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Self-phase modulation dominated supercontinuum generation employing cosh-Gaussian pulses in photonic crystal fibers Nitu Borgohain Swapan Konar

Self-phase modulation dominated supercontinuum generation employing cosh-Gaussian pulses in photonic crystal fibers Nitu Borgohain* and Swapan Konar Birla Institute of Technology, Department of Physics, Mesra, Ranchi 835215, India

Abstract. This paper presents an investigation of broadband supercontinuum (SC) generation in photonic crystal fibers using cosh-Gaussian optical pulses. It is found that the SC spectra of these cosh-Gaussian pulses are composed of several internal oscillations. The number of the oscillations increases with an increase in the value of the cosh parameter Ω0. The internal structure of the SC spectra shows an asymmetric behavior, possessing fewer oscillations as we move from the lower to higher wavelength region. Our results indicate that the SC generation dynamics is dominated 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; supercontinuum generation. Paper 14103 received Sep. 18, 2014; accepted for publication Dec. 9, 2014; published online Jan. 13, 2015.

1 Introduction Supercontinuum (SC) generation is characterized by the dramatic spectral broadening of an optical field which occurs when an intense narrow-band light pulse propagates through nonlinear media.1–5 The spectral broadening is contributed by a host of nonlinear optical processes such as self-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 dispersion of the medium and a cleaver dispersion design can significantly reduce power requirements3 and improve the quality of the generated SC.6,7 The first SC generation was experimentally demonstrated by Alfano and Shaprio in 1970 using borosilicate glass pumped by a picoseconds pulse.8 Although the SC generation has been experimentally achieved in different nonlinear media, photonic crystal fibers (PCFs) have emerged as the most popular nonlinear media for SC generation due to the feasibility of dispersion and nonlinearity engineering.1,5 The discovery of highly nonlinear soft glass PCFs has further enhanced their popularity as a nonlinear medium for successful SC generation.9 The SC spectra in a highly nonlinear PCF is usually generated by pumping femtosecond or picosecond pulses close to the zero dispersion point in the anomalous dispersion regime of the fiber. However, the SC can also be generated in the normal dispersion regime where the spectral broadening is mainly dominated through SPM, Raman scattering, and four wave mixing.10 Although the majority of available literature on the SC generation in PCFs involve only a single zero dispersion wavelength,11 a considerable effort has been made to study SC generation in PCFs with two zero dispersion wavelengths12–14 yielding enhanced bandwidth with improved flatness. Multiwavelength pumping has also been successfully attempted to achieve 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 liquid filled PCFs18 have also been employed to investigate SC generation. Most of the experimental and numerical investigations on SC generation, as elucidated above, 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

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Borgohain and Konar: Self-phase modulation dominated supercontinuum generation. . .

light pulses whose temporal profiles are very close to Gaussian or “sech” and the qualitative features 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 broadening depends both on the shape as well as on the chirp of the input pulse. A couple of years ago, Konar and Jana19 predicted that sinh-Gaussian pulses could lead to, under certain conditions, more efficient SPM-induced spectral broadening. Hence, it would be worth examining the SC generation phenomenon using optical pulses whose temporal shape is different from “sech” or Gaussian. Therefore, in this paper, we have studied SC generation employing cosh-Gaussian pulses (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, we first prefer to design the fiber, to study its optical properties which are essential for SC generation, and at the end, to study SC generation in this fiber using cosh-Gaussian pulses. Lead silicate glass PCFs offer many advantages because of their large Kerr nonlinearity and good thermal and crystallization stability. These advantages have motivated various researchers to modify the structure of the PCF to achieve higher nonlinearity16 and exploit this nonlinearity to generate a broad SC. Therefore, in the present paper, we first design an SF57 PCF with a small group velocity dispersion and large optical nonlinearity, and thereafter, study the SC generation in this fiber using cosh-Gaussian pulses. The cross section of the fiber has been depicted in Fig. 1(a). It has 11 rings of air holes arranged in a hexagonal lattice formation. The hole diameter is d and the hole pitch is Λ. One missing air hole in the fiber acts as the center of the fiber whose backbone is made of SF57 glass. The effective index of the fundamental optical mode can be calculated using neff ¼ ðλ∕2πÞβ, where β is the propagation constant. Chromatic dispersion of the fiber is composed of two factors, the first one is the waveguide dispersion Dw ðλÞ ¼ −ðλ∕cÞðd2 neff ∕dλ2 Þ, while the second contribution comes from the material and is known as the material dispersion Dm ðλÞ ¼ −ðλ∕cÞðd2 nSF57 ∕dλ2 Þ, where c is the velocity of light in vacuum and nSF57 is the refractive index of the material.20,21 The refractive index is given by the P well-known Sellmeier equation, n2SF57 ¼ 1 þ 3j¼1 ½Bj λ2 ∕ðλ2 − Cj Þ, where B1 ¼ 1.81651371, B2 ¼ 0.37375749, B3 ¼ 1.07186278, C1 ¼ 0.0143704198, C2 ¼ 0.0592801172 and C3 ¼ 121.419942. The contribution of material dispersion to the total dispersion of the periodic structure can be evaluated using this expression for nSF57. Total chromatic dispersion DðλÞ ¼ Dw ðλÞ þ Dm ðλÞ. There are a number of methods to model the propagation dynamics as well as the dispersion profiles of the PCFs such as full vector BPM modeling,22 full vector FEM modeling,23,24 etc. In the present investigation, we have used a software based on the finite difference 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 field of the fiber has been depicted in Fig. 1(b), which signifies that the optical field is completely localized in the central core of the fiber. The total chromatic dispersion of the fiber has been displayed in Fig. 2(a). In order to have an idea about the contribution of material dispersion, we have demonstrated its variation with the wavelength with the dash-dot line in Fig. 2(a). The zero total 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) mode field of the fiber. d ¼ 0.4Λ, Λ ¼ 1.4 μm. Journal of Nanophotonics

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

virtually flat. The value of the anomalous dispersion at 1.55 μm is about 20 ps∕nm∕km which is quite low, therefore we choose the pump wavelength at this value. The values of higher-order dispersion terms are evaluated by taking the higher-order differential of the propagation constant at the pump wavelength, which arises as a result of Taylor expansion of the frequency dependent propagation constant β. Typical values of different higher-order betas are β2 ¼ −2.50× 10−3 ps2 m−1 , β3 ¼2.07×10−5 ps3 m−1 , β4 ¼−1.02×10−7 ps4 m−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 ¼ 1.20 × 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 ∕λAeff Þ, where n2 is 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, it would be more appropriate to incorporate the frequency dependence of γ in the modeling for more accurate prediction. However, since modeling with a constant γ does produce results with a reasonable accuracy, we use a constant γ without much loss of accuracy. The variation of optical nonlinearity with wavelength has been demonstrated in Fig. 2(b). The value of nonlinearity gradually decreases with the increase in λ. This behavior is quite usual since with the increase in λ, the spot size of the optical field increases and consequently the value of nonlinearity 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 temporal 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 Schrödinger equation5 X inþ1 ∂n ∂ αðωÞ Aðz; TÞ ¼ − Aðz; TÞ þ βn Aðz; TÞ ∂z 2 n! ∂T n n≥2 Z∞ i ∂ þ iγ 1 þ Aðz; TÞ RðT 0 ÞjAðz; T − T 0 Þj2 dT 0 ; ω0 ∂T

(1)

−∞

where Aðz; TÞ is the envelope of the electric field of the pulse in a frame that is moving at the group velocity of the pulse along the z-direction. βn ¼ ðdn βÞ∕dωn is the usual dispersion coefficient at the central frequency ω0. RðTÞ is the nonlinear response function which includes both the instantaneous electronic response and the contribution of the delayed Raman response. It can be written as, RðTÞ ¼ ð1 − f R ÞδðTÞ þ f R hR ðTÞ, f R ¼ 0.1 for lead silicate PCF and hR ðTÞ is calculated using hR ðTÞ ¼ ½ðτ21 þ τ22 Þ∕τ1 τ22 e½− ðT∕τ2 Þ sinðT∕τ1 Þ, with τ1 ¼ 5.5 fs and τ2 ¼ 32 fs. The nonlinear Schrödinger equation is solved by using the “split-step Fourier method,” the details of which can be found elsewhere.4,5 Journal of Nanophotonics

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

4 Linear Properties of cosh-Gaussian Pulses and Nonlinear Frequency Shift Before proceeding to study the SC generation, it would be worth presenting a brief discussion on the temporal and spectral properties of Gaussian and cosh-Gaussian pulses. A slowly varying envelope Aðz; TÞ of the propagating cosh-Gaussian pulse can be expressed as Aðz; TÞ ¼ A0 expf−½ð1 þ iCÞT 2 ∕2T 20 g × coshðΩTÞ, where A0 is a constant that determines the power of the pulse, C is the chirp, and Ω is a parameter known as the cosh factor. The above pulse form is rewritten as Aðz; τÞ ¼ A0 expf−½ð1 þ iCÞ∕2τ2 g coshðΩ0 τÞ, where τ ¼ T∕T 0 , Ω0 ¼ ΩT 0 and T 0 is associated with the pulse duration. Ω0 ¼ 0 corresponds to the Gaussian pulses. The value of Ω0 determines the deviation from the Gaussian pulse. The temporal 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 a dip at the center if the value of Ω0 is greater than one. The Fourier transform of these pulses for different values of Ω0 are displayed in the same Fig. 3(b). Since the SPM plays a very important role in SC generation, before proceeding further, we investigate the nonlinear frequency shift caused 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 þ γjAj2 A ¼ 0: ∂z

(2)

The solution to the above equation reads as Uðz; τÞ ¼ Uð0; τÞ exp½iϕnl ðz; τÞ, where ϕnl ðz; τÞ ¼ jUð0; τÞj2 δ is the nonlinear phase shift due to the intensity dependent change in the nonlinear refractive index pffiffiffiffiffiffi characterized by the Kerr nonlinearity, δ ¼ γP0 z and for convenience, we have taken A ¼ P0 U 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 chirping. Thus, the instantaneous nonlinear frequency shift caused by SPM is given by δωðτÞ ¼ −

∂ϕnl ∂ ¼ δ ðjUð0; τÞj2 Þ: ∂τ ∂τ

(3)

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

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

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

for comparison. In a Gaussian pulse, the leading edge undergoes a red shift, whereas the trailing edge undergoes a blue shift. It should be noted that the behavior is qualitatively different for a cosh-Gaussian pulse. Both the leading and trailing edges of the pulse undergo red and blue shifts simultaneously. More specifically, part of the leading edge undergoes a red shift while the remaining portion of it undergoes a blue shift. Similarly, part of the trailing edge undergoes a 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 W peak power. The wavelengths of these pump pulses are 1.55 μm and the maximum length of the fiber is 10 cm. Fiber parameters such as γ, D, and βn 0 s have been mentioned in the previous section. We ignore fiber loss without any hesitation since the fiber length is very small, and will not significantly affect SC generation dynamics. We have numerically simulated SC generation for 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 the end of different fiber lengths has been demonstrated in Fig. 5(a). It is amply clear that a large wide band spectrum has been generated at the end of the fiber. The structure of the spectrum suggests that the onset of spectral broadening starts with modulation instability followed by other nonlinear processes such as Raman scattering, SPM, and four wave mixing. The resultant broadening is due to the interplay between these nonlinear and dispersive processes. In order to study the SC generation phenomena with cosh-Gaussian input pulses, we have carried out simulations using these pulses with a peak power of 200 W. Figure 5(b) shows the simulation result at 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 the sake of comparison. As the value of Ω0 increases, the generation of dispersive waves, self-steepening, and Raman scattering results in a disordered oscillatory SC generation and the SC spectrum becomes asymmetric. The phenomenon of the SC generation with cosh-Gaussian pulses suddenly changes with a further increase in the value of Ω0 . For example, the SC spectrum of these 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 of Ω0 , the behavior of the SC spectra is similar, except that the number of oscillations in the spectra increases with an increase in Ω0 . In addition, the oscillation frequency is higher in the high frequency side and lower in the low frequency side of the spectra. It has also been noticed that the internal peaks of the SC spectrum have shown an asymmetric behavior, possessing Journal of Nanophotonics

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a gradual increase in width from the lower to higher wavelength region. This asymmetric behavior is caused by the self-steepening effect associated with the nonlinearity term in the nonlinear Schrödinger equation. Overall, from the internal structure of the generated SC, we can safely conclude 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 examine the development at different lengths of the fiber in Fig. 5(c), we have demonstrated the SC generation phenomenon for different fiber lengths using cosh-Gaussian pulses with Ω0 ¼ 2.5. From the 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 significantly large optical nonlinearity. This fiber has been used to simulate SC generation dynamics employing cosh-Gaussian pulses. We found that, for high power Gaussian optical pulses, the SC dynamics is initially dominated by modulation instability. Later on, Raman scattering, SPM, and fourwave mixing played a significant role in the evolution of the SC spectra. The SC generation for cosh-Gaussian pulses is different from that of Gaussian pulses. We have found that the SC spectra for cosh-Gaussian input pulses are composed of several internal oscillations. These oscillations increase with the increase in the value of the cosh parameter Ω0. The SC spectra confirm that the SC dynamics is dominated by SPM.

Acknowledgments We thank referee for their valuable suggestions. This work is supported by the Department of Science and Technology (DST), New Delhi, through a R&D Grant SR/S2/LOP-17/2010 and by University 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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13. G. Genty et al., “Enhanced bandwidth of supercontinuum generated in microstructured fibers,” Opt. Exp. 12, 3471–3480 (2004). 14. K. R. Khan, M. Mahmood, and A. Biswas, “Coherent super continuum generation in PCF at visible and near infrared wavelengths,” IEEE J. Sel. Top. Quantum Electron. 20, 7500309 (2014). 15. A. Boucon et al., “Supercontinuum generation by nanosecond dual-pumping near the two zero-dispersion wavelengths of a photonic crystal fiber,” Opt. Commun. 284, 467–470 (2011). 16. M. Tiwari and V. Janwani, “Two-octave spanning supercontinuum in a soft glass photonic crystal fiber suitable for 1.55 μm pumping,” IEEE J. Lightwave Technol. 29, 3560–3565 (2011). 17. V. Pureur and J. M. Dudley, “Design of solid core photonic bandgap fibers for visible supercontinuum generation,” Opt. Commun. 284, 1661–1668 (2011). 18. K. Nithyanandan et al., “A colloquium on the influence of versatile class of saturable nonlinear responses in the instability induced supercontinuum generation,” Opt. Fiber Technol. 19, 348–358 (2013). 19. S. Konar and S. Jana, “Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity,” Opt. Commun. 236, 7–20 (2004). 20. “Schott AG lead-oxide optical fiberglass series,” http://www.schott.com (2013). 21. J. V. H. Price et al., “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol. 18, 327–344 (2012). 22. K. Saitoh and M. Koshiba, “Imperical relations for simple design of photonic crystal fibers,” Opt. Exp. 13, 267–274 (2005). 23. K. Saitoh et al., “Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion,” Opt. Exp. 11, 843–852 (2003). 24. F. Fogli et al., “Full vectorial BPM modeling of index guiding photonic crystal fibers and couplers,” Opt. Exp. 10, 54–59 (2002). 25. BandSOLVE, RSoft Design Group, Inc., New York (2006). 26. K. R. Khan et al., “Soliton switching and multi-frequency generation in a nonlinear PCF,” Opt. Exp. 16, 9417–9428 (2008). 27. L. B. Fu et al., “Investigation of self-phase modulation based optical regeneration in single mode As2 Se3 chalcogenide glass fiber,” Opt. Exp. 13, 7637–7644 (2005). 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, Birla Institute of Technology, Mesra, Ranchi, India. His research interests include photonic crystal fiber, nonlinear optics, and optoelectronics. Swapan Konar received his MPhil and PhD degrees, respectively, in 1987 and 1990. He is a professor in the Department of Physics, Birla Institute of Technology, Mesra, Ranchi, India. His area of interest is in the field of photonics and nonlinear optics. He is a senior associate of International Center for Theoretical Physics (ICTP), Italy (2007 to 2014). He has published 111 research papers and coauthored a book “Introduction to Non-Kerr Law Optical Solitons,” Taylor and Francis, New York, USA (2007).

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