Ultra-wide bandwidth all-solid specialty bandgap fiber for ultrashort pulse delivery

We present a specialty photonic bandgap fiber (PBG) with multiple concentric cores based on the one-dimensional (1D) photonic crystal geometry in an all-solid form. It comprises three sets of tailored bilayer thickness in which each set of bilayer forms an effective core region that allows confinement of specific range of wavelengths. Thus, the successive overlap of the wavelength ranges supported by each of these concentric cores effectively enhances the overall transmission bandwidth of the designed 1D PBG fiber. Moreover, the concept can be extended to form a large number of concentric cores that allows further enhancement of the fiber bandwidth. As a proof-of-concept, an ultra-wide low-loss bandwidth covering a wavelength range of ∼\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim$$\end{document} 1600 nm for the fundamental mode are achieved. Going beyond, an advanced level customization of the proposed fiber geometry enables further minimization of loss and enhancement in structural robustness. The propagation dynamics of an ultrashort pulse ∼\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim$$\end{document} 300 fs are investigated numerically in both the normal and the anomalous dispersion regime of the proposed specialty fiber in the presence of nonlinearity and loss. Eventually, such all-solid multicore large-bandwidth fiber is proposed as a promising candidate for the delivery of ultrashort optical pulses over long distance with minimum amount of distortion and wave-breaking possible.


Introduction
The ultrashort optical pulses with temporal duration ranging from a few picoseconds ( 10 −12 s) to femtoseconds ( 10 −15 s) are of immense interest for applications in micro-machining, material processing, time-resolved measurements of ultrafast phenomena and others.The generation of such short pulses are widely reported to date in various platforms including semiconductor lasers to fiber based light sources.However, the major hindrances to the stable propagation of ultrashort pulses are limited bandwidth and significant dispersion while propagate in the existing state-of-the-art fibers, which limit their wide applicability.In optical fibers, several theoretical papers on ultrashort pulse propagation are reported inclusive of all-possible dispersive and nonlinear effects (Mamyshev and Chernikov 1990; Van et al. 2010).Recently, photonic crystal fibers (PCF) have emerged as a versatile platform to overcome the existing challenges by virtue of its exceptional design flexibility (Liu and Chiang 2010;Bermudez 2016).In this context, photonic bandgap (PBG) fibers have opened up a new opportunity in fiber optics for designing application-specific specialty fibers (Joannopoulos et al. 2008).Essentially, their periodically layered cladding geometry functions like a one/two-dimensional Bragg mirror, forming the photonic bandgap, that confines light via reflections of specific wavelengths.This PBG guiding mechanism has enabled tailoring of fiber bandwidth, and dispersion to a certain extent.We may mention that achieving low loss in a hollow core fiber is easier rather than in all-solid form (Vienne et al. 2004;Wheeler et al. 2012;Shere et al. 2020).Towards this, ultrashort pulse propagation is numerically demonstrated in a Argon-filled hollow-core PCF with low-loss and controlled dispersion (Im et al. 2009).Moreover, an experimental work on high power femtosecond pulse delivery is reported in a hollow-core PCF with significantly low loss over a large spectral window (Wang et al. 2012).However, the limitations on bandwidth tunability still exist in such hollow-core geometries, and their structural fragility makes them less suitable for typical long distance applications.As an alternative approach, an analytical model of ultrashort pulse propagation in linear as well as nonlinear regime is demonstrated using the concept of local modes in a multimode fiber (Macho Ortiz et al. 2017).Although, such multicore platforms have potential to greatly enhance the transmission bandwidth, but the complexities in their design, analysis and fabrication pose a challenge.
In this context, Bragg fiber geometries, which are essentially 1D PBG fibers, offer a much easier platform to achieve bandwidth tailoring in an all-solid form.A unique fiber engineering scheme was reported that involved a transverse chirping of the air holes in the cladding of a PCF to obtain a well-controlled dispersive effect (Skibina et al. 2008).Later, the similar concept of chirped-cladding is exploited in a Bragg fiber design to enhance the overall low-loss spectrum of the fiber along with a flattened dispersion profile (Ghosh et al. 2010).Also, a bandgap fiber based on a quasi-periodic cladding geometry is reported to support the entire visible wavelength range through various cladding modes (Hu et al. 2012).The major advantage of such 1D PBG fibers are their unique light guiding mechanism via customized bilayers making them suitable for bandwidth tailoring.However, such 1D geometries are comparatively less explored to serve as a potential platform to address the existing challenges of ultrashort pulse delivery.
In this paper, a novel fiber design scheme based on multiple core formation in a 1D PBG fiber is proposed to achieve a low-loss wide spectral bandwidth and controlled dispersion for delivery of ultrashort pulses over long distances.To implement the scheme, a design of an all-solid 1D PBG fiber is presented which hosts three concentric effective cores formed consecutively via three adjacent selective bilayer sets.Each effective core supports light of a specific range of wavelengths that lies within the bandgap spawned by the deliberately chosen bilayer dimensions.Such consecutive core design makes the supported successive wavelength ranges partially overlapping, which leads to an enhancement in overall bandwidth of the fiber.The initial design of the specialty fiber in this direction was earlier reported in Biswas and Ghosh (2020).Here, we discuss the detailed scheme along with the fiber engineering process, their large bandwidth support, dispersion profile and the advanced level technique for loss minimization and structural robustness.Moreover, the delivery of an optical pulse as short as 300 fs over few kilometers of the designed fiber in normal as well as in anomalous dispersion regimes is studied.Effects of nonlinearity and loss on the pulse dynamics are also discussed in detail.We may mention that the optimized fiber designs and all the results of ultrashort pulse propagation presented here are numerically simulated.

Theory and proof-of-concept
In 1D PBG waveguides, the periodic arrangement of high and low index layers surrounding the core essentially act as Bragg mirrors that reflect a specific range of wavelength to be guided within the core.Such fundamental principle of bandgap guidance of light is the key to unlock the limitations on bandwidth tailoring in fibers.Exploiting the principle of bandgap formation, we propose a novel fiber design scheme based on multiple fiber core formations in a single 1D PBG fiber through selectively chosen sets of bilayers.To implement this scheme, an all-solid specialty bandgap fiber design, namely SBF-I, is proposed with three selective sets of bilayers.The bilayer dimensions are optimized in such a way that it would create three successive partially overlapping bandgaps to be supported by the three effective concentric cores.Thus, it enhances the overall fiber bandwidth.Moreover, this design scheme essentially focuses on the light transmission through core modes only, unlike the quasi-periodic fiber geometry presented in Hu et al. (2012) to capture light through cladding modes.In this context, a similar concept of concentric multiple cores in a conventional index-guided fiber is reported recently for optical communication to address the issue of capacity crunch (Nadeem and Choi 2016;Bairagi et al. 2019).
The optimization of the bilayer thickness for the SBF-I geometry is obtained by numerically solving the Bloch wave equation for periodic layered media.Solving the equation numerically to locate the stop band, we obtain three sets of bilayer thickness with three partially overlapping wavelength bands.As a proof-of-concept of our fiber design scheme, only three bilayers and three concentric effective cores are considered here.However, the design can be extended to several bilayers and corresponding number of cores.
The schematic of the SBF-I is depicted in Fig. 1a along with the refractive index profile shown in Fig. 1b.For better understanding, we use the following notations: Λ n to denote the thickness of bilayer sets where d 1 is the high index layer thickness, and d 2 the low index layer thickness; R n denotes the consecutive core radii; n is integer number.Also, the parameters of commercially available glasses SF6 and SF66 are used as the low and the high index material for a practical design of the fiber.Further, three operating wavelengths-0.98 μ m, 1.20 μ m, and 2.10 μ m are chosen as center wavelengths to obtain three successive partially overlapping wavelength bands.The corresponding sets of bilayer thickness ( d 1 , d 2 ) are (0.20, 0.80), (0.26, 1.00), and (0.20, 1.56) all in μ m, which are optimized 1260 Page 4 of 15 numerically in MATLAB ® .The innermost core radius ( R 1 ) is optimized at 10 μ m and the subsequent effective cores would include the next adjacent bilayer thickness to the inner core dimension, i.e.R n+1 = R n + d 1 + d 2 .The light will be guided through the next effec- tive core region when the wavelength will no longer be supported by the previous core.The modal analysis of the designed fiber is performed numerically using a transfer matrix approach, originally proposed for modal analysis of any leaky cylindrical structures (Nagaraju et al. 2010).Under scalar approximation, the modal field can be written as, where, j = 1, 2 … Q with Q as the total number of regions in the structure, m, n = 0, 1, 2, 3 … , is the propagation constant, and k j = [(n j ∕c) 2 − 2 ] 1∕2 with n j be the refractive index of the j th region.The index contrast between the two chosen glasses is such that the scalar approximation is valid for the modal analysis.The formation of the effective cores is investigated by solving equation (1) numerically for fundamental, and other higher order modes.Further, the design is verified in COMSOL Multiphysics ® and the supported fundamental modes (FM) by the three distinct effective cores at three operating wavelengths are shown in Fig. 2. The modes are shown in terms of the maximum power flow (in W∕m 2 ).
The aim of this design scheme is to achieve a low-loss guidance with a wider bandwidth, and that is indeed discernible in the loss spectrum of the fiber (Fig. 3).Here, only the confinement loss of the designed fiber is shown as it is crucial for the leaky modes in a bandgap fiber; also the total optical loss including the materials contribution to loss can be included using the available absorption data of materials.However, for the enhancement of effective bandwidth of the proposed fiber design to establish the novelty of the fiber design for the targeted application, the confinement loss management is the key.Accordingly, throughout the paper, we focus on the spectral dependence of confinement loss to achieve an ultra-wide bandwidth.The red curve in Fig. 3 shows the confinement loss of the fiber for the FM only, exhibiting a spectacularly large low-loss spectral range covering nearly 1600 nm wide range of wavelengths, and should be extendable further with judicious optimization owing to its design flexibility.The minimum confinement loss of 1.0 dB/km is achieved at 1.49 μ m within the spectral range above 0.70 μ m, whereas a loss as low as 0.05 dB/km is obtained at the lower wavelength range (0.40 − 0.70 μ m) for the FM.Moreover, two small kinks with maximum confinement losses of 3.04 dB/km and 2.87 dB/km are observed at 0.98 μ m and 1.56 μ m, respectively and also denoted by black downward arrows in Fig. 3.These kinks are the hallmark of the overlapping of three consecutive wavelength bands.However, such kinks in the loss spectrum are reducible by increasing the number of bilayers which will be discussed in the next subsection.To appreciate the usefulness of the proposed geometry over conventional bandgap fibers, the confinement loss of the fiber is compared with that of a standard Bragg fiber having identical index contrast along with a 10 μ m core as the SBF-I and three periodic cladding bilayers of same dimension as the first set of bilayer in the SBF-I design.The confinement loss of the counterpart Bragg fiber for FM is depicted by the black curve in Fig. 3, which covers the lower wavelength range from 0.40 to 0.97 μ m with loss values very close to the losses of the designed fiber within that spectral range.It is to note that the extension of the fiber loss window is in the longer wavelength side covering from visible to near mid-infrared range.A quantitative analysis reveals that a ∼75% enhancement in the low-loss spectrum with confinement losses around 1 dB/ km can be achieved through the effective core management mechanism compared to its counterpart standard Bragg fiber.This eventually results in added advantage in contrast to the loss and bandwidth enhancement realizable through a chirped-clad Bragg fiber (Ghosh ).The performance of this specialty fiber can further be enhanced by accommodating more number of effective cores in the structure i.e., effectively adding more wavelength ranges, and thereby creating plenty of room for a nearly unlimited enhancement in the fiber bandwidth.Such an ultra-wide low loss bandwidth of the fiber is rare for an allsolid bandgap fiber and also very interesting for new generation high bandwidth all-optical networks, and spectroscopic investigations.
Further, the confinement losses of higher order modes (HOM) are calculated at the three operating wavelengths and also in the lower wavelength range.The confinement losses obtained for the 1st higher order mode were: 3.7 dB/km at 0.98 μ m, 8.7 dB/km at 1.20 μ m, and 10.1 dB/km at 2.10 μ m; whereas, losses obtained for the 2nd higher order mode are: 6.8 dB/km at 0.98 μ m, 9.2 dB/km at 1.20 μ m, and 15.2 dB/km at 2.10 μ m, respectively.Also, in the low wavelength range ( ≤ 0.70 μ m) these HOMs have confinement loss values an order-of-magnitude larger than those realizable for the FM.Such losses of HOMs are less likely to induce large differential loss to the fiber which may limit its single mode operation over the wavelength range resulting in a complicated ultrashort pulse dynamics in multimode platform.An advanced design is discussed in the next subsection to overcome this challenge.

Advanced scheme and fiber design
Here, we describe the technique to reduce the overall confinement loss further as well as to give the structural strength to such fiber geometry without limiting its attractive attributes.It is noteworthy here that for an all-solid bandgap fiber structure, the choice of material is very crucial which controls the inherent loss and the fiber dispersion to a certain extent.For a wavelength range covering near to mid-infrared, there are few soft glasses which exhibits excellent transmission property.LLF1 (low-index) and SF6 (high-index) are two such glasses that are considered in the advanced design scheme, which covers a spectral range from 0.40 to 2.50 μ m.However, the optical losses of these glasses above 1.30μ m are large enough to limit the long distance delivery of ultrashort light pulses.Further, sets of three bilayers are optimized from the point of view of bandgap estimation, which together form the specialty fiber geometry.The optimized dimensions of the individual bilayer thicknesses Λ n are (0.20, 0.60), (0.25, 0.95), and (0.28, 1.22), all in μ m, respectively.The FM profiles in various cores of the fiber obtained through numerical estimation are presented in Fig. 4 along with the corresponding PBG fiber's refractive index profile.As can be seen from the refractive index profile shown in the lower panel of Fig. 4, a specific bilayer set of thickness (0.6, 1.80) in μ m is appended periodically after the third optimized bilayer, and is denoted as buffer layer.The function of this extended cladding is to reduce the confinement loss and also to increase the robustness of the fiber.This advanced fiber geometry is named SBF-II and will be used throughout the discussion.
In SBF-II geometry, a 4 μ m radius low-index core is chosen as the innermost core (namely, Core 1).In order to reduce potential number of HOMs, core dimension in this second design is chosen to be the less than half of that in the SBF-I.However, such small cores could have a detrimental effect on the FM confinement.As it could be explicitly seen from Fig. 4 that the penetration of the mode field tails into the bilayer of the subsequent core.Moreover, the comparatively increasing oscillations at the tail region signifies the reduction in mode confinement in the outermost core.However, the presence of buffer layers in the extended cladding enhances the mode confinement in the outermost core, as well as mitigate the adverse effect (increased loss) of the small dimension of the innermost core to a good extent.In this context, we observed previously that a wider loss spectrum for the SBF-I without any buffer layer but with a larger innermost core which in turn increases the number of HOMs.Here, introduction of the buffer layers not only significantly reduces the leakage loss but also enables us to choose a small innermost core and hence helps in eliminating the HOMs that makes the fiber suitable for effective single mode operation.
The modal analysis and the estimation of confinement losses of the SBF-II are done in COMSOL Multiphysics ® only, as the glasses involved in the designed PBG fiber results in high index contrast ( ∼ 0.23), and the transfer matrix approach is no longer valid.The FM ( HE 11 ) and the corresponding loss spectrum of the SBF-II are depicted in Fig. 5. Confinement loss as low as 0.001 dB/km for HE 11 mode was achieved over a wavelength range from 1.25 to 1.8 μ m, with an overall spectral range of ∼ 800 nm exhibiting confinement losses below 1 dB/ km for the all-solid bandgap geometry.Moreover, the confinement loss of the HE 11 mode sup- ported by the third core was 1-4 dB/km covering a total spectral range up to 2.2 μ m.However, this confinement loss value can be reduced further below 1 dB/km by adding more selective sets of bilayers, which will obviously increase the number of cores.The kinks present near 1.4 and 1.95 μ m are the signatures of bandgap overlap.The loss profile of the innermost core and the first kink are shown as the inset of Fig. 5. Also, the present fiber geometry supports two HOMs, TE 01 and HE 12 .The confinement losses of TE 01 and HE 12 were estimated as well and have found to have a loss of at least 10 dB/km on an average over the entire spectral range.This also indicates the high differential mode loss between the FM and HOMs that would result in an effective single-mode behavior of SBF-II over a wide wavelength range.Thus, only the FM is considered henceforth for the study of pulse delivery.
In addition, fiber performance parameters such as dispersion coefficients (second and third order) and nonlinearity of the SBF-II fiber are estimated carefully over the low-loss spectral range for the HE 11 mode.The variation of dispersion coefficient D and group velocity disper- sion (GVD) parameter 2 over the wavelength range are depicted in Fig. 6.The dispersion curve shows a zero dispersion wavelength (ZDW) at 1.605 μ m.Moreover, this dispersion pro- file is the characteristic of the second fiber core which is of interest for pulse propagation.The dispersion curves for the innermost and the outermost cores show the similar features, however overlapping of the spectral regions results in kinks in the dispersion curves too similar to the loss curve.

Ultrashort pulse propagation: results and discussions
The dynamics of pulse propagation through the designed SBF-II fiber are studied by solving the nonlinear Schrödinger equation (NLSE) of the form (Agrawal 2013), where, A(z, t) is a slowly varying pulse envelope, is the loss parameter, 2 is the second order dispersion coefficient which includes the group velocity dispersion (GVD), 3 the third order dispersion (TOD) coefficient, and is the nonlinear parameter, which includes the effect of self-phase modulation (SPM) in the process.Moreover, we will use the second and third order dispersion lengths L D , L ′ D , and nonlinear length L NL during the discussion of the results.Here, P 0 is the peak-power of the pulse, and T 0 is the pulse width and is related to the pulse full-width-at-half-maximum (FWHM) by the relation T FWHM = 2 √ ln(2)T 0 for Gaussian input. (2) The spectral variation of dispersion coefficient D and GVD parameter 2 of the designed SBF-II Equation ( 2) is solved numerically using split-step Fourier method (SSFM) in MATLAB ® .SSFM transforms the NLSE in operator format, where the dispersion operator and nonlinear operator act independently over the ultrashort longitudinal step.Thus, one can neglect the nonlinear term during propagation in linear regime and consider only the loss and dispersion terms without loss of generality.

Pulse propagation in linear regime
The propagation of ultrashort pulses are studied numerically through the designed SBF-II fiber.In linear regime of pulse propagation, we consider the effects of fiber dispersion only.The aim is to observe the effects of dispersion on pulses traversing through the specialty fiber and to estimate an optimum fiber length of propagation that would not affect the stability of the ultrashort pulse.An unchirped Gaussian pulse of 500 fs full-width-at-halfmaximum (FWHM) (thus, pulse width T 0 is 303 fs) is chosen to be delivered through the fiber.Furthermore, the dispersion of the SBF-II as shown in Fig. 6 is considered.The pulse dynamics is studied at two operating wavelengths-1.59μ m at which the fiber shows nor- mal dispersion and 1.62 μ m at which it exhibits anomalous dispersion.The choice of the operating wavelengths is dictated by the exhibition of low dispersion parameter (D) away from the ZDW, as well as its occurrence near the center of the low-loss wavelength span of SBF-II design, and can be launched in practice by using commercially available broadband sources.At 1.59 μ m, D = −33 ps/km/nm, 2 = 4.5×10 −29 s 2 /m, and third order dispersion (TOD) parameter 3 = 4.15×10 −42 s 3 /m.All these values are directly calculated from the numerical analysis of the fundamental mode propagation in our designed fiber geometry.Also, the value of GVD taken here is the total dispersion of the designed fiber that include the material and waveguide dispersion.The temporal and the spectral evolution of the input Gaussian pulse in normal dispersion regime is shown in Fig. 7a, b, respectively.For a 303 fs input pulse width, the dispersion lengths L D and L ′ D are ∼ 2 km and 6.7 km, respectively.Thus, we have considered the propagation of the pulse over a kilometer long fiber only.The negligible effect of dispersion on the propagating pulse over 1 km can be clearly observed in Fig. 7a.Also, the accumulation of dispersive phases is evident from the frequency chirp at 1 km fiber length shown in Fig. 7a.Here the chirp is increasing due to normal dispersion regime.As dispersion has no effect on pulse spectrum, thus we can observe no significant change in Fig. 7b and the pulse spectra at different fiber lengths are found to be overlapped.
Similarly, the pulse dynamics in the anomalous dispersion regime is shown in Fig. 7c,  d where the unchirped Gaussian pulse with 500 fs FWHM centered at 1.62 μ m is allowed to propagate through the 1 km long fiber.At 1.62 μ m, D = 25 ps/km/nm, 2 = − 3.5×10 −29 s 2 /m, and 3 = 3.83×10 −42 s 3 /m.The values of L D and L ′ D are ∼ 2.5 km and 7 km, respec- tively.Here also in Fig. 7c we can observe no significant dispersive effect on pulse dynamics up to 1 km.Moreover, the accumulated frequency chirp at the fiber end is shown in the same figure with a decreasing pattern owing to anomalous dispersion regime.The spectra at various fiber lengths are shown in Fig. 7d which are exactly overlapped.
The stable delivery of a 300 fs pulse through the kilometer long specialty fiber is possible owing to the supporting environment for ultrashort pulses enabled by the unique multiple concentric core management scheme.Moreover, the SBF-II design exhibits better control on dispersion and loss management compared to the SBF-I.It is also assumed that addition of effective core numbers will further broaden the low-loss spectrum with a sufficiently flat dispersion profile leading to a reduced dispersive effect.In that case, pulses as short as 100 fs can be delivered over long distances without any distortion.

Pulse propagation in nonlinear regime
Further, the propagation dynamics of ultrashort pulses is investigated through the designed SBF-II in the presence of both nonlinearity and dispersion.At first, a 500 fs FWHM Gaussian pulse centered at 1.62 μ m is assumed to be fed at the input end of the designed fiber and allowed to propagate over kilometers of distance.To appreciate the significant effect of the nonlinearity in presence of anomalous dispersion regime, the peak-power of the pulse is optimized deliberately to form fundamental soliton during its propagation.Considering the Kerr nonlinearity, the nonlinear parameter at 1.62 μ m is = 1.273W −1 /km as calculated from the modal analysis, whereas the dispersion parameters 2 and 3 are the same as stated in subsection A for the anomalous dispersion regime.The evolution of an unchirped Gaussian pulse with 500 fs input FWHM ( T 0 303 fs) and 0.3 W peak power is shown in Fig. 8a through 10 km length of the designed SBF-II.The input pulse parameters are chosen in such a way that L D = L NL , i.e., both are ∼ 2.5 km, where L NL is the nonlinear length.We can observe that the input Gaussian pulse (gray dashed curve) reshapes into a fundamental soliton beyond 2 km and propagates smoothly up to 6 km length of the fiber (green dashdotted curve).On further propagation, the effect of TOD gradually becomes prominent and results in an asymmetric pulse form at 10 km fiber length (red curve) which is beyond the TOD length L ′ D .Here, we also can observe an exact scaling between the pulse peak power and the pulse width characteristic to soliton formation.In this study, we consider the lossless propagation of the pulse.However, the effect of loss in the fiber is included and shown in Fig. 8b.In this context, a total optical loss of 2 dB is considered over a length of 4 km, which includes both the confinement and the material contribution.The typical loss effect on the soliton can be observed in Fig. 8b at the end of 2 km and 4 km fiber length which shows a drastic change in pulse forms due to the combined effect of loss and dispersion.It eventually shows a faster decay of power within only 4 km length and also speeds up the onset of TOD that results in a slightly tilted output pulse shape at the end of 4 km.The pulse propagation results with the inclusion of loss are also compared with the pulse form at the end of 4 km with no loss inclusion.Here, we observe that the soliton propagation can sustain up to 2 km of the loss-included specialty fiber design.However, such a large amount of loss is taken into account as we have chosen glasses that exhibits high material loss at this wavelength.On choosing lower loss materials, we can achieve stable ultrashort pulse delivery over several kilometers in the presence of both dispersion and nonlinearity through this all-solid specialty fiber.In Fig. 8c, the corresponding output spectra at the end of 2 km and 4 km are shown along with the spectrum without loss inclusion.Here, we can observe the spectral broadening due to the presence of nonlinearity and also the effect of loss which reduces the spectral intensities mainly in the lower wavelength side.Moreover, an asymmetric spectral broadening can be observed which is the result of the TOD inclusion.Notable is the performance of the designed SBF-II during formation and propagation of the fundamental soliton over kilometer long distance, from which we can foresee a wide range of applications in this direction.
Further, we investigate the pulse dynamics in the normal dispersion regime through the designed fiber.For this, again an unchirped Gaussian pulse of 500 fs FWHM is considered centered at 1.59 μ m.The nonlinear parameter at 1.59 μ m is 1.28 W −1 /km, whereas the dispersion parameters 2 and 3 are same as stated in subsection A. Moreover, the pulse power is chosen to be 4W only to get a slow occurrence of nonlinear effect, which results in a steady-state reshaping.In this case, L NL becomes 200 m.With these pulse and fiber specifications, we observe a steady state reshaping of the input Gaussian pulse into a parabolic one.The pulse evolution is shown in Fig. 9a over a 2 km fiber length.We can explicitly observe the parabolic pulse shape (the red curve shown in Fig. 9b) along with the monotonously linear chirp developed during the pulse propagation.The output parabolic pulse broadens to acquire a maximum output FWHM of 2 ps.Also the corresponding output spectrum at 2 km is depicted in Fig. 9c, which also shows nearly parabolic profile.This reshaping into parabolic pulse is termed as a steady state reshaping as it requires a longer propagation length exceeding the wave-breaking length for the dispersion and nonlinearity to be counterbalanced, and also it keeps its parabolic pulse shape intact over longer distances without going into wave-breaking.Moreover, such pulse reshaping in normal dispersion regime remains stable even after the inclusion of loss (both the confinement and the material loss) as shown in Fig. 9b,  c.Moreover, the quality of the evolved parabolic pulses through SBF-II is checked by computing the evolution of the misfit parameter (M 2 ) between the pulse temporal inten- sity profile |A| 2 and the parabolic fit |p| 2 of the same energy defined as, (3) where the expression for the parabolic fit with peak power P p and pulse duration T p is given by: The smaller value of M 2 indicates the better fit to the targeted parabolic waveform.Here, M 2 < 0.04 is considered as sufficient for a pulse to be parabolic.Figure 9d shows the evo- lution of M 2 over 2 km long fiber.Also, the red circle at 2 km denotes the minimum misfit point corresponding to M 2 = 0.0194 which indicates a significant pulse reshaping.Also in Fig. 9d, the occurrence of the minimum misfit point (the red circle) at the end of the fiber length explicitly indicates the steady state pattern of the reshaped pulse where the pulse continues to reshape into a precise parabolic shape without going into wave-breaking.It is interesting that an ultrashort pulse indeed survives in an all-solid framework without any significant amount of temporal broadening or wave-breaking.Essentially, the effective core management mechanism employed in the designed PBG fiber improves the fiber loss, bandwidth, and the dispersion to a good extent, as compared to that obtained from known effect of introducing a chirping in the cladding, resulting in a distortion free transmission of ultrashort pulses over several kilometers.It is noteworthy, during this entire study we restricts the pulse FWHM at 500 fs which corresponds to a pulse width of 303 fs where we can neglect the effect of higher order nonlinearities like Raman scattering etc. as the choice of material and the pulse width are not very likely to encounter the same.Also, the effect of Brillouin scattering is neglected during pulse propagation, as a careful analysis reveals that the threshold power level required to excite the same is much higher (several kW) than the input pulse power used for these studies.
In addition, The fabrication feasibility of this prototype fiber design is also plausible as analyzed from the tolerance study.A ± 5% tolerance check on the dimensions of d 1 and d 2 shows good stability of the overall performance of the fiber.The entire low-loss spec- tral range remains unaltered with variation of a minimum of 6% to a maximum of 30% in specific loss values at various wavelengths.Also, the positions of the kinks in the loss spectrum undergoes shift by 2%, whereas the spectral ranges remains unaltered around the wavelengths of operation.Moreover, there are insignificant changes in the dispersion of the fiber over the entire low-loss spectrum, which keeps the dynamics of ultrashort pulse delivery through the fiber unperturbed.Also, fabrication of such specialty photonic bandgap fibers with layer dimensions in sub-micrometer range are practically feasible via the state-of-the-art extrusion method which can use soft glasses such as LLF1 and SF6 (Feng et al. 2009).

Conclusion and prospects
A novel fiber design scheme to support an ultra-wide spectral window of light through concentrically formed multiple effective cores via selective bilayer customization is proposed for ultrashort pulse delivery.Based on the scheme, an all-solid specialty 1D PBG fiber design is demonstrated.The simple 1D geometry of the fiber and the fundamental mechanism of bandgap formation are the key to provide an excellent degreeof-freedom to customize the fiber structure such that it inherently forms a number of concentric cores to support a wider range of wavelength.As a proof-of-concept, an ultra-wide bandwidth for the fundamental mode ranging from 0.4 to 2.2 μ m is achieved with confinement loss in the range of 1-3 dB/km.However, the controlled loss spectrum of the first fiber design is the result of a larger innermost core diameter which includes the higher order modes with losses closer to that obtained for the fundamental mode.This issue is further addressed by introducing additional buffer layers as a novel design procedure in the cladding which have minimized the confinement loss to a range ∼10 −3 dB/km over 700 nm spectral range in presence of a smaller core dimension and also provides structural strength to the fiber.An optimization of fiber parameters in an advanced level design makes the differential mode loss higher, thereby exhibiting an effective single-mode operation of the designed fiber over long wavelength range.Such wide bandwidth of the fiber particularly makes it an extremely promising candidate for ultrashort pulse delivery over long distances.It is noteworthy here that a hollow-core fiber is best suited for loss minimization, however their dispersive nature and structural fragility poses limitations on the bandwidth tailoring process as well as on long distance delivery of light through them.Moreover, the proposed scheme of bandwidth enhancement via fiber core customization is not possible in hollow-core geometries and requires only an all-solid platform.Eventually, the prototype design is proposed to deliver optical pulses as short as 300 fs over several kilometers of the all-solid fiber length without any distortion or wave-breaking.Implementation of the proposed scheme with further optimization will make the fiber design suitable for delivery of pulses shorter than 300 fs.It is strongly envisaged that the proposed all-solid bandgap fiber design scheme has potential to alleviate the bandwidth limitation in optical fibers to a great extent, which would facilitate not only the ultrashort pulse delivery but also many all-fiber based specialty applications.Furthermore, the proposed design scheme has the possibility to open up a new versatile specialty fiber design platform providing an alternative to the multicore fiber geometry with the major advantage of possible straightforward splicing between fibers owing to their structural simplicity.Thus, we also envisage that with appropriate customization, the scheme will result in an endless-core fiber design which has immense potential to be used in emerging high bandwidth local-area network and related specialty applications as an alternative to the contemporary fibers.

Fig. 1 a
Fig. 1 a The schematic representation of the designed SBF-I showing three consecutive concentric cores namely Core 1, Core 2, and Core 3, where each core is guided by a set of specific combination of bilayer composed of a high index material followed by a low index material; b the cross-sectional refractive index profile is shown using two materials with refractive indices denoted by n high and n low .Λ 1 , Λ 2 , and Λ 3 are the chosen thicknesses of the bilayers which form the corresponding concentric cores having radii R 1 , R 2 , and R 3 , respectively

Fig. 2
Fig. 2 The fundamental mode profiles at the three operating wavelengths, 0.98 μ m, 1.20 μ m, and 2.10 μ m, of the designed SBF-I Fig. 3 The confinement loss of the fundamental mode of the SBF-I geometry along with that of the counterpart standard Bragg fiber as a function of the wavelength.(Color figure online)

Fig. 4 Fig. 5
Fig. 4 The fundamental mode profiles denoted by blue solid, dashed and dash-dotted lines guided in the three consecutive concentric cores, respectively are shown in the upper panel along with the refractive index profile of the SBF-II geometry shown in the lower panel

Fig. 7
Fig. 7 The temporal and spectral evolution of the input Gaussian pulse through the SBF-II design in the normal dispersion regime are shown in a, b, whereas their evolution through the same fiber in the anomalous dispersion regime are shown in c, d, respectively

Fig. 8 a
Fig. 8 a The lossless evolution of a 500 fs FWHM Gaussian pulse through the designed SBF-II in presence of anomalous dispersion and nonlinearity, b the time-domain output profiles of the pulse, and c the corresponding spectral profiles without loss and with loss at various fiber lengths.(Color figure online)

Fig. 9 a
Fig. 9 a The lossless evolution of a 500 fs FWHM Gaussian pulse through the SBF-II in the presence of normal dispersion and nonlinearity, b the time-domain output pulse profiles and the chirp, and c the corresponding spectral profiles without loss (red curve) and with loss included (blue dotted curve) at the end of 2 km of the fiber length, d the evolution of the misfit parameter M 2 over the fiber length along with the minimum misfit point shown by the red circle.(Color figure online)