Synopsis on optical attributes of chalcogenide glass solid-core hexagonal microstructured optical fibers in infrared regime

Materials Chemistry and Physics 243 (2020) 122632

Contents lists available at ScienceDirect

Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys

Synopsis on optical attributes of chalcogenide glass solid-core hexagonal microstructured optical fibers in infrared regime Dinesh Kumar Sharma a, *, Saurabh Mani Tripathi a, b a b

Center for Lasers and Photonics, Indian Institute of Technology Kanpur, Kanpur, 208016, India Department of Physics, Indian Institute of Technology Kanpur, Kanpur, 208016, India

H I G H L I G H T S

G R A P H I C A L A B S T R A C T

� Optical attributes of chalcogenide glass H-MOFs is explored in infrared regime. � Near-and-far-field intensity profiles are evaluated for air/chalcogenide H-MOFs. � MFD is examined by employing a vari­ ety of techniques for chalcogenide glass MOFs. � Chalcogenide H-MOF as a potential candidate for slow-light generation is explored. � Fractional power coupled to core is investigated for evanescent field devices.

A R T I C L E I N F O

A B S T R A C T

Keywords: Chalcogenide glasses Microstructured optical fibers Optical materials Infrared (IR) Single-mode fibers

Chalcogenide glasses are furnished with several beneficial features such as high nonlinear coefficient, and their low bulk material losses as compared to silica in mid-infrared (IR) spectral regime makes them doubtlessly as one of the excellent candidates with highly tunable optical characteristics. They have wide transparency in IR domain and can be drawn into fine optical fibers; moreover, possessing potential applications in nonlinear optics realm. We investigated the elementary modal characteristics of solid-core air/chalcogenide hexagonal microstructured optical fibers (H-MOFs) such as mode-index, near-field and far-field intensity patterns, mode-field diameters (MFDs), dispersion and nonlinear characteristics with different geometrical configurations by implementing an alternative theoretical model. Chalcogenide glass H-MOFs as a potential candidate for slow-light generation in IR regime is explored. Also, the utility of such fibers for evanescent field based sensing applications is examined. The strength of the model has been checked by comparing the results with experimental and those which are based on numerical simulation, as available in the literature. Relative errors are also quoted.

1. Introduction In recent years nonsilica materials (or glasses) with cultured nanoand micro-structured [1–9] have enticed much scrutiny because they can paved an alternative route for development of compact and the

robust all-fiber optical devices [10–16] for prospective applications. The latest developments in micro and nano technologies, and their potential applications in biomedical regime [1–3] has led to evolution of new techniques in domain of clinical diagnostics, food safely, health moni­ toring, pharmaceutical analysis and elementary biological research

* Corresponding author. E-mail addresses: [email protected] (D.K. Sharma), [email protected] (S.M. Tripathi). https://doi.org/10.1016/j.matchemphys.2020.122632 Received 3 June 2019; Received in revised form 26 November 2019; Accepted 7 January 2020 Available online 10 January 2020 0254-0584/© 2020 Elsevier B.V. All rights reserved.

D.K. Sharma and S.M. Tripathi

Materials Chemistry and Physics 243 (2020) 122632

[11–15]. It is conspicuous that bulk materials lack some unique physi­ ochemical characteristics such as reactivity, adsorption and high surface-to-volume ratio [17,18], which led to the development of cultured (or structured) materials [1–9] for analytical applications where they can provide not only high sensitivity but also furnish a valuable platform for analysis of single molecular activity [5–7]. Research interest in such structured materials has been substantially increased in past few years due to their high active surface area and regular size. With distinctive properties, e.g., large surface area, non­ toxicity, piezoelectricity, environmentally friendliness, good conduc­ tivity, stability and excellent biocompatibility [1–5], such structural materials can adhere phenomenally in physical, optical, chemical, photo catalytic and electronic sensors [4–10]. The fabrication of compact, affordable, swift and more efficient devices is never-ending quest in sensor community which may require seamless assimilation of biolog­ ical and electronic systems [13–18]. As a source for infrared (IR) radiation fibre lasers can offer advan­ tages of high power and high beam quality in a compact package. The operating wavelength of currently available rare-earth doped silica fibre and silica fibre Raman lasers is limited to 1–2 μm [11–15]. There are two main drawbacks which limit the operating range of such fibre lasers (i) silica fibre is limited in the transmission to ~2 μm for both lasers and (ii) relatively high phonon energy of silica glass host for rare-earth doped fibre lasers, quenches the transitions of rare earth ion in mid-IR (which covers 2–20 μm) spectral range [12–16]. Therefore, alternate glass compositions (or materials) are the choice of the first priority for fibre lasers operating beyond 2 μm [13,14]. Optimization of optical characteristics and the microstructures of nonsilica glasses [16–19] can be considered as an applaud research area, and the popularity has been driven by the demand to develop efficacious devices with additional advanced possessions, enhanced accuracy, lower detection limits and prominent reproducibility [17,20]. In terms of healthcare applications, biosensors based on such structured mate­ rials can possess great promise but the popularity and adoption of such sensing probes depends on their fidelity, perception, cost, elegance, availability and marketing [10,13]. The determined efforts in the development of such sensors for healthcare are still unavailable for end users and are restricted to laboratory scale. Most of the researched structured materials to fabricate optical waveguides (or fibers) are oxide based glasses such as fused silica (which is one of the best choice for all-optical switching [12,13]) since it is a mature and well-grounded technology; however, such glasses exhibit very limited properties to­ wards nonlinear and mid-IR applications [12–16]. Low nonlinearity offered by silica glass demands extremely long length (e.g., a few kilo­ meters) of the fiber; therefore, other promising optical medium with higher nonlinearity is required, and it is desirable that medium should have high power density and provide sufficiently long interaction length [14–16]. Recently, it has been disclosed that chalcogenide materials (or glasses) based optical waveguides possessed the encouraging charac­ teristics to be used as a nonlinear medium, and the research on fabri­ cating such types of optical waveguide dates back to the early 1960s [21]. Chalcogenide glasses are so named as they contain one (or more) of chalcogen elements (such as S, Se and Te [13,15]) and their rare-earth doped compositions [22–25], and comprised of heavy elements, bonded covalently facilitating to unique characteristics in IR regime [21–27]. The vibrational energies of these bonds are low which means that such glasses are transparent into the mid-IR zone. It is to be noteworthy that their inter-atomic bonds are weaker relative to those in oxides and their band gaps are red shifted to visible (or near-IR) region of the spectrum [24–27]. One of the most striking feature of chalcogenide glasses resides in their low phonon energy of (~300-450 cm 1) as compared to silica glasses (~1100 cm 1) and the fluoride glasses (~400-650 cm 1), which allow many mid-IR rare earth transitions as host for rare-earth ion­ s/dopants [22,23] that are normally quenched in silica and fluoride glasses. Thus, possessing strong potential to encourage rare-earth doped

fibre lasers [26–28] for wider IR transmission window. Chalcogenide glasses are also ideal host for luminescent rare-earth ions and their structure can be measured by neutron and/or x-ray diffraction method [22–25] to confer the type of site-specific structural details which is requisite in development of practical structural models [22]. Further­ more, high nonlinearity of several other types of chalcogenide glass compositions is beneficial for producing the photonic devices with po­ tential applications in the mid-IR regime [13–15]. Rare-earth doped chalcogenide and highly nonlinear chalcogenide glass optical fibers have been reviewed by Sanghera et al. [14], and the novel applications of chalcogenide glasses such as all-optical switching [28] have been reported by Zakery and Elliott [27]. The thermoelectric elucidation of amorphous semiconductors has been anticipated to nominate the relevant information about the elec­ trical transport mechanism. Among amorphous semiconductors, chal­ cogenides have attracted much attention due to their versatility in synthesis, thermal, optical as well as electrical properties; therefore, the thin films of chalcogenides possessed wider spectrum of applications such as thermoelectric, electronic and optoelectronic devices, and the advanced energy conversion and its storage [29–32]. Chalcogenides are also investigated to form the diverse series of superb thermoelectric materials [29], and demonstrate potential candidature to refine the electric performance; moreover, such type of glassy materials is being actively investigated worldwide due to their potential applications in civil, medical and military countermeasures [11–14]. In spite of that they have attractive linear and nonlinear optical properties (which overshadow the other glasses) such as very high nonlinear coefficients, high transparency in mid-IR window including two atmospheric win­ dows (lying from 3.0 to 5.0 μm and 8.0–12.0 μm) making them prom­ ising candidates for light-imaging detection and ranging (LIDAR) spectroscopy [15–21]. Depending on glass composition they can be transparent from visible region up to mid-IR (with an ultimate limit close to 25 μm), and have presented strong potential for applications in power transmission, remote imaging as well as spatial interferometry [16–32]. However, the hallmark property of chalcogenide glasses is their photosensitivity [13], which has been widely studied in domain of photo induced effects during last decades [33]. The synthesis of such structured materials with distinguishing attributes and functionality has facilitated an advantageous platform to establish the novel analytical probes, which upon exposure to targeted molecules, trigger the pro­ duction of analytical signals that can be measured [2–8]; moreover, they have demonstrated different mechanisms of interaction with bio­ molecules, resulting in intensified sensing ability [17,18]. In the present scenario, various configurations of the structured materials have been synthesized by simple and cost-effective techniques [1–9], which can be beneficial for immobilization of analytes such as DNA and tumor markers, leading to alternate methods for clinical measurement (or diagnosis) facilitating to fabrication of feasible com­ mercial biosensor. For point of care measurement, it is noticeable that reproducibility of the fabricated sensing probe is another topic of concern obligatory to be addressed that would assist in the development of miniaturized devices. Commercialization and tactile application in biosensors and healthcare field are way far and demand adequate knowledge to explore the performance of such structured materials in fiber form. Single-mode guidance (at telecom wavelength) is the matter of great interest due to their potential applications in the nonlinear waveguiding regime [34,35] but in literature, only a few studies are available out of them some are concern to classical step-index fibers, specialty fibers with porous structure such as microstructured optical fibers (MOFs) [36–40], and others are about integrated optical waveguides [41,42]. For single-mode fibers, small mode-field diameter (MFD) (which is useful to minimize the nonlinear effects [13,28]) or large MFD (to enable the enhancement of the nonlinear effects [43]) are difficult to achieve since the precision required for core and cladding refractive index are incompatible with bulk glass forming techniques [26,32]. A 2

D.K. Sharma and S.M. Tripathi

Materials Chemistry and Physics 243 (2020) 122632

possible solution may be found in MOFs also refereed as “photonic crystal fibers” [35,42], which have created new opportunities for the emerging applications including microfluidics, biophotonics and biomedical [34–36]. There are several common techniques for producing the MOFs [13, 18] but “stack-and-draw” is the most versatile technique [34, 345] which is used to fabricate the fine silica fibers as well as the chalco­ genide glass MOFs [44–47]. Brilland et al. [37] illustrated that “stack-­ and-draw” is a useful tool to realize the complex and regular chalcogenide glass (Sb10S65Ge20Ga5) MOFs with several rings of air-holes. Liao et al. [45] also fabricated a highly nonlinear composite fiber with chalcogenide glass core surrounded by tellurite glass micro­ structure cladding by using the same method. It is notable that MOFs fabricated by this method possessed high optical losses [47]. Direct fabrication of high quality chalcogenide MOF is a tedious task; therefore, an efficient alternative route would be required to combine the mature silica MOF technology with chalcogenide glasses [43–46]. To avoid in­ homogeneities and to prepare low-loss chalcogenide MOFs, casting method has been successfully used [16,46]. Another potential technique is extrusion [47,48], which seems simpler but it is challenging when aiming the preform with complex microstructure patterns. To produce preform for preparing the high quality MOFs such as As2S3 chalcogenide glasses [49] and polymers [50] another powerful technique exploiting the mechanical drilling is also used. In literature, only a few chalco­ genide MOFs have already been reported, and the first report concerned an irregular structure (based on Gallium Lanthanum Sulphide (Ga-L-S) glasses) obtained by extrusion method [47]. Chalcogenide glasses can also be used to fabricate hollow-core photonic band gap omniguide optical fibres, and have proven as an effective medium for transporting high power CO2 laser radiation for applications, e.g., surgery [12,14]. For improving the optical transmission in mid-IR regime [51–56] chal­ cogenide glass hollow-core MOFs [57] of negative curvature are pre­ sented by Shiryaev [51]. Recently, supercontinuum generation in chalcogenide fibre taper [56] with an ultra-high numerical aperture has been reported by Wang et al. [54]. The analytic expression for the fundamental mode (which is the allowed solution to the boundary value problem of propagation [58,59]) of traditional step-index fibres (SIFs) and the solid-core MOFs is crucial in predicting the splicing loss and free space to fibre coupling efficiency [60]. For SIFs, mode field can be approximated by a Gaussian function and has been explored extensively [61–67]. Recently, MOFs have been explored in stellar interferometers [68–70] but due to their intricate geometry, it is cumbersome to have an analytical expression for the principal mode (which is strong function of air-hole structure in the cladding). Various numerical techniques had been utilized for investi­ gating the fundamental propagation characteristics and for studying the coupling efficiency for MOFs. It is not always admirable to use the simple Gaussian function solely to approximate the MOF’s mode-profile [71–73]. In 2004, it has been reported by Hirooka et al. [60] that approximating the electrical field profile for principal mode of solid-core MOFs with small air-hole diameter to pitch ratios, d=Λ (i.e., for weakly guiding structures) by Gaussian function [64,65] brought out large er­ rors; therefore, they proposed that it should be replaced by hyperbolic-secant (sech) function. They also suggested that Gaussian function should still be used for MOFs with large d=Λ values (i.e., for strongly guiding structures). For the fundamental mode of index-guiding H-MOFs an analytical field model was proposed by Sharma and Chauhan [74] in 2009, and they utilized the variational principle for studying the modal properties of such fibres. Ghosh et al. [75] also used variational approach for estimating the effective indices and group-velocity dispersion characteristics of hexagonal MOFs and the square lattice MOFs with different configurations. In 2012, Zhang et al. [76] proposed a constructed function for approximating the fundamental mode-profile precisely in index-guiding H-MOFs by using the least square error criteria. In this paper, we target to systematically study the propagation

characteristics of solid-core air/chalcogenide hexagonal MOFs (i.e., HMOFs) with different structures in IR realm by using our earlier devel­ oped simplified theoretical model, which provides qualitatively accu­ rate results as obtained by other experimental and numerical methods. Effective index for the fundamental and the next higher-order core modes, and the mode-profiles of fibers are evaluated by using an analytical model which is based on scalar wave approximation [58,59]. Also, we evaluated the mode-field diameters (MFDs) by using different approaches and the accuracy of the simulated results is scrutinized with experimental as well as numerical method (such as those based on the multipole method). Optical performance of chalcogenide glasses based fiber as a candidate for slow-light generation and air/light overlap is also investigated. The high material nonlinearity of chalcogenides incorporated with strong confinement and dispersion engineering attainable in optical fiber, assembles them fascinating as fast nonlinear devices. Moreover, this makes them a good platform for ultrafast nonlinear optics and an indispensable technology for future ultrahigh bandwidth optical communication systems. For an optical waveguide, strength of the nonlinear response can be designated by a nonlinear parameter. Therefore, we explored the nonlinear characteristics, elab­ oration of single-mode guidance regime and the engineerable chromatic dispersion for chalcogenide H-MOFs. The paper is organized as follows: we described the chalcogenide MOFs design to be investigated and briefly outlined the theoretical approach adopted for modeling in section 2. In section 3 simulated re­ sults are explored. In section 4, conclusions are briefed. 2. Theoretical modeling For solid-core H-MOFs, we have developed theoretical model (details have been reported in Refs. [77–80]) providing a vital theoretical basis for evaluating the fundamental propagation characteristics of the chal­ cogenide H-MOFs. We chose the following trial electric field (with α; A; α1 and σ as the field parameters) for the principal core mode of H-MOF [77–80]; � � � Ψ t1 ðr; ϕÞ ¼ exp αr2 A expf α1 ðr Λσ Þg2 � ð1 þ cos 6 ϕÞ (1) where Λ denotes the lattice spacing. We used variational technique for scalar modes [81,82] to obtain the optimized values of propagation constant (using MATLAB® optimization toolbox), and the variables involved in Eq. (1), at a given wavelength (details have been reported in Refs. [77–80]). It is noticeable that the essence of variational technique resides in well-judged choice of the trial field [83–85]. Furthermore, we have evaluated the propagation constant for principal cladding mode [86] by using scalar effective index method (and the details have been reported in Refs. [77–80]). 3. Optical characteristics The chalcogenide MOFs possessed interesting optical characteristics depending on their opto-geometrical parameters, defined by the normalized air-hole diameter ratio, i.e., d=Λ. MOFs have a regular arrangement of air-holes which is stretched along the entire length of fiber and can have different size and shape around the central defect site (i.e., solid-core), acting as the cladding. Such structures have strong potential to harness the optical properties over a wide range of wave­ length due to their inherent flexibility in design [87–89]. In spite of that single-mode propagation (which is required for application to ultrafast all-optical switching [55]) and multimode wave guidance regime of chalcogenide MOFs enables numerous applications including laser power delivery, scanning near-field microscopy/spectroscopy, chemical sensing, temperature monitoring, thermal and hyper-spectral imaging, fiber IR sources/lasers, and the amplifiers [51,89]; therefore, it is the subject of enormous interest to explore their wave guidance regimes. In the following, we explored fundamental guiding characteristics of 3

D.K. Sharma and S.M. Tripathi

Materials Chemistry and Physics 243 (2020) 122632

air/chalcogenide H-MOFs (and their structural parameters are summa­ rized in Table 1, at 1.55 μm) in IR spectrum with different configurations. 3.1. Mode field profiles Gaussian propagation of the light through optical fiber is required in various applications such as interferometry and other nonlinear phe­ nomena [14–16]. Different designs of optical fibers have been investi­ gated to obtain single-mode guidance regime, and the several complex and regular fibers from Ga5Ge20Sb10S65 (2S2G) chalcogenide glass have been realized. We considered 2S2G H-MOF1 (cladding parameters are listed in Table 1) with d=Λ ¼ 0.63, indicating the multimode guiding [37,90] at 1.55 μm. The refractive index is 2.25 at 1.55 μm and the nonlinear coefficient is 120 times greater than that of silica [37], counteracting the disadvantage of optical losses, and facilitate to have applications in compact nonlinear devices working in mid-IR regime [54,55]. In the following we evaluated the near-field (NF) intensity profiles for solid-core H-MOFs. Fig. 1 illustrates the transverse intensity profile for the near-field of LP01 mode of the solid-core H-MOF1 at 1.55 μm, which is in-line with experimental observations [37]. It can be depicted (from Fig. 1) that most of the optical energy is localized in the central solid region of the fiber, and the inset illustrates SEM image of the dielectric cross-section for the solid-core 2S2G H-MOF1. For obtaining a single-mode guidance condition (regardless of the wavelength of light), for an MOF consisting of regular array of low-index inclusions (usually air-holes) running parallel to the axis of the fiber [43, 86], the core radius has to be below amax (depending on refractive index of the core and cladding), according to the following relation as [91]: amax ¼ 2:405ðλ =2πNAÞ; where NA is the numerical aperture, which is a crucial parameter for the applications in nonlinear domain [54]. Nu­ merical aperture of an optical fiber is the measure of its light gathering capacity and is directly related to the effective mode area, Aeff [92], qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi which can be evaluated as [93]: NA � 1 = ð1 þ πAeff =λ2 Þ: We evaluated

Fig. 1. Near-field intensity profile for LP01 mode of air/chalcogenide H-MOF1 with pitch of 7.7 μm and d=Λ ¼ 0.63 at 1.55 μm (Inset illustrates SEM picture of the cross-section for 2S2G H-MOF1 [37]). Table 2 Numerical aperture for the chalcogenide H-MOF at 1.55 μm. Fiber ID

Ref. [94]

Theoretical model

Relative error (%)

2S2G H-MOF1

0.15

0.14

6.66

The NF-intensity profile, i.e., It1 ðr; ϕÞ ¼ jΨ t1 ðr; ϕÞj2 , for 2SG (Sb20S65Ge15) H-MOF2 (with refractive index of 2.37 at 1.55 μm [40], and the nonlinear refractive index of 2SG glass is very similar to that of 2S2G glass [37]) as a function of radial distance in ϕ ¼ 0 (as illustrated by solid line) and ϕ ¼ π=6 (as illustrated by dashed line) directions at 1.55 μm, is displayed in Fig. 2, and the inset depicts SEM picture of the fiber’s dielectric cross-section. We included the experimental data as shown by solid circle in Fig. 2 (based on Gaussian approximation [40]), which is matching well to the simulated intensity profile (in ϕ ¼ 0 direction). It is notable that deformity and uncertainty can be the technical problems in the NF measurement process while far-field (FF) radiation pattern is less susceptible. Therefore, it can be measured more precisely [82]. In the following, we evaluated the FF-intensity pattern for the solid-core chalcogenide H-MOFs.

by using the following expression [92]: Aeff ¼ Aeff R 2π R ∞ 2 R 2π R ∞ 1 2 4 ð 0 0 jΨ t1 ðr; ϕÞj rdrdϕÞ ð 0 0 jΨ t1 ðr; ϕÞj rdrdϕÞ , and λbeing the wavelength (for details, see appendix A). The optical characterization of IR transmitting core-cladding chalcogenide glasses and mid-IR guiding step-index fibers offering single-mode guidance has been reported by Houizot et al. [36]. We investigated the unique and versatile optical properties of solid-core chalcogenide H-MOFs (e.g., numerical aperture and the endlessly single-mode propagation in IR domain) by using the field model [77–80] for Ga-Ge-Sb-S system [37], and evaluated the numerical aperture (for details, see Ref. [79]). We made comparison (see Table 2) with those as reported in Ref. [94], and reported the relative error. We would like to highlight that the utilization of larger numerical aperture chalcogenide fibers can have appreciable impact on long wavelength extension of generated supercontinuum [95,96] by reducing the mode-field diameter and increasing the nonlinearity. Using the calculated value of NA (in the aforementioned relation), we obtained amax �9 μm, which indicates that to elaborate the singlemode guidance (in Ga-Ge-Sb-S system based H-MOF) the dimension of the core must be lower then amax . Thus, it can be emphasized that chalcogenide MOFs can be either endlessly single-mode or provide large/small mode volumes, are an appealing alternative to classical stepindex fiber configurations [13].

Fig. 2. Radial distribution of near-field intensity profile (in arb. units) for 2SG H-MOF2 at λ ¼ 1.55 μm in ϕ ¼ 0 and ϕ ¼ π=6 directions (Inset shows the scanning electron microscope image of the fiber cross-section [40]).

Table 1 Summary of MOF structural parameters at 1.55 μm. Fiber ID

Glass composition

Refractive index

Geometrical parameters (μm)

Refs.

H-MOF1 H-MOF2

2S2G (Ga5Ge20Sb10S65) 2SG (Sb20S65Ge15)

2.25 2.37

d ¼ 4.85 and Λ ¼ 7.7 d ¼ 1.07 and Λ ¼ 2.5

[37] [40]

4

D.K. Sharma and S.M. Tripathi

Materials Chemistry and Physics 243 (2020) 122632

The field amplitude, δðθÞ can be evaluated by using Fraunhoffer’s Z 2π Z ∞ diffraction formula [97,98]: δðθÞ ¼ C0 Ψ t1 ðr; 0

0

ϕÞexpðik0 r sin θ cos ϕÞrdrdϕ; where C0 is a constant, and Ψ t1 ðr; ϕÞis the near-field profile (as given by Eq. (1)). IðθÞ ¼ j~δðθÞj2 denotes the normalized FF-intensity, and ~δðθÞ ¼ δðθÞ =δð0Þ is used to denote the relative FF-amplitude (at the far-field angle of θ). The angular spectrum of FF-intensity profile (with low-intensity satellite peaks) for chalco­ genide H-MOF1 and H-MOF2 at 1.55 μm in ϕ ¼ 0 (as shown by dashed-dotted line) and ϕ ¼ π=6 (as displayed by solid line) is illustrated in Fig. 3, and the inset depicts the SEM images of their cross-section. It can be observed (from Fig. 3(a)), that for an air/chalcogenide 2S2G H-MOF1 (d ¼ 4.85 μm and Λ ¼ 7.7 μm), FF-intensity profile for the principal mode (without intensity satellite peaks) can be assumed to have overall Gaussian profile; however, the same is not true for 2SG H-MOF2 with d ¼ 1.07 μm and Λ ¼ 2.5 μm (see Fig. 3(b)), illustrating the strong non-Gaussian nature of mode-profile. We would like to mention that our results are analogous to those as reported by Morten­ sen and Folkenberg [99] for FF-intensity profile of the air/silica solid-core H-MOF. The MOF can be considered to be single-mode when the secondorder mode is not confined due to large propagation losses in the core of the fiber [16,100], which is also reserved for large structures including chalcogenide glasses [48]. Recently, we have presented the trial field with η, B, η1 and ρ as the set of variables, to evaluate the effective index of LP11 mode for the solid-core H-MOFs [80]; � � � B expf η1 ðr ΛρÞg2 � ð1 þ cos 6 ϕÞ � cos ϕ Ψ t2 ðr; ϕÞ ¼ r exp ηr2

Fig. 4. NF-intensity profile for LP11 mode of 2S2G H-MOF1 with Λ ¼ 7.7 μm and d=Λ ¼ 0.63 at the wavelength of 1.55 μm.

1.55 μm for LP11 core mode (and the procedure has been reported in Ref. [80]). Also, we reported the numerical values of the variables (as involved in Eq. (2)): B ¼ 0.085244, η ¼ 0.125949, η1 ¼ 1.623226, and ρ ¼ 0.704456, at the wavelength of 1.55 μm.

(2)

3.2. Mode-index

Fig. 4 illustrates the simulated intensity profile for the scalar secondorder mode (LP11) at 1.55 μm for air/chalcogenide 2S2G H-MOF1 with Λ ¼ 7.7 μm and d=Λ ¼ 0.63. The simulated mode-profile is seen to be very similar to second-order mode of traditional step-index fiber. For chalcogenide H-MOF1, we reported the effective index of 2.241833 at

We evaluated the effective mode-index (see Table 3) for the propa­ gating lowest-order core (and the procedure used for optimization has been reported in Ref. [79]) and the cladding mode (details has been given in Refs. [79,80]), for chalcogenide H-MOFs at 1.55 μm. Recently, we have reported the effective index for the principal core mode [101] (over the range of wavelength) of tellurite (TeO2) glass solid-core H-MOF with different structural parameters, and have checked the ac­ curacy of our simulated results with multipole method. Coherent agreement between the results has been demonstrated (for details, see Ref. [101]). We are in faith that our model possessed significant accu­ racy in evaluating the effective indices for guided core modes. Typical numerical values of the field variables (as involved in Eq. (1), which is obtained, during the process of optimization) are also tabulated in Table 3, at 1.55 μm. Thus, we can infer that chalcogenide H-MOFs can offer a very high-index contrast as compared to traditional fiber facili­ tating to enhance the confinement characteristics, and can be considered as the paramount intention of the fiber designing [16,27]. 3.3. Mode-field diameters (MFDs) Mode-field diameter (MFD) is one of the fundamental parameter that characterizes the mode field profile in the radial direction, for singlemode optical fiber (or waveguide) against the core diameter. In addi­ tion, it is a critical parameter to determine the splice loss and the coupling efficiency [56,66]. Traditional step-index single-mode fibre possessed circular symmetry; therefore, mode-profile can be approxi­ mated by Gaussian function with adequate accuracy [61–65]; however, the same is not true always for MOF as the mode pattern is strong function of air-holes configuration [71–73]. Several definitions of the mode-field diameters (which are treated as different MFDs) have been reported in literature. Petermann-1 spot-size and the Petermann-2 spot-size are the two main definitions which have been widely used for evaluating the MFD (which is twice of modal spot-size) [97,98]. Petermann-1 MFD is defined as [82,97]: dP1 ¼

Fig. 3. Far-field intensity profiles in ϕ ¼ 0 and ϕ ¼ π=6 directions for H-MOFs at 1.55μm. (Inset illustrates the SEM image of the dielectric cross-section for (a) 2S2G H-MOF1 [37], and (b) 2SG H-MOF2 [40]). 5

D.K. Sharma and S.M. Tripathi

Materials Chemistry and Physics 243 (2020) 122632

Table 3 Effective indices for principal mode and the field parameters at 1.55 μm. Fiber ID

Effective cladding index

Effective core index

A

α

α1

σ

H-MOF1 [37] H-MOF2 [40]

2.238706 2.340627

2.247504 2.356642

0.046743 0.103061

0.071963 0.304533

0.471595 2.567798

0.728474 0.886244

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2π R ∞ R 2π R ∞ 1 and 2 2ð 0 0 jΨ t1 ðr; ϕÞj2 r2 r dr dϕÞð 0 0 jΨ t1 ðr; ϕÞj2 r dr dϕÞ , Petermann-2 MFD, can be evaluated as [82,97]: dP2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2π R ∞ R 2π R ∞ 1 2 2ð 0 0 jΨ t1 ðr; ϕÞj2 r dr dϕÞð 0 0 ðdΨ t1 =drÞ2 r dr dϕÞ ; where Ψ t1 ðr; ϕÞ is given by Eq. (1). One can also define a spot-size, weff (by assuming the effective area of the principal core mode [71,92] to be circular) such that, Aeff ¼ πw2eff , and a new MFD can be defined as,

dAeff ¼ 2weff . The mode-field diameter of H-MOF is 1=e2 of the maximum intensity, measured by Gaussian curve fitted to the intensity pattern [40] (which is averaged over all azimuthal directions) facilitating to define the Gaussian MFD. We evaluated the different MFDs by using the theoretical model at the wavelength of 1.55 μm, for chalcogenide glass 2SG H-MOF2 (clad­ ding parameters are summarized in Table 1), and the results are tabu­ lated in Table 4. It can be inferred that there is difference between dP1 and dP2 indicating the departure of the field from Gaussian distribution (as for Gaussian profile dP1 and dP2 are equal [97]); therefore, it can be emphasized that triangular 2SG H-MOF2 has non-Gaussian field distri­ bution. In addition, the values of dAeff and dP1 are close enough sug­ gesting either of them can be used for evaluating the near-field based MFD. The specific azimuthal variation of NF-pattern in between two ex­ tremes (i.e., ϕ ¼ 0 and ϕ ¼ π=6) promote to define two MFDs, corre­ sponding to MFD along x-axis and y-axis, respectively (as shown schematically in Fig. 5). We evaluated the MFD for chalcogenide glass 2SG H-MOF2 (using the opto-geometrical parameters as listed in Table 1) by fitting the Gaussian function in ϕ ¼ 0 and ϕ ¼ π=6 direction, and the results are summarized in Table 5. A notable difference in MFD values (see Table 5) can be observed which are obtained from fitting the Gaussian function in ϕ ¼ 0 and ϕ ¼ π=6, and the average of these two values is close to MFD which is achieved by 2D Gaussian fitting (see Table 4). Our results are coherent to those obtained from using the multipole method [40], suggesting that our methodology can be implemented for quick evaluation of MFDs for solid-core chalcogenide H-MOFs. Theo­ retically simulated results are in good agreement with experimentally measured values (which have been obtained by using the Gaussian fitting) [40] and with those based on the multipole method [40]. For quantifying the accuracy of our results, we evaluated the relative error by using the following expression: Δw ¼ jðw1 w2 Þ =w2 j � 100; where w1 and w2 are the MFDs obtained from employing the theoretical model and those taken from Ref. [40]. We reported the smaller values of Δw (see Table 5) justifying the robustness of our theoretical model in evaluating the MFDs for chalcogenide H-MOF in IR spectral regime.

Fig. 5. Near-field intensity profile (shown schematically) along ϕ ¼ 0 and ϕ ¼ π=6 directions for solid-core H-MOF.

material dispersion (occurs due to wavelength dependency of refractive index of the material [82,98]) and waveguide dispersion (arises due to wavelength dependence of the propagation constant on the mode), respectively [82,98]. The material dispersion is defined as [98]: Dm ðλÞ ¼ ðλ =cÞd2 n =dλ2 ; where ‘n’ is function of the wavelength (evaluated by using the Sellemier formula), c is the velocity of light in vacuum and λ being the operating wavelength. Material dispersion is independent of opto-geometrical parameters of the fiber; therefore, it is same for different lattice structures of the designed MOF. MOFs possessed novel dispersion characteristics in contrast to stan­ dard optical fibers such as high negative dispersion, ultra-flattened dispersion and zero group-velocity dispersion (GVD) due to flexibility in tailoring the geometrical parameters (i.e., d=Λ and Λ); moreover, GVD is strongly influenced by the cladding configuration of MOFs [87]. In 2013, Shabahang et al. [102] have reported the results of systematic measure­ ments of GVD and demonstrated many possibilities for dispersion man­ agement, and nonlinearity enhancement in chalcogenide glass fibers stemming from the flexibility of the fiber fabrication methodology [103–106]. The dispersion coefficient can be evaluated as [88,97]: DðλÞ ¼ ðλ =cÞd2 neff =dλ2 ; where neff denotes the refractive index of the principal mode. For studying the GVD characteristics of selenide chalcogenide (As2Se3) glass, Sellmeier formula is applied for evaluating wavelength qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dependent refractive index [107]: nðλÞ ¼ 7:56 þ 1:03=λ2 þ 0:12=λ4 . The wavelength dependence for refractive index of fused silica is

considered by using the well-known Sellmeier expression [82]: nðλÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffi 1 þ λ2 0:6961663λ . þ λ2 0:4079426λ þ λ20:8974794λ ð0:0684043Þ2 ð0:1162414Þ2 ð9:896161Þ2

3.4. Dispersion characteristics The chromatic dispersion limits the data bit rate and plays a pivotal role in estimating the information carrying capacity of transmission fi­ bers (or optical waveguides) [87,88]. The total dispersion, DðλÞ can be expressed as: DðλÞ ¼ Dm ðλÞ þ Dw ðλÞ; where Dm ðλÞ and Dw ðλÞ are the

The spectral dependence of GVD (in ps/km-nm), which is increasing monotonically with increase in wavelength is illustrated in Fig. 6, for the solid-core As2Se3 H-MOF for d=Λ ¼ 0.30, 0.50, 0.70 and 0.90 with

Table 4 Mode-field diameters (MFDs) calculated by different methods at λ ¼ 1.55 μm. MFD (μm)

dP1 (based on Petermann-1 spot-size)

dP2 (based on Petermann-2 spot-size)

dAeff (based on Effective mode area)

2D Gaussian Fitting

H-MOF2

3.28

3.08

3.32

3.47

6

D.K. Sharma and S.M. Tripathi

Materials Chemistry and Physics 243 (2020) 122632

Table 5 Mode-field diameters for 2SG H-MOF2 at λ ¼ 1.55 μm. Fiber ID

MFD (μm)

Multipole method [40]

Experimental data [40]

Theoretical model

Relative error (%)

H-MOF2

x-axis (orϕ ¼ 0 )

3.3

3.5

3.24

7.42

y-axis (orϕ ¼ π=6)

3.5

3.7

3.62

2.16

different pitch values of 2.0 μm, 3.0 μm and 4.0 μm. For better illus­ tration, we have considered two wavelength windows from 1.2 to 2.5 μm and from 2.5 to 5.0 μm, for studying the dispersion characteristics (i. e., negative and zero dispersion) of MOF with different geometrical parameters. Dispersion causes broadening (or widening) of optical pulses resulting reduction in volume of data transmission in spite of that data retrieval becomes difficult [103–106]; therefore, in order to minimize the spreading of optical signals, dispersion should be compensated (through a short length of specially designed fiber). In literature, several techniques has been reported for achieving the dispersion compensation [87] but the technique based on dispersion compensating fibers (DCFs) is widely utilized [88,104]. DCFs should have negative dispersion co­ efficients (for reducing the length of fibers and hence to reduce the cost) [87,108]; moreover, dispersion and dispersion slope of the transmission fibers should be compensated simultaneously [108–110]. Traditional DCFs are designed to have a high negative dispersion coefficient which is not acceptable in broadband communication systems for dispersion compensation. Recently, Liu et al. [111] proposed a novel DCF based on hybrid MOFs, compensating multiple dispersion wavelengths at the same time. It has been also illustrated in Ref. [88] that chalcogenide MOFs can be used as DCF due to their high negative dispersion slope. In Fig. 7, we have displayed the negative chromatic dispersion for As2Se3 glass MOF for the fixed pitch of Λ ¼ 2.0 μm and d=Λ ¼ 0.30 (as illustrated by solid line), 0.50 (as illustrated by dash-dotted line), 0.70 (as illustrated by dotted line) and 0.90 (as illustrated by dashed line) in the wavelength range of 1.2–2.5 μm. It can be depicted from Fig. 7 that when d=Λ ratio increases for a given value of pitch, negative GVD also increases which is supported by the tight modal field confinement due to

Fig. 7. Negative dispersion curve for chalcogenide glass H-MOF against the wavelength for a given pitch and at different d=Λ ratios.

decrease in core dimension. Fig. 8 illustrates the negative GVD against wavelength for air/chal­ cogenide (As2Se3) H-MOF and the conventional air/silica H-MOF with Λ ¼ 3.0 μm and d=Λ ratios of 0.20, 0.30 and 0.40 in the wavelength range of 1.25–1.65 μm, and it has been observed that negative dispersion in As2Se3 glass H-MOF is extremely higher in contrast to silica based HMOF. It can be inferred that As2Se3 glass H-MOFs have high-index contrast which can be obtained by tailoring the cladding parameters over a broad wavelength range; therefore, chalcogenide MOF possessed strong potential to be used as a DCF in telecommunication industry [103,105]. It is to be notable that air/chalcogenide H-MOF with high negative GVD can also be achieved from chalcogenide compounds as

Fig. 6. Group-velocity dispersion curve for high-index core air/chalcogenide MOF for d=Λ ratios of 0.30 (as shown by dashed line), 0.50 (as shown by solid-line), 0.70 (as shown by dotted line) and 0.90 (as shown by dash-dotted line). 7

D.K. Sharma and S.M. Tripathi

Materials Chemistry and Physics 243 (2020) 122632

H-MOF can be shifted (from infrared (IR) to ultraviolet (UV) regime which is difficult to attain in traditional step-index fibers) to desired value simply by altering the air-hole diameter and the separation be­ tween them. Next, we considered another example of As2Se3 glass based H-MOF structure with d=Λ ¼ 0.40 (to assure that fiber is endlessly single-mode [86]), and the pitch is varying from 3.0 μm to 6.0 μm. For evaluating the refractive index of As2Se3 glass, we have used Thompson’s Sellmeier equation [113]: nðλÞ ¼ 1=2

½1 þ λ2 ½A20 =ðλ2 A21 Þ þ A22 =ðλ2 192 Þ þ A23 =ðλ2 4 � A21 Þ�� ; where the Sellmeier coefficients are, A0 ¼ 2.234921, A1 ¼ 0.24164, A2 ¼ 0.347441, and A3 ¼ 1.308575, respectively. We plotted the dispersion (see Fig. 10) for As2Se3 MOF for pitch values of 3.0 μm, 4.0 μm, 5.0 μm and 6.0 μm at d=Λ ¼ 0.40. The trend of variation is same as reported in Ref. [113] for different pitch values, and it can be depicted (from Fig. 10) that a flat dispersion profile can be obtained for a large wave­ length range with a double zero dispersion wavelength by choosing the appropriate value of the pitch. At the end of this section, we can emphasized that endlessly singlemode characteristics of As2Se3 MOF, and the flat dispersion profile associated with high nonlinearity of chalcogenide glasses can aid supercontinuum broadening process in IR domain [114–116]. 3.5. Nonlinear characteristics Fig. 8. Spectral dependence of negative dispersion (a) silica and (b) chalco­ genide glass MOF with Λ ¼ 3.0 μm and d=Λ ¼ 0.20, 0.30 and 0.40.

In telecommunications, nonlinear materials play a vital role [113–117] and for those applications single-mode waveguides are of prime requiremnt. A first obvious choice is the preparation of conven­ tional step-index structure (with single-mode guidnace) from two dif­ ferernt compositions exhibiting compatible thermal and the optical characteristics [36–41], and the second possibilty is provided by MOF structures [34,35]. The reduction in mode area and subsequent enhancement of nonlinearity is ultimately limited by the index contrast in case of traditonal single-mode fibers [117], and the structural di­ mensions in case of MOFs [118]. Ability to enhance (or reduce) the effective nonlinear coefficients and at the same time to control the magnitude and wavelength dependence of GVD, facilitates MOF as a potentail candidate for exploring the nonlinear effects [117–120]. Chalcogenides are one of the promising materials for second or third order nonlinear processes in near-IR spectral regime [13,115] despite their intrinsic characteristics together with microstructure allow exac­ erbating the extraordinary optical properties such as dispersion and the nonlinearity [43,49]. The effective nonlinearity can be improved either by altering their structure for reducing effective area of the core, Aeff [117,121] or by chosing the materials with higher Kerr nonlinearity [113,117] or both [43,45]. In spite of that multi-component glasses also play critical role as they have typical values an order of magnitude or more higher, and hollow channels can of course be filled with another

they possessed high refractive index contrast [102–104] but the fabri­ cation of such type of compound fibers is a difficult task [106]. One possible solution to overcome the problem can be made by incorporating them into air/silica MOFs. Thus, DCFs with large negative dispersion can be created by infusing air-holes in air/silica H-MOFs with chalco­ genide glasses [31,112]. For evaluating the zero GVD as a function of wavelength in hexag­ onal lattice As2Se3 glass MOF, we considered that the pitch is varied from 2.0 μm to 6.0 μm (in steps of 2.0 μm) while d=Λ ratio is 0.90. From Fig. 9, it is obvious that for considered pitch values, zero group-velocity dispersion can be achieved at the following wavelengths ~3.3 μm, 3.9 μm and 4.6 μm for Λ ¼ 2.0 μm, 4.0 μm and 6.0 μm, respectively, at d=Λ ¼ 0.90. It is clear from the figure that for a given d=Λ ratio, zero dispersion wavelength (ZDW) can be shifted (which occurs due to strong wavelength dependent of cladding index) to higher value with increase in pitch. A suspended core chalcogenide (As2S3) MOF with a ZDW around 2 μm (whereas the ZDW of bulk glass is around 5 μm) was realized by El-Amraoui et al. [49] by tailoring the core diameter and the microstructured geometry. We can argue that a ZDW in chalcogenide

Fig. 9. Zero chromatic dispersion against the wavelength for hexagonal chal­ cogenide MOF with d=Λ ¼ 0.90 and for different pitch values of 2.0 μm (as shown by dash-dotted line), 4.0 μm (as shown by dotted line) and 6.0 μm (as shown by solid line).

Fig. 10. Spectral variation of dispersion for As2Se3 glass based triangular MOF with different values of pitch: Λ ¼ 3.0 μm (dashed line), 4.0 μm (dotted line), 5.0 μm (dashed-dotted line), and 6.0 μm (solid line) and d=Λ ¼ 0.40. 8

D.K. Sharma and S.M. Tripathi

Materials Chemistry and Physics 243 (2020) 122632

Table 6 Effective nonlinearity at the wavelength of 1.55 μm. Fiber ID H-MOF2

n2 � 10 3:2

18

ðm2 =WÞ

Aeff (μm2) [40]

Aeff (μm2) (Theoretical model)

~10

8.65

γ1 ðW

1

m

1

Þ[40]

~1.3

ðcβ =kÞ

1

m

1

Þ(Theoretical model)

1.4

�Z

gas with different nonlinear coefficient [32,114]. The effective kerr type nonlinearlity (with n2 as nonlinear refractive index) can be evalauted as [114,121]: γ ¼ ðωn2 =cAeff Þ ¼ 2π =λðn2 =Aeff Þ; where ω is the frequency. We evaluated the effective mode area (by employing the analytical field model, and the details are specified in appendix A) for H-MOF2, which is 8.65 μm2, nearly five times smaller than H-MOF1 (having mode area of ~40.98 μm2) at 1.55 μm (see Table 6). Effective mode area, quantifying the effective area of the propagating principal core mode for an optical waveguide is presumed as one of the crucial field parameter which has one-to-one correspondence with mode-field diameter that can be controlled by the refractive index-profile of the respective waveguide [121,122]. Further, we evaluated the effective nonlinearity coefficient (as listed in Table 6) for solid-core 2SG H-MOF2 (with d ¼ 1.07 μm and Λ ¼ 2.5 μm) at 1.55 μm, which is fairly matching to those as quoted from Ref. [40]. Such high values of γ permit the short device lengths and when integrated with small dispersion, this can leads to nonlinear devices with bandwidth of several terahertz [14]. Also, we have evaluated: Δγ ¼ jðγ 1 γ2 Þ =γ1 j � 100; where γ2 is obatined from utilizing the theoretical model (with Ψ t1 ðr; ϕÞ as the transverse field amplitude, which is given by Eq. (1)) while γ 1 is quoted from Ref. [40], for appraising relative error of Δγ ¼ 7:7% (see Table 6), justifying the strength of theoretical model in evaluating the nonline­ arity coefficient for chalcogenide H-MOF in IR spectral regime. From Table 6, it can be depicted that γ2 is close to maximal theo­ retical value as computed by Magi et al. [123], for an As2Se3 traditional fiber taper. Comparison will be more appreciating if we consider the length, L of the devices. For a meter length of the MOF, the product ½γL�MOF leads to 1.4W 1, whereas the product ½γL�taper for the taper length of ~20 mm results in 3.3W 1 [123]. Thus, it can be inferred that such type of structure can support to have enhanced nonlinear processes at the low threshold power levels [102,117]. In spite of that chalcogenides can recommand exclusive set of attributes among optical glasses which will make them an excellent choice for mid-IR sciences and nonlinear optics [10,13].

γ2 ðW

2π

Z

∞ 0

0

jΨ t1 ðr; ϕÞj2 ds

Relative error (%) 7.7

�� Z 0

2π

Z

∞

0

� 1 n2 ðr; ϕÞjΨ t1 ðr; ϕÞj2 ds ;

where dsð¼ rdrdϕÞ is an area element. Fig. 11 illustrates how the normalized group velocity, ðυg =cÞ of the mode varies with wavelength spanning from 1.0 μm to 10 μm for chalcogenide As2Se3 H-MOF with d=Λ ¼ 0.40 and Λ ¼ 3.0 μm (as displayed by dashed line), 4.0 μm (as displayed by dotted line), 5.0 μm (as displayed by dash-dotted line) and 6.0 μm (as displayed by solid line). It can be depicted from Fig. 11, that group ve­ locity of the mode is following the trend (i.e., it is slower for shorter wavelengths than that of higher wavelengths) as reported in Ref. [125]. The typical values of normalized group velocity, ðυg =cÞ and the variables (as involved in Eq. (1)) for As2Se3 H-MOF at 1.55 μm have listed (in Table 7) at d=Λ ¼ 0.40. 3.7. Air/light overlap The emergence of MOFs was a revolution in the fibre technology, not properly in domain of optical transmission where traditional single-mode fibre possesses miraculous performance but particularly in regime of optical sensing [126–130]. MOFs are the suitable candidate for designing the optical fibre components and devices with remarkable optical char­ acteristics, and one of the natural choices as suggested by the fibre sensing community [126]. MOFs essentially endorse a substantial increase in design flexibility due to additional degrees of freedom as compared to conventional fibres, in optical fibre sensing realm by providing the effectively long optical path length [125–128]. This is particularly true in fluid sensing where the possibility of fluid (liquid or gas) to occupy the air-holes brings qualitatively better performances [119,126] via evanescent field effect with gases and liquids [127,129]. Relying on this phenomenon, a sensitive detection of acetylene using an MOF with a length of 75 cm has been demonstrated by Hoo et al. [129]. For gas detection with enhanced characteristics in MOFs based on evanescent field interaction and absorption has been reported by Monro et al. [127] and Pickrell et al. [128]. Fractional power associated with principal core mode, PFcore of H-MOF can be evaluated as [79]: PFcore ¼ ðPco =Pt Þ � �� Z 2π Z Λ �� Z 2π Z ∞ � 1� 100 ¼ � jΨ t1 ðr; ϕÞj2 rdrdϕ jΨ t1 ðr; ϕÞj2 rdrdϕ

3.6. Performance of chalcogenide MOF as a candidate for slow-light generation

0

0

0

0

100; where, Pco denotes the power carried by the core mode, and the total power associated with propagating mode is Pt ð ¼ Pco þ Pcl Þ. Fig. 12 depicts PFcore (which is decreasing monotonically) against the wavelength (ranging from 1.0 μm to 10 μm) for chalcogenide H-MOF for

One of the main components of all optical processors is the optical buffers, which control the delay of optical signals and have one of the important applications to prevent the data collisions, facilitated by delaying one of the pulses by activating the slow-light (refers to possi­ bility of controlling the group velocity) medium in one of the branches so that there is no chance for collision [124,125]. Slow-light in optical fibres, particularly in MOFs offers many advantages and possessed strong potential in information and communication technology, microwave-photonics and photonics. Oskooi et al. [52] demonstrated that chalcogenide glass MOF can be designed to have a complete gap which opens up a regime for guiding zero group-velocity modes, and could be used for enhancing the nonlinear interactions. In 2016, slow-light in chalcogenide photonic crystal square-lattice fibers with uniform geometry was proposed and theoretically investigated by Hou et al. [53]. We evaluated the group velocity, υg (which is the first derivative of the frequency with respect to the wave vector [97,125]) for the guided mode for chalcogenide glass H-MOF. Group velocity is used for characterizing the optical performance of waveguides, and can be evaluated as [82]: υg ¼

Fig. 11. Variation of normalized group velocity against wavelength for chal­ cogenide MOF with Λ ¼ 3.0 μm, 4.0 μm, 5.0 μm and 6.0 μm for d=Λ ¼ 0.40. 9

D.K. Sharma and S.M. Tripathi

Materials Chemistry and Physics 243 (2020) 122632

Table 7 Numerical values of ðυg =cÞ for H-MOF with d=Λ ¼ 0.40 at 1.55 μm. Λ (μm)

υg =c

A

α

α1

σ

3.0 4.0 5.0 6.0

0.352398 0.352733 0.352897 0.352988

0.100236 0.091677 0.084101 0.077611

0.222511 0.136778 0.094055 0.069367

1.841734 1.041974 0.667971 0.465396

0.876628 0.858711 0.844319 0.832599

different values of pitch 3.0 μm (as illustrated by dashed line), 4.0 μm (as illustrated by dotted line), 5.0 μm (as illustrated by dash-dotted line) and 6.0 μm (as illustrated by solid line) for a given d=Λ ¼ 0.40. In MOFs, air/light overlap in IR domain can be tailored by judicious choice of the structural parameters, and it is illustrated that increase in pitch (e.g., for Λ ¼ 6.0 μm) can offer to PFcore as high as ~99.88% (see Fig. 12). It is impressive that on decreasing the pitch values (e.g., for Λ ¼ 3.0 μm) air/ light overlap can be increased in IR waveguiding regime. At this point, we can argue that the preliminary poling results based on theoretical model for chalcogenide glass H-MOFs can enable to explore the supercontinuum generation, high power delivery, sensors and spatial filters operating in IR domain. It is to be noted that indexprofile of chalcogenide MOFs can be designed to tailor GVD at desired wavelength, and several meters of uniform and reproducible optical fibre can be produced by standard “stack-and-draw” technique [40]. Despite, air-holes facilitate to create the dramatic morphological changes merely by altering their size [126] and that can be done by collapse or inflation [45,46] of air-holes during the process of fabrica­ tion. Chalcogenide MOFs provide an alternative route of tailoring the effective nonlinearity over at least five orders of magnitude [119] and demonstrated strong potential for enhanced IR fiber optics sensing ap­ plications; moreover, high emission cross-section with high quantum efficiencies enabling their candidature for efficient fibre lasers in IR wavelength regime [129,130]. The research outlined here is not only led to a much better under­ standing of the optical performance of chalcogenides but also exempli­ fying the momentum in the area that can serve to the development of high performance chalcogenide waveguide devices. We are in faith that in foreseeable future, chalcogenides with improvements in optical quality and new compositions will assist a platform spanning the diverse areas of science and technology; moreover, it can be believed that chalcogenide glass fibre optics will progress due to its innumerable potential applications accelerated by low loss single-mode (and/or multimode) fibres [89].

Fig. 12. Fractional power associated with fundamental mode, PFcore of chal­ cogenide H-MOF with d=Λ ¼ 0.40 against the wavelength for Λ ¼ 3.0 μm, 4.0 μm, 5.0 μm and 6.0 μm.

highlighting the large negative dispersion of air/chalcogenide H-MOFs which makes them potential candidate to be used as DCF in optical communication network. It is observed that theoretical results, which engineering GVD through index-profile design may enable zero disper­ sion high-index contrast chalcogenide glass based MOFs. The effect of inter-hole spacing (or the pitch) on PFcore of air/chal­ cogenide MOF in IR wavelength regime for a given d=Λ ratio is explored for evanescent field applications, and it is observed that an adequate amount of field is located within the air-holes. We observed that chal­ cogenide glass MOFs can be used for accessing the benefits of both small and large effective mode area in IR domain. The effective nonlinearity coefficient is also evaluated at wavelength of telecommunication inter­ est, and comparison has been made with those as available in literature. We are assured that our theoretical method is elegant and present itself as an alternative candidate to those numerically intensive and time consuming propagation methods, employed for analysis and design of chalcogenide H-MOFs in IR spectral regime. In addition, our model can be utilized as an efficient theoretical tool for realization of various photonic components and devices in infrared waveguiding domain. Declaration of competing interest The authors Dinesh Kumar Sharma and Saurabh Mani Tripathi whose names are listed in the manuscript with title, “Synopsis on optical attributes of chalcogenide glass solid-core hexagonal microstructured optical fibers in infrared regime,” certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patentlicensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.

4. Conclusions We studied the fundamental modal characteristics of solid-core air/ chalcogenide H-MOFs with circular air-holes such as mode-index and the field profiles by using the theoretical model. Chalcogenide glass HMOF possessed low intensity peaks in the radiated field; moreover, for the intensity profile (above 1=e2 level) in both principle directions of observations no significant deviation from Gaussian distribution is observed. We also evaluated the MFDs for chalcogenide MOFs by using different approaches such as those based on Petermann-1 and Petermann-2 spot-size and the effective mode area. Simulated results are in favor to experimental and those based on the multipole method. The effect of structural parameters on group velocity is investigated for realization of compact slow-light devices. The GVD characteristics are also studied in details by using the scalar modal analysis,

Acknowledgments D.K. Sharma is grateful to IIT Kanpur for providing the Institute Postdoctoral Fellowship (PDF-102). The author wishes to thank Prof. A. Sharma, IIT Delhi for his fruitful conversation and Dr. S.M. Tripathi for his kind encouragement and technical support.

Appendix B. dSupplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.matchemphys.2020.122632.

10

Materials Chemistry and Physics 243 (2020) 122632

D.K. Sharma and S.M. Tripathi

Appendix A. Effective mode area �Z

Z

2π

∞

Aeff ¼

�2 , � Z

0

0

2π

2 jΨ t1 ðr; ϕÞj rdrdϕ

Z

∞

0

0

� � 4 jΨ t1 ðr; ϕÞj rdrdϕ ¼ I 21 I2

(A1)

where Ψ t1 ðr; ϕÞ is the scalar near-field as given by Eq. (1). 2α r 2

jΨ t1 ðr; ϕÞj2 ¼ e

þ A2 e

2α1 ðr σΛÞ2

ð1 þ cos 6 ϕÞ2

αr2 α1 ðr σ ΛÞ2

2Ae

(A2)

ð1 þ cos 6 ϕÞ

We can express integral, I1 as the sum of six integrals, which is given below; Z 2π Z ∞ I1 ¼ jΨ t1 ðr; ϕÞj2 r dr dϕ ¼ I11 þ I12 þ I13 0

(A3)

0

where, Z I11 ¼

Z

2π

e

2αr2

2π

I12 ¼

∞

rdr ¼ 2π

e

0

0

Z

Z

∞

dϕ

2αr2

rdr

0

Z

Z

∞

ð1 þ cos 6 ϕÞ2 dϕ

2α1 ðr σ ΛÞ2

A2 e

e

0

0

Z

2α1 ðr σΛÞ2

Z

∞

ð1 þ cos 6 ϕÞdϕ

2Ae

αr2 α1 ðr σ ΛÞ2

∞

4π A

rdr ¼

e

0

0

rdr

0

Z

2π

I13 ¼

∞

rdr ¼ 3πA2

αr2 α1 ðr σΛÞ2

rdr

0

Integral, I2 can be expressed as the sum of five integrals which is given as; Z 2π Z ∞ jΨ t1 ðr; ϕÞj4 r dr dϕ ¼ I21 þ I22 þ I23 þ I24 þ I25 I2 ¼

(A4)

0

0

where, Z I21 ¼

Z

2π

dϕ

e

4αr2

2π

I22 ¼

e

4αr2

rdr

0

Z

∞

ð1 þ cos 6 ϕÞ4 dϕ

A4 e

0

Z

∞

rdr ¼ 2π

0

0

Z

Z

∞

4α1 ðr σ ΛÞ2

rdr ¼

0 2π

I23 ¼

Z ð1 þ cos 6 ϕÞ2 dϕ

Z

2π

Z ð1 þ cos 6 ϕÞ3 dϕ

2αr2 2α1 ðr σ ΛÞ2

Z ð1 þ cos 6 ϕÞdϕ

0

6A2 e

Z

∞

rdr

2αr2 2α1 ðr σ ΛÞ2

rdr

αr2 3α1 ðr σ ΛÞ2

∞

4A3 e

rdr ¼ 5π

αr2 3α1 ðr σ ΛÞ2

rdr

0

Z

∞

4Ae 0

∞

rdr ¼ 3π

0 2π

4α1 ðr σ ΛÞ2

A4 e 0

0

4A3 e

0

Z

∞

0

I24 ¼

I25 ¼

Z

Z

∞

6A2 e

0

35π 4

3αr2 α1 ðr σ ΛÞ2

∞

rdr ¼ 2π

4Ae

3αr2 α1 ðr σ ΛÞ2

rdr

0

References

[6] ChV. Reddy, I.N. Reddy, B. Akkinepally, K.R. Reddy, J. Shim, Synthesis and photoelectrochemical water oxidation of (Y,Cu) codoped α-Fe2O3 nanostructure photoanode, J. Alloy. Comp. 814 (2020), 152349-1-152349-8. [7] N.P. Shettia, S.J. Malodea, D.S. Nayaka, G.B. Bagihallia, S.S. Kalanur, R. S. Malladi, ChV. Reddy, T.M. Aminabhavi, K.R. Reddy, Fabrication of ZnO nanoparticles modified sensor for electrochemical oxidation of methdilazine, Appl. Surf. Sci. 496 (2019), 143656-1-143656-8. [8] ChV. Reddy, I.N. Reddy, V.V.N. Harish, K.R. Reddy, N.P. Shetti, J. Shim, T. M. Aminabhavi, Efficient removal of toxic organic dyes and photoelectrochemical properties of iron-doped zirconia nanoparticles, Chemosphere 239 (2020), 124766-1-124766-13. [9] P.S. Basavarajappa, B.N.H. Seethya, N. Ganganagappa, K.B. Eshwaraswamy, R. R. Kakarla, Enhanced photocatalytic activity and biosensing of gadolinium substituted BiFeO3 nanoparticles, Chemistry 3 (2018) 9025–9033. [10] D. Pudo, E.C. Magi, B.J. Eggleton, Long-period gratings in chalcogenide fibers, Opt. Express 14 (2006) 3763–3766. [11] J.A. Savage, Optical properties of chalcogenide glasses, J. Non-Cryst. Solids 47 (1982) 101–116. [12] J.S. Sanghera, I.D. Aggarwal, Active and passive chalcogenide glass optical fibers for IR applications: a review, J. Non-Cryst. Solids 256&257 (1999) 6–16.

[1] R. Cai, Y. Du, D. Yang, G. Jia, B. Zhu, B. Chen, Y. Lyu, K. Chen, D. Chen, W. Chen, L. Yang, Y. Zhao, Z. Chen, W. Tan, Free-standing 2D nanorafts by assembly of 1D nanorods for biomolecule sensing, Nanoscale 11 (2019) 12169–12176. [2] N.P. Shetti, S.D. Bukkitgar, K.R. Reddy, ChV. Reddy, T.M. Aminabhavi, ZnObased nanostructured electrodes for electrochemical sensors and biosensors in biomedical applications, Biosens. Bioelectron. 141 (2019), 111417-1-111417-12. [3] N.P. Shettia, S.D. Bukkitgar, K.R. Reddy, ChV. Reddy, T.M. Aminabhavi, Nanostructured titanium oxide hybrids-based electrochemical biosensors for healthcare applications, Colloids Surfaces B Biointerfaces 178 (2019) 385–394. [4] ChV. Reddy, I.N. Reddy, K.R. Reddy, S. Jaesool, K. Yoo, Template-free synthesis of tetragonal Co-doped ZrO2 nanoparticles for applications in electrochemical energy storage and water treatment, Electrochim. Acta 317 (2019) 416–426. [5] V.N. Rao, N.L. Reddy, M.M. Kumari, P. Ravi, M. Sathish, K.M. Kuruvilla, V. Preethi, K.R. Reddy, N.P. Shetti, T.M. Aminabhavi, M.V. Shankar, Photocatalytic recovery of H2 from H2S containing wastewater: surface and interface control of photo-excitons in Cu2S@TiO2 core-shell nanostructures, Appl. Catal. B Environ. 254 (2019) 174–185.

11

D.K. Sharma and S.M. Tripathi

Materials Chemistry and Physics 243 (2020) 122632

[13] B.J. Eggleton, B. Luther-Davies, K. Richardson, Chalcogenide photonics, Nat. Photonics 5 (2011) 141–148. [14] J.S. Sanghera, L.B. Shaw, I.D. Aggarwal, Chalcogenide glass-fiber-based mid-IR sources and applications, IEEE J. Sel. Top. Quant. 15 (2009) 114–119. [15] R. Frerichs, New optical glasses with good transparency in the infrared, J. Opt. Soc. Am. 43 (1953) 1153–1157. [16] C. Conseil, Q. Coulombier, C. Boussard-Pledel, J. Troles, L. Brilland, G. Renversez, D. Mechin, B. Bureau, J.L. Adam, J. Lucas, Chalcogenide step index and microstructured single mode fibers, J. Non-Cryst. Solids 357 (2011) 2480–2483. [17] P. Lucas, G.J. Coleman, S. Jiang, T. Luo, Z. Yang, Chalcogenide glass fibers: optical window tailoring and suitability for bio-chemical sensing, Opt. Mater. 47 (2015) 530–536. [18] Mohajerani, S.-Q. Hui, M. Mirzababaei, A. Arulrajah, S. Horpibulsuk, A.A. Kadir, Md T. Rahman, F. Maghool, Amazing types, properties, and applications of fibres in construction materials, Mater 12 (2019), 2513-1-2513-45. [19] S. Xing, D. Grassani, S. Kharitonov, A. Billat, C.-S. Bres, Characterization and modeling of microstructured chalcogenide fibers for efficient mid-infrared wavelength conversion, Opt. Express 24 (2016) 9741–9750. [20] Y. Wu, M. Meneghetti, J. Troles, J.-L. Adam, Chalcogenide microstructured optical fibers for mid-infrared supercontinuum generation: interest, fabrication, and applications, Appl. Sci. 8 (2018), 1637-1-1637-11. [21] N.S. Kapany, R.J. Simms, ‘‘Recent developments of infrared fiber optics, Infrared Phys. 5 (1965) 69–80. [22] J.W.E. Drewitt, P.S. Salmon, A. Zeidler, C.J. Benmore, A.C. Hannon, Structure of rare-earth chalcogenide glasses by neutron and x-ray diffraction, J. Phys. Condens. Matter 29 (2017), 225703-1-225703-12. [23] Z. Tang, D. Furniss, N.C. Neate, T.M. Benson, A.B. Seddon, “Low gallium-content, dysprosium III-doped, Ge-As-Ga-Se chalcogenide glasses for active mid-infrared fiber optics, J. Am. Ceram. Soc. 102 (2019) 195–206. [24] J. Zheng, L. Li, H. Yin, Y. Wang, H. Zeng, J. Wei, G. Chen, Structural study of chalcogenide glasses in the Bi5As35Se60-xTex system, J. Non-Cryst. Solids 523 (2019), 119606-1-119606-4. [25] Pethes, R. Chahal, V. Nazabal, C. Prestipino, S. Michalik, J. Darpentigny, P. Jovaria, Chemical order in Ge-Ga-Sb-Se glasses, J. Non-Cryst. Solids 484 (2018) 49–56. [26] Yi, P. Zhang, F. Chen, S. Dai, X. Wang, T. Xu, Q. Nie, Fabrication and characterization of Ge20Sb15S65 chalcogenide glass for photonic crystal fibers, Appl. Phys. B 116 (2014) 653–658. [27] Zakery, S.R. Elliott, Optical properties and applications of chalcogenide glasses: a review, J. Non-Cryst. Solids 330 (2003) 1–12. [28] T. Cardinal, K.A. Richardson, H. Shim, A. Schulte, R. Beatty, K. Le Foulgoc, C. Meneghini, J.F. Viens, A. Villeneuve, Non-linear optical properties of chalcogenide glasses in the system As-S-Se, J. Non-Cryst. Solids 256 & 257 (1999) 353–360A. [29] Dahshan, P. Sharma, K.A. Aly, Semiconducting quaternary chalcogenide glasses as new potential thermoelectric materials: an As-Ge-Se-Sb case, Dalton Trans. 44 (2015) 14799–14804. [30] A.S. Hassanien, A.A. Akl, Effect of Se addition on optical and electrical properties of chalcogenide CdSSe thin films, Superlattice Microstruct. 89 (2016) 153–169. [31] Dahshan, H.H. Hegazy, K.A. Aly, P. Sharma, Semiconducting Ge-Se-Sb-Ag chalcogenides as prospective materials for thermoelectric applications, Physica B 526 (2017) 117–121. [32] P. Zhang, J. Zhang, P. Yang, S. Dai, X. Wang, W. Zhang, Fabrication of chalcogenide glass photonic crystal fibers with mechanical drilling, Opt. Fiber Technol. 26 (2015) 176–179. [33] Markos, Photo-induced changes in a hybrid amorphous chalcogenide/silica photonic crystal fiber, Appl. Phys. Lett. 104 (2014), 011114-1-011114-4. [34] P. St, J. Russell, Photonic crystal fibers, J. Light. Technol. 24 (2006) 4729–4749. [35] P. St, J. Russell, Photonic crystal fibers, Science 299 (2003) 358–362. [36] P. Houizot, C. Boussard-Pledel, A.J. Faber, L.K. Cheng, B. Bureau, P.A. V. Nijnatten, W.L.M. Gielesen, J.P. do Carmo, J. Lucas, Infrared single mode chalcogenide glass fiber for space, Opt. Express 15 (2007) 12529–12538. [37] L. Brilland, F. Smektala, G. Renversez, T. Chartier, J. Troles, T.N. Nguyen, N. Traynor, A. Monteville, Fabrication of complex structures of holey fibers in chalcogenide glass, Opt. Express 14 (2006) 1280–1285. [38] E.M. Dianov, V.G. Plotnichenko, Y.N. Pyrkov, I.V. Smolnikov, S.A. Koleskin, G. G. Devyatykh, M.F. Churbanov, G.E. Snopatin, I.V. Skripachev, R. M. Shaposhnikov, Single-mode As-S glass fibers, Inorg. Mater. 39 (2003) 627–630. [39] R. Mossadegh, J.S. Sanghera, D. Schaafsma, B.J. Cole, V.Q. Nguyen, P.E. Miklos, I. D. Aggarwal, Fabrication of single-mode chalcogenide optical fiber, J. Light. Technol. 16 (1998) 214–217. [40] F. Desevedavy, G. Renversez, L. Brilland, P. Houizot, J. Troles, Q. Coulombier, F. Smektala, N. Traynor, J.-L. Adam, Small-core chalcogenide microstructured fibers for the infrared, Appl. Opt. 47 (2008) 6014–6021. [41] N. Ho, M.C. Phillips, H. Qiao, P.J. Allen, K. Krishnaswami, B.J. Riley, T.L. Myers, N.C. Anheier, Single-mode low loss chalcogenide glass waveguides for the midinfrared, Opt. Lett. 31 (2006) 1860–1862. [42] T.A. Birks, P.J. Roberts, P. St, J. Russell, D.M. Atkin, T.J. Shepherd, Full 2-D photonic bandgaps in silica/air structures, Electron. Lett. 31 (1995) 1941–1943. [43] K. Schuster, J. Kobelke, S. Grimm, A. Schwuchow, J. Kirchhof, H. Bartelt, A. Gebhardt, P. Leproux, V. Couderc, W. Urbanczyk, Microstructured fibers with highly nonlinear materials, Opt. Quant. Electron. 39 (2007) 1057–1069. [44] F. Desevedavy, G. Renversez, J. Troles, L. Brilland, P. Houizot, Q. Coulombier, F. Smektala, N. Traynor, J.-L. Adam, Te-As-Se glass microstructured optical fiber for the middle infrared, Appl. Opt. 48 (2009) 3860–3865.

[45] M. Liao, C. Chaudhari, G. Qin, X. Yan, C. Kito, T. Suzuki, Y. Ohishi, M. Matsumoto, T. Misumi, Fabrication and characterization of a chalcogenidetellurite composite microstructure fiber with high nonlinearity, Opt. Express 17 (2009) 21608–21614. [46] Q. Coulombier, L. Brilland, P. Houizot, T. Chartier, T.N. N’Guyen, F. Smektala, G. Renversez, A. Monteville, D. Mechin, T. Pain, H. Orain, J.-C. Sangleboeuf, J. Troles, Casting method for producing low-loss chalcogenide microstructured optical fibers, Opt. Express 18 (2010) 9107–9112. [47] T.M. Monro, Y.D. West, D.W. Hewak, N.G.R. Broderick, D.J. Richardson, Chalcogenide holey fibres, Electron. Lett. 36 (2000) 1998–2000. [48] X. Feng, A.K. Mairaj, D.W. Hewak, T.M. Monro, Nonsilica glasses for holey fibers, J. Light. Technol. 23 (2005) 2046–2054. [49] M. El-Amraoui, G. Gadret, J.C. Jules, J. Fatome, C. Fortier, F. Desevedavy, I. Skripatchev, Y. Messaddeq, J. Troles, L. Brilland, W. Gao, T. Suzuki, Y. Ohishi, F. Smektala, Microstructured chalcogenide optical fibers from As2S3 glass: towards new IR broadband sources, Opt. Express 18 (2010) 26655–26665. [50] M.V. Eijkelenborg, M. Large, A. Argyros, J. Zagari, S. Manos, N. Issa, I. Bassett, S. Fleming, R. McPhedran, C.M. Sterke, N.A. Nicorovici, Microstructured polymer optical fibre, Opt. Express 9 (2001) 319–327. [51] V.S. Shiryaev, Chalcogenide glass hollow-core microstructured optical fibers, Front. Mater. 2 (2015) 1–10. [52] A.F. Oskooi, J.D. Joannopoulos, S.G. Johnson, Zero-group-velocity modes in chalcogenide holey photonic crystal fibers, Opt. Express 17 (2009) 10082–10090. [53] J. Hou, D.S. Citrin, Z. Cao, C. Yang, Z. Zhong, S. Chen, Slow light in square-lattice chalcogenide photonic crystal holey fibers, IEEE J. Sel. Top. Quantum Electron. 22 (2016) 4900108. [54] Y. Wang, S. Dai, X. Peng, P. Zhang, X. Wang, C. You, Mid-infrared supercontinuum generation spanning from 1.9 to 5.7μm in a chalcogenide fiber taper with ultra-high NA, Infrared Phys. Technol. 88 (2018) 102–105. [55] M. Asobe, Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching, Opt. Fiber Technol. 3 (1997) 142–148. [56] A.N. Ghosh, M. Meneghetti, C.R. Petersen, O. Bang, L. Brilland, S. Venck, J. Troles, J.M. Dudley, T. Sylvestre, Chalcogenide-glass polarization-maintaining photonic crystal fiber for mid-infrared supercontinuum generation, J. Phys.: Photonics 1 (2019), 044003-1-044003-12. [57] V.S. Shiryaev, Chalcogenide glass hollow-core microstructured optical fibers, Front. Mater. 2 (2015) 1–10. [58] V. Lakshminarayanan, S. Varadharajan, Approximate solutions to the scalar wave equation: the decomposition method, J. Opt. Soc. Am. A 15 (1998) 1394–1400. [59] P. Varga, P. Torok, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation, Opt. Commun. 152 (1998) 108–118. [60] T. Hirooka, Y. Hori, M. Nakazawa, Gaussian and Sech approximations of mode field profiles in photonic crystal fibres, IEEE Photonics Technol. Lett. 16 (2004) 1071–1073. [61] R.E. Wagner, W.J. Tomlinson, Coupling efficiency of optics in single-mode fibre components, Appl. Opt. 21 (1982) 2671–2687. [62] Ruilier, F. Cassaing, Coupling of large telescope and single mode waveguides: application to stellar interferometry, J. Opt. Soc. Am. A 18 (2001) 143–149. [63] Y. Dikmelik, F.M. Davidson, Fibre-coupling efficiency for free space optical communication through atmospheric turbulence, Appl. Opt. 44 (2005) 4946–4952. [64] Marcuse, Gaussian approximation of the fundamental modes of graded-index fibers, J. Opt. Soc. Am. 68 (1978) 103–109. [65] Gloge, Weakly guiding fibers, Appl. Opt. 10 (1971) 2252–2258. [66] Marcuse, Loss analysis of single-mode fiber splices, Bell Syst. Tech. J. 56 (1977) 703–718. [67] Chen, H. Yang, H. Wang, S. Tong, Y. Lou, Coupling plane wave received by an annular aperture into a single-mode fibre in the presence of atmospheric turbulence, Appl. Opt. 50 (2011) 307–312. [68] J.C.W. Corbett, J.R. Allington-Smith, Coupling starlight into single-mode photonic crystal fibre using a field lens, Opt. Express 13 (2005) 6527–6540. [69] J. Corbett, A. Dabirian, T. Butterley, N.A. Mortensen, J.R. Allington-Smith, The coupling performance of photonic crystal fibres in fibre stellar interferometry, Mon. Not. R. Astron. Soc. 368 (2006) 203–210. [70] J.C.W. Corbett, T.J. Morris, J.R. Allington-Smith, Tip-tilt requirements for coupling starlight into single-mode photonic crystal fibres using a lenslet: a first analysis, New Astron. Rev. 49 (2006) 675–680. [71] M. Koshiba, K. Saitoh, Structural dependence of effective area and mode field diameter for holey fibres, Opt. Express 11 (2003) 1746–1756. [72] N.A. Mortensen, J.R. Folkenberg, M.D. Nielsen, K.P. Hansen, Modal cut-off and the V parameter in photonic crystal fibres, Opt. Lett. 28 (2003) 1879–1881. [73] J.R. Folkenberg, N.A. Mortensen, K.P. Hansen, T.P. Hansen, H.R. Simonsen, C. Jakobsen, Experimental investigation of cut off phenomena in nonlinear photonic crystal fibres, Opt. Lett. 28 (2003) 1882–1884. [74] Sharma, H. Chauhan, A new analytical model for the field of microstructured optical fibres, Opt. Quant. Electron. 41 (2009) 235–242. [75] Ghosh, S. Roy, S.K. Bhadra, Determination of modal effective indices and dispersion of microstructured fibres with different configurations: a variational approach, J. Mod. Opt. 57 (2010) 607–620. [76] L. Zhang, Z. Wu, S. Gao, M. Cui, Study of a constructed function for approximating mode field in photonic crystal fibres, Opt. Eng. 51 (2012), 065003-1 - 065003-11. [77] D.K. Sharma, A. Sharma, Characteristic of microstructured optical fibers: an analytical approach, Opt. Quant. Electron. 44 (2012) 415–424.

12

D.K. Sharma and S.M. Tripathi

Materials Chemistry and Physics 243 (2020) 122632

[78] D.K. Sharma, A. Sharma, Splicing of index-guiding microstructured optical fibers and single-mode fibers by controlled air-hole collapse: an analytical approach, Opt. Quant. Electron. 46 (2014) 409–422. [79] D.K. Sharma, A. Sharma, S.M. Tripathi, Thermo-optic characteristics of hybrid polymer/silica microstructured optical fiber: an analytical approach, Opt. Mater. 78 (2018) 508–520. [80] D.K. Sharma, A. Sharma, S.M. Tripathi, Microstructured optical fibers for terahertz waveguiding regime by using an analytical field model, Opt. Fiber Technol. 39 (2017) 55–69. [81] Sharma, Optimal Variational Method for Rectangular and Channel Waveguides,” Guided Wave Optics; Selected Topics, Viva Books, New Delhi, 2005. [82] A.W. Snyder, J.D. Love, Optical Waveguide Theory, Chapman & Hall, London, 1983. [83] Sharma, J.-P. Meunier, On the scalar modal analysis of optical waveguides using approximate methods, Opt. Commun. 281 (2007) 592–599. [84] Sharma, S.I. Hosain, A. K Ghatak, The fundamental mode of graded-index fibers: simple and accurate variational methods, Opt. Quant. Electron. 14 (1981) 7–15. [85] Sharma, A.K. Ghatak, A variational analysis of single mode graded-index fibers, Opt. Commun. 36 (1981) 22–24. [86] T.A. Birks, J.C. Knight, P. St, J. Russell, Endlessly single-mode photonic crystal fiber, Opt. Lett. 22 (1997) 961–963. [87] P.J. Roberts, B.J. Mangan, H. Sabert, F. Couny, T.A. Birks, J.C. Knight, P. St, J. Russell, Control of dispersion in photonic crystal fibers, J. Opt. Fiber Commun. Rep. 2 (2005) 435–461. [88] M. Seifouri, M. Dekamina, S. Olyaee, A new circular chalcogenide/silica hybrid microstructured optical fiber with high negative dispersion for the purpose of dispersion compensation, Optik 126 (2015) 3093–3098. [89] J.S. Sanghera, L.B. Shaw, I.D. Aggarwal, Applications of chalcogenide glass optical fibers, C.R. Chimie 5 (2002) 873–883. [90] Y. Guimond, J.L. Adam, A.M. Jurdyc, H.L. Ma, J. Mugnier, B. Jacquier, Optical properties of antimony-stabilised sulphide glasses doped with Dy and Er ions, J. Non-Cryst. Solids 256 & 257 (1999) 378–382. [91] M. Koshiba, K. Saitoh, Applicability of classical optical fiber theories to holey fibers, Opt. Lett. 29 (2004) 1739–1741. [92] N.A. Mortensen, Effective area of photonic crystal fibers, Opt. Express 10 (2002) 341–348. [93] N.A. Mortensen, J.R. Folken, P.M.W. Skovgaard, J. Broeng, Numerical aperture of single-mode photonic crystal fibers, IEEE Photonics Technol. Lett. 14 (2002) 1094–1096. [94] J. Troles, F. Smektala, L. Brillant, P. Houizot, F. Desevedavy, G. Boudebs, T. Chartier, N. Traynor, Preparation of chalcogenide step-index and microstructured fiber for telecommunication, Tu.C2 4 (2006) 112–115. ICTON. [95] Theberge, N. Berube, S. Poulain, S. Cozic, S. Chatigny, L.-R. Robichaud, L.P. Pleau, M. Bernier, R. Vallee, Infrared supercontinuum generated in concatenated InF3 and As2Se3 fibers, Opt. Express 26 (2018) 13952–13960. [96] K. Guo, R.A. Martinez, G. Plant, L. Maksymiuk, B. Janiszewski, M.J. Freeman, R. L. Maynard, M.N. Islam, F.L. Terry, R. Bedford, R. Gibson, F. Chenard, S. Chatigny, A.I. Ifarraguerri, Generation of near-diffraction-limited, high-power supercontinuum from 1.57μm to 12μm with cascaded fluoride and chalcogenide fibers, Appl. Opt. 57 (2018) 2519–2532. [97] A.K. Ghatak, K. Thyagarajan, Introduction to Fiber Optics, Cambridge University Press, Cambridge, 1998. [98] Ghatak, K. Thyagarajan, Optical Electronics, Cambridge University Press, Cambridge, 1989. [99] N.A. Mortensen, J.R. Folkenberg, Near-field to far-field transition of photonic crystal fibers: symmetries and interference phenomena, Opt. Express 10 (2002) 475–481. [100] Renversez, F. Bordas, B.T. Kuhlmey, Second mode transition in microstructured optical fibers: determination of the critical geometrical parameter and study of the matrix refractive index and effects of cladding size, Opt. Lett. 30 (2005) 1264–1266. [101] D.K. Sharma, S.M. Tripathi, A. Sharma, Modal analysis of high-index core telluirite glass microstructured optical fibers in infrared regime, J. Non-Cryst. Solids. 511 (2019) 147–160. [102] S. Shabahang, G. Tao, J.J. Kaufman, A.F. Abouraddy, Dispersion characterization of chalcogenide bulk glass, composite fibers, and robust nanotapers, J. Opt. Soc. Am. B 30 (2013) 2498–2506. [103] M.S. Habib, M.S. Habib, M.I. Hasan, S.M.A. Razzak, R.R. Mahmud, Y. Namihira, Microstructured holey fibers as wideband dispersion compensating media for high speed transmission system, J. Opt. 124 (2013) 4984–4988. [104] M.S. Habib, M.S. Habib, S.M.A. Razzak, Y. Namihira, M.A. Hossain, M.A.G. Khan, Broadband dispersion compensation of conventional single mode fibers using microstructured optical fibers, J. Opt. 124 (2013) 3851–3855.

[105] M.I. Hasan, M.S. Habib, M.S. Habib, S.M.A. Razzak, Highly nonlinear and highly birefringent dispersion compensating photonic crystal fiber, J. Opt. Fiber Technol. 20 (2014) 32–38. [106] M.S. Habib, M.S. Habib, S.M.A. Razzak, M.A. Hossain, Proposal for highly birefringent broadband dispersion compensating octagonal photonic crystal fiber, J. Opt. Fiber Technol. 12 (2013) 461–467. [107] Boudebs, S. Cherukulappurath, M. Guignard, J. Troles, F. Smektala, F. Sanchez, Linear optical characterization of chalcogenide glasses, Opt. Commun. 230 (2004) 331–336. [108] M.M. Haque, M.S. Rahman, M.S. Habib, M.S. Habib, S.M.A. Razzak, A new circular photonic crystal fiber for effective dispersion compensation over E to L wavelength bands, J. Microwaves Optoelectron. Electromagn. 12 (2013) 1074–2179. [109] Begum, Y. Namihira, S.M.A. Razzak, S. Kaijage, N.H. Hai, T. Kinjo, K. Miyagi, N. Zou, Novel broadband dispersion compensating photonic crystal fibers: applications in high-speed transmission systems, J. Opt. Laser Technol. 41 (2009) 679–686. [110] M.I. Hasan, M.S. Habib, M.S. Habib, S.M.A. Razzak, Design of hybrid photonic crystal fiber: polarization and dispersion properties, J. Photonics Nanostruct. Fundam. Appl. 12 (2014) 205–211. [111] Y. Liu, J. Wang Li, R. Wang, J. Li, X. Xie, A novel hybrid photonic crystal dispersion compensating fiber with multiple windows, J. Opt. Laser Technol. 44 (2012) 2076–2079. [112] N. Granzow, P. Uebel, M.A. Schmidt, A.S. Tverjanovich, L. Wondraczek, P. St, J. Russell, Bandgap guidance in hybrid chalcogenide-silica photonic crystal fibers, Opt. Lett. 36 (2011) 2432–2434. [113] W. Yuan, 2-10μm mid-infrared supercontinuum generation in As2Se3 photonic crystal fiber, Laser Phys. Lett. 10 (2013), 095107-1 - 095107-6. [114] Quemard, F. Smektala, V. Couderc, A. Barthelemy, J. Lucas, Chalcogenide glasses with high nonlinear optical properties for telecommunications, J. Phys. Chem. Solids 62 (2001) 1435–1440. [115] J.T. Gopinath, M. Solajcic, E.P. Ippen, Third order nonlinearities in Ge-As-Se based glasses for telecommunications applications, J. Appl. Phys. 96 (2004) 6931–6933. [116] J.M. Harbold, F.O. Ilday, F.W. Wise, J.S. Sanghera, V.Q. Nguyen, L.B. Shaw, I. D. Aggarwal, Highly nonlinear As-S-Se glasses for all-optical switching, Opt. Lett. 27 (2002) 119–121. [117] M.A. Foster, K.D. Moll, A.L. Gaeta, Optimal waveguide dimensions for nonlinear interactions, Opt. Express 12 (2004) 2880–2887. [118] H.C. Nguyen, B.T. Kuhlmey, M.J. Steel, C.L. Smith, E.C. Magi, R.C. McPhedran, B. J. Eggleton, Leakage of the fundamental mode in photonic crystal fiber tapers, Opt. Lett. 30 (2005) 1123–1125. [119] T.M. Monro, H. Ebendorff-Heidepriem, Progress in microstructured optical fibers, Annu. Rev. Mater. Res. 36 (2006) 467–495. [120] J.S. Sanghera, C.M. Florea, L.B. Shaw, P. Pureza, V.Q. Nguyen, M. Bashkansky, Z. Dutton, I.D. Aggarwal, Non-linear properties of chalcogenide glasses and fibers, J. Non-Cryst. Solids 354 (2008) 462–467. [121] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, 2001. [122] Laegsgaard, N.A. Mortenson, J. Riishede, A. Bjarklev, Material effects in airguiding photonic bandgap fibers, J. Opt. Soc. Am. B 20 (2003) 2046–2051. [123] E.C. Magi, L.B. Fu, H.C. Nguyen, M.R.E. Lamont, D.I. Yeom, B.J. Eggleton, Enhanced Kerr nonlinearity in sub-wavelength diameter As2Se3 chalcogenide fiber tapers, Opt. Express 15 (2007) 10324–10329. [124] R. Cherif, M. Zghal, L. Tartara, Characterization of stimulated Brillouin scattering in small core microstructured chalcogenide fiber, Opt. Commun. 285 (2012) 341–346. [125] K.R. Khan, K. Mnaymneh, H. Awad, I. Hasand, Slow light propagation in tunable nanoscale photonic crystal cavity filled with nematic liquid crystal, Opt. Eng. 53 (2014), 102705-1 - 102705-7. [126] O. Frazao, J.L. Santos, F.M. Araujo, L.A. Ferreira, Optical sensing with photonic crystal fibers, Laser Photonics Rev. 2 (2008) 449–459. [127] T.M. Monro, D.J. Richardson, P.J. Bennett, Developing holey fibres for evanescent field devices, Electron. Lett. 35 (1999) 1188–1189. [128] Pickrell, W. Peng, A. Wang, Random-hole optical fiber evanescent wave gas sensing, Opt. Lett. 29 (2004) 1476–1478. [129] Y.L. Hoo, W. Jin, H.L. Ho, D.N. Wang, R.S. Windeler, Evanescent wave gas sensing using microstructure fiber, Opt. Eng. 41 (2002) 8–9. [130] C.M.B. Cordeiro, M.A.R. Franco, G. Chesini, E.C.S. Barretto, R. Lwin, C.H.B. Cruz, M.C.J. Large, Microstructured core optical fiber for evanescent sensing applications, Opt. Express 14 (2006) 13056–13066.

13