nonlinearrefractive index characterized by the Kerr nonlinearity, δ ¼ γP0z and for convenience, we havetaken A ¼ ffiffiffiffiffiffi

P0

pU such that P0 determines the power of the pulse and U determines its form. The

spectral broadening is the consequence of the time dependence of ϕnl and the frequency chirp-ing. Thus, the instantaneous nonlinear frequency shift caused by SPM is given by

δωðτÞ ¼ −∂ϕnl

∂τ¼ δ

∂∂τ

ðjUð0; τÞj2Þ: (3)

The temporal variation of the frequency chirp due to the SPM has been demonstrated inFig. 4. The corresponding variation for a Gaussian pulse is also depicted in the same figure

Fig. 3 Temporal and spectral profiles of cosh-Gaussian pulses at different cosh factor Ω0 withidentical peak power. Note that Ω0 ¼ 0 corresponds to Gaussian pulse. (a) Temporal and(b) Fourier transformed spectra.

Fig. 4 Variation of nonlinear frequency shift δω of Gaussian and cosh-Gaussian pulses.

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for comparison. In a Gaussian pulse, the leading edge undergoes a red shift, whereas the trailingedge undergoes a blue shift. It should be noted that the behavior is qualitatively different for acosh-Gaussian pulse. Both the leading and trailing edges of the pulse undergo red and blue shiftssimultaneously. More specifically, part of the leading edge undergoes a red shift while theremaining portion of it undergoes a blue shift. Similarly, part of the trailing edge undergoesa blue shift, whereas the remaining portion suffers a red shift.

5 Supercontinuum Generation

In this section we investigate SC generation in the fiber that was designed in the previous section.For SC generation, we use cosh-Gaussian pulses with 50 fs FWHM pulse duration and 200 Wpeak power. The wavelengths of these pump pulses are 1.55 μm and the maximum length of thefiber is 10 cm. Fiber parameters such as γ, D, and βn

0s have been mentioned in the previoussection. We ignore fiber loss without any hesitation since the fiber length is very small, and willnot significantly affect SC generation dynamics. We have numerically simulated SC generationfor different lengths of the fiber for better understanding and clarity. We first set Ω0 ¼ 0 (i.e.,Gaussian pulse) and study the SC dynamics. The spectral dynamics of the Gaussian pulses at theend of different fiber lengths has been demonstrated in Fig. 5(a). It is amply clear that a largewide band spectrum has been generated at the end of the fiber. The structure of the spectrumsuggests that the onset of spectral broadening starts with modulation instability followed byother nonlinear processes such as Raman scattering, SPM, and four wave mixing. The resultantbroadening is due to the interplay between these nonlinear and dispersive processes. In order tostudy the SC generation phenomena with cosh-Gaussian input pulses, we have carried out sim-ulations using these pulses with a peak power of 200 W. Figure 5(b) shows the simulation resultat the end of a 10 cm fiber for a different cosh parameter Ω0. The top panel is for Gaussian pulses(Ω0 ¼ 0), whose SC generation dynamics were discussed earlier; this has been introduced for thesake of comparison. As the value of Ω0 increases, the generation of dispersive waves, self-steep-ening, and Raman scattering results in a disordered oscillatory SC generation and the SC spec-trum becomes asymmetric. The phenomenon of the SC generation with cosh-Gaussian pulsessuddenly changes with a further increase in the value of Ω0. For example, the SC spectrum ofthese pulses at Ω0 ¼ 2.0 consists of periodic oscillations throughout the length of the spectra.These oscillations are only produced in the SPM dominating regime.26,27 For higher values ofΩ0, the behavior of the SC spectra is similar, except that the number of oscillations in the spectraincreases with an increase in Ω0. In addition, the oscillation frequency is higher in the highfrequency side and lower in the low frequency side of the spectra. It has also been noticedthat the internal peaks of the SC spectrum have shown an asymmetric behavior, possessing

Fig. 5 (a) SC spectra for Gaussian pulses at different fiber lengths, (b) SC spectrum of cosh-Gaussian pulses at the end of 10 cm fiber for different cosh factor Ω0, and (c) SC spectra atthe end of the fiber with different lengths for Ω0 ¼ 2.5. Peak power is 200 W.

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a gradual increase in width from the lower to higher wavelength region. This asymmetric behav-ior is caused by the self-steepening effect associated with the nonlinearity term in the nonlinearSchrödinger equation. Overall, from the internal structure of the generated SC, we can safelyconclude that the SC generation process is dominated by SPM if cosh-Gaussian pulses are used.The effect of SPM is more pronounced with the increase in the value of Ω0. In order to examinethe development at different lengths of the fiber in Fig. 5(c), we have demonstrated the SC gen-eration phenomenon for different fiber lengths using cosh-Gaussian pulses with Ω0 ¼ 2.5. Fromthe figure, it is evident that the SC dynamics is dominated by SPM.

6 Conclusion

In the present paper, we have designed a lead silicate PCF that promises to yield a significantlylarge optical nonlinearity. This fiber has been used to simulate SC generation dynamics employ-ing cosh-Gaussian pulses. We found that, for high power Gaussian optical pulses, the SC dynam-ics is initially dominated by modulation instability. Later on, Raman scattering, SPM, and four-wave mixing played a significant role in the evolution of the SC spectra. The SC generation forcosh-Gaussian pulses is different from that of Gaussian pulses. We have found that the SC spec-tra for cosh-Gaussian input pulses are composed of several internal oscillations. These oscilla-tions increase with the increase in the value of the cosh parameter Ω0. The SC spectra confirmthat the SC dynamics is dominated by SPM.

Acknowledgments

We thank referee for their valuable suggestions. This work is supported by the Department ofScience and Technology (DST), New Delhi, through a R&D Grant SR/S2/LOP-17/2010 and byUniversity Grants Commission, Bahadur Shah Zafar Marg, New Delhi, through Major Project F.No. 41-909/2012 (SR). Authors thank DST and UGC for financial support.

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Nitu Borgohain received his MSc degree in physics from Indian School of Mines, Dhanbad,India, in 2012. He is currently working toward his PhD degree in department of physics, BirlaInstitute of Technology, Mesra, Ranchi, India. His research interests include photonic crystalfiber, nonlinear optics, and optoelectronics.

Swapan Konar received his MPhil and PhD degrees, respectively, in 1987 and 1990. He is aprofessor in the Department of Physics, Birla Institute of Technology, Mesra, Ranchi, India. Hisarea of interest is in the field of photonics and nonlinear optics. He is a senior associate ofInternational Center for Theoretical Physics (ICTP), Italy (2007 to 2014). He has published111 research papers and coauthored a book “Introduction to Non-Kerr Law OpticalSolitons,” Taylor and Francis, New York, USA (2007).

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