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Abstract
We propose a new approach to developing of graded-index chalcogenide fibers. Since chalcogenide glasses are incompatible with current vapor deposition techniques, the arbitrary refractive index gradient is obtained by means of core nanostructurization by the effective medium approach. We study the influence of graded-index core profile and the core diameter on the fiber dispersion characteristics. Flat, normal dispersion profiles across the mid-infrared transmission window of the assumed glasses are easily obtained for the investigated core nanostructure layouts. Nonlinear propagation simulations enable to expect 3.5-8.5 µm spectrum of coherent, pulse preserving supercontinuum. Fabrication feasibility of the proposed fiber is also discussed.
© 2017 Optical Society of America
1. Introduction
Chalcogenide glass optical fibers are of keen interest in context of supercontinuum generation (SCG) light sources for a variety of applications such as fluorescence microscopy [1], optical coherence tomography [2], spectroscopy [3] or early cancer diagnostics [4], due to the potential to cover the biological and chemical “fingerprint” region of the electromagnetic spectrum. The most advantageous optical properties of chalcogenide glasses are very high nonlinearities (several orders of magnitude higher than silica) and very broad transmission range, even up to 25 μm [5]. However their use is limited by the complexity of currently available high power pump lasers operating at wavelengths compatible with their transmission window. Since the zero-dispersion wavelength (ZDW) of bulk chalcogenide glass, e.g. As2Se3, is located at around 7.4 μm, it is challenging to fabricate fibers with ZDW that matches any robust and commercially available high peak-power laser sources. Only few types of step-index and all-solid microstructured chalcogenide fibers were successfully fabricated and used for SCG.
First chalcogenide step-index fibers were demonstrated already in 1980’ [6,7]. More recently Gattass et al. presented a step-index As2S3 chalcogenide fiber in the normal dispersion SCG application [8]. The spectrum, which spanned 1.9 ÷ 4.8 μm, was generated by an initial pulse with 2.45 μm center wavelength, 40 ps duration and 200 mW average power. In 2014 Petersen et al. presented a step-index chalcogenide fiber made of Ge10As23.4Se66.6 and As40Se60 (atomic %) with elliptical core 16.5x15.7 μm, along with SCG measurements [9]. However the ellipticity of the core had no significant impact on fiber dispersion and the fiber had anomalous dispersion at the 6.3 μm pump wavelength used in the SCG experiment. Yu et al. presented a step-index chalcogenide fiber made of GeAsSe and GeAsS glasses along with SC experiment, using a pump pulse with 330 fs duration, 3 kW peak power and 4 μm center wavelength [10]. Though the obtained SCG spanned more than two octaves (1.8 ÷ 10 μm), its coherence properties were poor due to anomalous dispersion of the fiber. Godin et al. presented the influence of tapering the chalcogenide step-index fiber made of As2Se3 on fiber chromatic dispersion [11]. By changing the taper waist from 3 μm to 16 μm, they were able to shift the dispersion by 200 ps/nm/km into normal regime, moving the ZDW beyond 4 μm. Recently, Robichaud et al. presented a normal dispersion SCG spanning 3 ÷ 8 μm, generated in a commercially available step-index AsSe fiber [12]. The fiber was pumped with an in-amplifier SC source extending up to 4.2 μm, leading to SC with an average output power of 1.5 mW. Yet in another work, Nagasaka et al. reported a double-clad chalcogenide fiber with normal dispersion, made of As2Se3, AsSe2 and As2S5 glasses, along with SCG experiments [13] The SC spanning 2 ÷ 14 μm has been obtained with 200 fs pulse centered at 10 μm wavelength and maximum coupled peak power of 1.3 MW.
One of the first fabricated microstructured all-solid chalcogenide glass fibers, which successfully transmitted light, was presented by Lian et al. in 2009 [14]. The fiber, made of Ge10As23.4Se66.6 and As40Se60, had 12.5 μm core diameter and seven 7 μm glass inclusions. In 2016 Liu et al. reported on an all-normal (ANDi) all-solid chalcogenide fiber made of AsSe2-As2S5 glasses with four 3.5 μm glass inclusions and 2.4 μm core [15]. By pumping a 2 cm long sample at 2.7 μm wavelength, they obtained SCG spanning wavelengths from 2.2 μm to 3.3 μm. The pumping pulse had 200 fs duration, 80 MHz repetition rate and 5.2 kW peak power.
The use of chalcogenide glasses for optical fiber fabrication and SCG is still limited by technological difficulties i.e. glass susceptibility to crystallization, glass toxicity, stringent purity requirements for the raw materials and low mechanical resistance of the final fibers [16]. Despite technological difficulties, many chalcogenide fiber structures including step-index fibers, suspended-core photonic crystal fibers (PCF) or regular lattice PCFs are now routinely fabricated and offered commercially by several vendors [17,18]. One of such step-index fibers were used by Kedenburg et al. for SCG experiments [19]. Using pump pulses centered at 3.83 μm wavelength, 300-450 fs duration and with 550 mW average power, a SCG in normal dispersion regime was obtained, spanning 2.9 ÷ 4.8 μm. In general, SCG in normal dispersion regime requires engineered, flattened curve of fiber dispersion to be efficient [20,21]. However, step-index structures allow only limited dispersion engineering margins. On the other hand, PCF fiber structures, typically with many rings of small glass inclusions (e.g. hexagonal lattice PCFs), allow a higher degree of dispersion shaping, but are difficult to fabricate. Therefore, new methods of chalcogenide fiber dispersion engineering and fiber fabrication are needed.
Graded-index fibers are the well-known alternative approach to dispersion and mode area engineering in fiber optics, enabling e.g. enhanced modal loss control and shaping of fundamental mode dispersion in large (~50 μm) core fibers, compared to step-index fibers [22]. Silica graded-index fibers are commonly fabricated with vapor deposition (VD) techniques [22]. Chalcogenide glass, on the other hand, is incompatible with current VD. Therefore a different method of obtaining refractive index gradient in the core is necessary for the chalcogenide glass platform.
In this work we present a novel approach to the development of chalcogenide glass graded-index fibers. We propose a graded-index core chalcogenide fiber made of thermally matched AsSe-GeAsSe glasses. Refractive index gradient of the fiber core is obtained by means of nanostructurization, where effective medium theory (EMT) is applied [23]. Using numerical simulations we show that mid-infrared, broad and flat dispersion profiles in chalcogenide graded-index fibers are possible in the normal range of values. Influence of various graded-index core profiles of refractive index and various core diameters on the shape of fiber dispersion characteristics is discussed. With these results we perform nonlinear propagation simulations and obtain flat, coherent supercontinuum spectra covering 3.5 ÷ 8.5 µm under 6.3 µm femtosecond pumping. Practical aspects of fabrication of the proposed fiber with nanostructured core are also discussed.
2. Graded-index core chalcogenide fiber concept
The advantage of the graded-index fiber approach is the possibility to extend the degree of freedom in dispersion profile engineering, as compared to step-index fibers. Chromatic dispersion of a step-index fiber made of one particular pair of glasses can be engineered only by varying the core diameter. Chromatic dispersion of graded-index core fibers on the other hand, can be engineered also by changing the refractive index profile inside the fiber core.
Our discussion is focused on a graded-index core chalcogenide fiber structure made of thermally matched Ge10As23.4Se66.6 and As40Se60 (% mass) glasses. Material dispersion n(λ) and dispersion D(λ) of both glasses, calculated based on Sellmeier coefficients [24] (Table 1), are presented in Figs. 1(a) and 1(b) respectively. Dispersion profiles of both considered glasses have ZDW around 7.0-7.5 μm - Fig. 1(b) inset, for which the difference between their refractive indices is Δn = 0.2. The nonlinear refractive indices of the glasses measured at 1064 nm wavelength are n2(As40Se60) ≈18·10−18 m2/W and n2(Ge10As23.4Se66.6) ≈14·10−18 m2/W [25]. This particular pair of glasses has been chosen, because it has already been reported, that they are suitable for fiber drawing and SCG experiments [9,14].
Table 1. Sellmeier coefficients of the AsSe and GeAsSe glassesa.
Fig. 1 Calculated (a) material dispersion and (b) dispersion D of Ge10As23.4Se66.6 and As40Se60 chalcogenide glasses.
The means to reach the proposed graded-index fiber core is to introduce nanostructure and the effective medium approach [23]. Physically, the fiber preform is obtained by the stack-and-draw method, similarly to classic microstructured fibers. We have recently validated this approach with demonstration of physically fabricated, nanostructured core graded-index fibers on the silicate glass platform [26]. In the following study of the chalcogenide glass graded-index fiber, the diameter of the core dc has been varied from 6 to 20 μm and four different profiles of refractive index have been assumed: referential step-index, as well as linear, parabolic and x4 profiles, as shown in Figs. 2(a)-2(d), respectively. Flattened, normal dispersion characteristic extending over the glass transmission window has been set as the design criterion. The graded-index profiles shown in Figs. 2(b)-2(d), have been assumed in a way, that the center of the circular core contains only the higher-index glass AsSe and outer rim of the core (at distance dc/2) contains only the lower-index glass GeAsSe. Four distinct sets of fiber structures were then used in linear simulations of chromatic dispersion in fibers, as well as in nonlinear simulations of SCG, which follow later.
Fig. 2 Scheme of effective refractive index profile of the nanostructured fiber core. (a) step-index, (b) linear, (c) parabolic, (d) x4.
3. Results of linear and nonlinear simulations
Linear simulations of the dispersion profiles of the considered graded-index fibers were performed with finite difference method (FDE) [27]. First, dispersion characteristics of chalcogenide fibers made of AsSe-GeAsSe glasses with four profiles of refractive index – shown in Figs. 2(a)-2(d) – and with core diameter of dc = 16 μm, were calculated. The core diameter was assumed to be the same as core of a chalcogenide step-index fiber presented in the work of Petersen et al. [9]. The results of calculations are shown in Fig. 3. Dispersion profile of the step-index fiber, investigated here to validate the used model, has a ZDW at 5.8 μm, and at longer wavelengths the dispersion increases monotonically in the whole wavelength range of calculation. It matches the shape of the step-index fiber dispersion reported physically earlier [9], hence the result proves the correctness of the dispersion simulations and of the used FDE method. All four dispersion profiles shown in Fig. 3 are similar, and the fiber with the linear refractive index profile has the most red-shifted ZDW in the line-up. Increasing the exponent in the refractive index profile function from x1 (linear) up to x4 results in blue-shifting of the ZDW and shifts the dispersion towards the dispersion of the step-index fiber. In case of the analyzed refractive index profiles, the lowest exponent function describing the index gradient – the linear profile in Fig. 3 – results in the broadest flattened part of the dispersion curve. Therefore, changing the profile of refractive index inside fiber core shifts position of the ZDW and changes the slope of the dispersion curve.
Fig. 3 Comparison of a graded-index core chalcogenide fiber chromatic dispersion. Assumed step-index, linear, parabolic and x4 profiles of refractive index of the core with dc = 16 μm.
In the next step, the dispersion characteristics of chalcogenide fibers with various core diameters were calculated for the four refractive index profiles, and results are presented in Fig. 4. The core diameter dc has been varied in the range of 6 ÷ 20 μm. In all cases of refractive index profile shapes the increase of nanostructured core diameter dc from 6 μm to 20 μm shifted the dispersion into the anomalous range of values, and it broadened the flattened part of the curve. Moreover, the long-wavelength part of the dispersion characteristic shifts its position, initially toward the longer wavelengths, but above a certain diameter (≥10 μm) this is reversed and the dispersion profile shifts back towards shorter wavelengths with increasing nanostructured core diameter. The explanation of this effect is as follows. In case of a fiber with a small core diameter of e.g. 6 μm, with the increasing effective mode area for longer wavelengths, an increasing amount of energy of the propagating mode leaks out of the core into the cladding area. An increasing influence of the material dispersion of the cladding glass (GeAsSe) at the longer wavelengths causes a dramatic change of fiber dispersion. In the case of fibers with dc = 10 μm all energy of the mode, despite the increase of the effective mode area , is still contained inside the nanostructured core. This results in dispersion, which is flat in a broader range of wavelengths, than dispersion of fibers with smaller cores. In case of graded-index core fibers with increasing core diameter dc > 16 μm, the propagating mode is mainly influenced by material of the core, therefore, the shape of chromatic dispersion is similar to the dispersion of the AsSe core glass.
Fig. 4 Numerically obtained dispersion characteristics of chalcogenide fiber with nanostructured graded-index core for various core diameters dc. (a) step-index, (b) linear, (c) parabolic and (d) x4 profile of refractive index.
Results of dispersion simulations for various nanostructured core diameters dc and refractive index profiles have shown, that there is an optimal value of dc, for which the flattened part of normal dispersion is the broadest. Chalcogenide graded-index core fibers with parabolic profile and dc = 10 μm and with x4 profile and dc = 8 μm have broad flat part of normal dispersion characteristic – spectrally the broadest from among the investigated gradient profiles of refractive index – which suggest interesting properties for coherent, pulse preserving SC generation.
These properties are discussed by numerically solving the Generalized one-dimensional Nonlinear Schrödinger Equation (GNLSE) with use of the split-step Fourier method [28] for each of the gradient index profiles shown in Fig. 2. The range of core diameters was limited to dc = 8 and 10 μm for which the most attractive dispersion characteristics were obtained in linear simulations. The step-index profile has been treated as a reference, because experimental results of SCG in such fibers are available [9]. For the purpose of SCG simulations a nonlinear refractive index of the core equal n2 ≈16·10−18 m2/W has been assumed, as a mean value of n2 of both chalcogenide glasses. Assumed pump pulse had 6.3 μm central wavelength, 200 fs duration, 100 kHz repetition rate and 1 nJ input energy, which was two orders of magnitude lower than in [9] – 229 nJ .
The pump wavelength of 6.3 µm for SCG has been chosen based on the shape of the curves of material dispersion of the investigated glasses – Fig. 1(b), as well as shape of the curve of chromatic dispersion of the final graded-index core fiber – Fig. 4. For efficient SCG the pump wavelength should be located in the vicinity of the wavelengths, for which dispersion curve of the fiber is flat. However, the chalcogenide glasses for GRIN fiber fabrication need to be thermally and rheologically matched, i.e. thermal expansion coefficient and viscosity of both glasses has to be similar for the temperature in which the drawing process is performed. Therefore, the shape of the chromatic dispersion of the fiber, hence the optimum position of the pump wavelength, is limited by the choice of the glasses. Based on a successful all-solid chalcogenide fiber fabrication presented in [9], the AsSe and GeAsSe glasses have been selected for GRIN fiber fabrication, since it is proved that they are thermally matched. Moreover, the pump wavelength of 6.3 μm, used by Petersen et al. in [9] should be optimal in case of the investigated GRIN fiber from the point of view of the fiber’s dispersion profile (dispersion curve is flat around 6.3 μm wavelength – Fig. 12 in the revised version of the manuscript). Although admittedly, the current state-of-the-art in femtosecond laser technology necessitates use of a complex OPA system as the pump source.
Nonlinear simulations of SCG in the exemplary investigated fiber with dc = 10 μm and parabolic profile of refractive index were performed assuming pump pulses centered at 2.8 μm – source reported by [29] and – 3.5 μm, source reported by [30], which is feasible with an erbium-doped ZBLAN fiber laser with parabolic profile of refractive index. The comparative results of SCG for pump wavelength of 2.8 μm, 3.5 μm and 6.3 μm are presented in Fig. 5. The pump pulse had 1 nJ input energy, 200 fs duration and 100 kHz repetition rate.
Fig. 5 SC spectra obtained in nanostructured graded-index core chalcogenide fiber, pumped with 200 fs duration, 1 nJ input energy and 100 kHz repetition rate for various pump wavelengths.
SC spectra for all pump wavelengths, presented in Fig. 5, are spanning over one octave. As expected, the spectrum obtained for 6.3 μm pump wavelength (red curve in Fig. 5) is the broadest. It is also the flattest of the three spectra, because of the most favorable dispersion properties of the fiber around this pump wavelength. Our numerical SC results are comparable to the spectral width obtained for a similar wavelength (3.83 μm) experimentally by Kedenburg et al. [19], although that particular experiment involved a dozen nJ pump pulse energy coupled into a commercial chalcogenide step-index fiber. While the longest assumed pump wavelength enabled the broadest numerical SC spectrum, even the two shorter pump wavelengths produced coherent SC spectra in the proposed GRIN fibers, which covered wavelengths attractive for infrared spectroscopy applications. Our nonlinear simulations further suggested, that with the proposed GRIN chalcogenide fiber, these SC pulses would be feasible with a factor of 10 smaller in-coupled energy, compared to experimental results with chalcogenide step-index fibers [19]. This in turn is of importance for the perspective of considering the emerging mid-infrared Er:ZBLAN mode-locked lasers as pump sources for our fiber [30].
The results of SCG simulations in 2 cm long chalcogenide nanostructured graded-index core fibers sample with dc = 8 μm and 10 μm were presented in Figs. 6(a) and 6(b) respectively. Additionally, nonlinear coefficients γ calculated for the selected fiber with dc = 10 μm and all four discussed profiles of refractive index are presented in Fig. 7. The characteristic length scales for fiber with e.g. parabolic profile and 10 μm core are: dispersion length LD = 1.6 m, nonlinearity length LNL = 0.4 mm and MI length LMI ~16LNL = 6.4 mm [28].
Fig. 6 SC generated in graded-index core chalcogenide fibers with three gradient profiles of refractive index in the core, step-index fiber as a reference and with core diameter (a) dc = 8 μm and (b) dc = 10 μm. Sample length 2 cm, pump pulse: central wavelength 6.3 μm, 1 nJ input energy, 200 fs duration, 100 kHz repetition rate.
Fig. 7 Nonlinear coefficient γ of graded-index core chalcogenide fibers with three gradient profiles of refractive index in the core, step-index fiber as a reference and dc = 10 μm.
In both analyzed cases of fiber core diameter, shown in Figs. 6(a) and 6(b), the referential step-index fiber – shown with black curve – exhibits anomalous dispersion for the 6.3 μm pump wavelength. Due to very high nonlinearity – Fig. 6, even for low pulse energies a characteristic structure of SC intensity can be seen, i.e. modulation instability (MI) sidebands around 5.5-5.7 μm and 6.8 μm wavelengths and a dispersive wave at 4-4.4 μm. Changing the profile of refractive index inside the core from a step-index to a gradient, shifts the dispersion into the normal dispersion range. SC generated in fibers with linear and parabolic refractive index profiles have clear self-phase modulation (SPM) spectral components in the central part of the spectrum and spectral side wings, generated by optical wave breaking (OWB) induced four-wave mixing (FWM). Due to quite short 200 fs pulse and very short fiber sample length, the slow effect of stimulated Raman scattering (SRS) does not occur in the presented case of SCG in the normal dispersion regime. However, increasing the order of the refractive index profile function from linear to x4 shifts the dispersion back towards the anomalous regime. Fiber with dc = 8 μm and x4 profile inside the core – blue curve in Fig. 6(a) – has very small dispersion near the pump wavelength (a few ps/nm/km). Under conditions of very low group delay, the FWM gain bands can be expected to be very broad. Therefore, a beating between FWM and OWB spectral components can be seen already at the stage, where otherwise (i.e. larger group delay) spectral broadening would still occur solely due to SPM. Dispersion of fiber with dc = 10 μm and x4 profile, shown with a blue trace in Fig. 6(b), has two ZDWs. In this case the pump wavelength was located between the ZDWs in anomalous dispersion, which is also used for generation of coherent SC under femtosecond pumping [31,32], albeit a more complex (than pure ANDi operation) phase profile in this scenario would typically preclude any pulse recompression applications.
In both chalcogenide graded-index core fiber designs with dc = 8 or 10 μm and parabolic x2 refractive index gradient, numerically obtained SC spanned over one octave, as shown in Fig. 8. The fiber with dc = 10 μm enabled the broadest SC generation, which extended from 3.2 μm to 8.5 μm in 20 dB dynamic range. To summarize, in the investigated case of nanostructured graded-index core chalcogenide fibers, made of AsSe-GeAsSe glasses, parabolic profile of refractive index inside the fiber core ensured the broadest SCG, while fiber core with dc = 10 μm was an upper limit for SCG in normal dispersion regime for the assumed pump wavelength.
Fig. 8 SC generated in graded-index core chalcogenide fibers with parabolic profile of refractive index inside the core and core diameter dc = 8 or 10 μm. Sample length 2 cm, pump pulse: central wavelength 6.3 μm, 1 nJ input energy, 200 fs duration, 100 kHz repetition rate.
Hyperspectral, femtosecond pulse pumped supercontinuum generation was recently studied numerically in a proposed air-hole lattice chalcogenide PCF [33]. Albeit our spectrum covered a narrower wavelength range (3.2 ÷ 8.5 μm against 2 ÷ 15 μm in [33]), we assumed a factor of 4 longer pump pulse (200 fs against 50 fs) with a comparable peak power (5 kW against 3.5 kW). Further, the nanostructured air-hole-less (all-solid) fiber, proposed in this work, would have lesser thermal isolation of the core, as compared to an air-hole lattice PCF, facilitating improved pump power handling.
Temporal coherence of SC in the analyzed graded-index core chalcogenide fibers has been investigated by calculating the modulus of the complex degree of first-order coherence [34]:
where angle brackets denote an ensemble average over independently generated pairs of SC spectra with one photon per mode noise [35]. The has been calculated for a set of 20 individual pairs of SC spectra, generated in a graded-index core chalcogenide fiber with a parabolic profile of refractive index inside the core and dc = 10 μm, as shown in Fig. 8, and presented in Fig. 9. Obtained spectrum has very high temporal coherence with. The small decrease in temporal coherence located around the pump wavelength is probably caused by low group delay conditions and previously mentioned beating between OWB and FWM spectral components.Fig. 9 Complex degree of first-order coherence calculated from a set of 20 independent pairs of SC spectra in graded-index core chalcogenide fiber with parabolic profile of refractive index inside the core and dc = 10 μm.. Pump pulse used: 6.3 μm central wavelength, 1 nJ input energy, 200 fs duration, 100 kHz repetition rate.
4. Fiber core nanostructurization
A chalcogenide graded-index core fiber is impossible to fabricate with existing VD techniques. We propose a fabrication method by means of nanostructurization, where effective medium theory (EMT) is applied and the stack-and-draw approach is assumed as the possible route to fabrication of a physical structure. According to the EMT, a continuous refractive index distribution can be substituted by a given discrete subwavelength distribution of two types of glass rods with distinct refractive indices. By stacking two types of glass rods with different refractive index into a designed matrix, a type of “preform” can be assembled [36]. The rods are then downscaled to sub-wavelength size at a fiber drawing tower. Since each individual rod in the final fiber is much smaller (< λ/2π) than the wavelength of the propagating light, the fiber core has an effective gradient refractive index distribution. It is important to note, that the proposed EMT has been already successfully used for nanostructured graded-index core silicate fibers fabrication [23,26]. Moreover, the stack-and-draw method has been already successfully used for standard chalcogenide step-index and microstructured photonic crystal fiber fabrication [e.g 14,15.]. In what follows, we explain the designing detail of the nanostructure layout of a chalcogenide glass fiber with a parabolic gradient of refractive index in the core. We showed earlier, that a parabolic index gradient is physically feasible with the silicate glass platform [26]. Other profiles of index gradient, including those discussed here earlier, can be designed using exactly the same mathematical procedure.
Statistically, each glass rod composing the structure of the core preform can be of one of two states – made of either AsSe or GeAsSe glass. Since there are 2n possible distributions of rods, where n is total number of rods in the core matrix, direct calculation of their arrangement corresponding to the desired distribution of the effective refractive index would be time consuming. Therefore a stochastic Simulated Annealing (SA) optimization method has been used, which is a mixing rule inspired by the Maxwell-Garnet formula [36]
where εeff denotes effective electric permittivity, εe is the electric permittivity of glass inclusions, εi is the electric permittivity of the dielectric medium surrounding the inclusions and f is the ration between number of both types of rods. Since the difference of both permittivities is , an approximation given by the first order Taylor expansion of formula (2) can be used. Therefore, the effective permittivity εeff can be simplified to a form:The SA method involves calculating an effective refractive index distribution stemming from the distribution of the rods, by using formula (3), then calculating a cost function H(S), which represents the difference between the calculated effective refractive index distribution and the ideal (target) refractive index distribution for all states S of the rods at positions i and j:
The optimization algorithm aims to minimize Eq. (4). When the value of H(S) is close to zero, then the effective refractive distribution is similar to the ideal (target) refractive index distribution.
The nanostructured core of the discussed graded-index chalcogenide fiber with 10 μm core has been calculated with the SA method. The discretized core area with the target parabolic effective refractive index distribution and the central cross-section of refractive index distribution are presented in Fig. 10(a) to the left and to the right, respectively. The average parabolic distribution of effective refractive index inside the fiber core and its central cross-section, which are the result of SA optimization algorithm, are presented in Fig. 10(b) to the left and to the right respectively. The obtained average refractive index distribution is in agreement with the distribution assumed as target.
Fig. 10 Left: Refractive index distribution in the fiber core; Right: central cross-section of the refractive index distribution. (a) Discretized target assumed parabolic distribution of refractive index, (b) averaged parabolic distribution of refractive index inside the core calculated using the SA algorithm.
A discretized distribution of chalcogenide glass rods in the fiber core has been calculated, and the result is shown in Fig. 11. The obtained matrix contains over 7500 glass rods, each 0.5 mm in diameter. There are 101 rods at the maximal diameter. The higher index rods, shown in red in Fig. 11, denote AsSe glass, while lower index rods (blue) - the GeAsSe glass. The size of each glass rod in the resulting fiber with outer diameter of 125 μm is expected to be below 200 nm.
Fig. 11 Glass rods distribution of the graded-index core chalcogenide fiber, calculated with effective medium approach. Fiber made of AsSe (higher refractive index) and GeAsSe (lower refractive index) glasses and core with dc = 10 μm.
The process of creating GRIN fiber preform consists of two main steps. In the first step, two sets of rods made of each of the glasses are fabricated, in numbers sufficient for preform stacking (almost 8000 pieces altogether). Due to our current technological limitations i.e. number of drawing towers, this step lasts two days, but can be performed even in one day. The second step involves stacking together the two sets of rods in the previously designed matrix. Currently, this step is performed manually, which takes approximately 10 working days, imposed by limitations related to human fatigue. However, works on automatization of the stacking process are ongoing. Successful automatization of the stacking process will shorten the time needed for stacking the preform down to one or two days.
The proposed method of preform preparation allows for efficient fabrication of few kilometers of graded-index chalcogenide fiber from one GRIN preform. Since the chalcogenide glasses are highly nonlinear (n2 = 14-18·10−18 m2/W), only a dozen centimeters of the fiber is sufficient for broad SCG. Therefore, one GRIN fiber preform can provide for thousands of fiber samples to be used in physical SCG setups.
To validate the used method of structure discretization, dispersion of discrete graded-index core fiber structure as shown in Fig. 11, has been calculated. Since FDE method does not allow precise or efficient calculations of structures with small elements <200 nm, the finite element method (FEM) has been used for dispersion calculation [37]. The dispersion of the graded-index core fiber structure with ideal parabolic profile of refractive index and the dispersion of discrete fiber structure are presented in Fig. 12.
Fig. 12 Dispersion of an ideal parabolic and the discrete grade-index core chalcogenide fiber structures with dc = 10 μm.
Both dispersion profiles are well-matched in the spectral range of interest. The dispersion of the discrete fiber structure at the short wavelengths differs from the dispersion of the ideal structure by 8 ps/nm/km. In the long wavelengths the difference is comparable, of about 10 ps/nm/km. The difference between the dispersion characteristics stems from imperfection of the discretization process, i.e. the step-index-like (“sharp”) contribution of each individual rod to the resulting refractive index profile assumed in the calculation. Therefore, the calculated effective refractive index distribution, shown in Fig. 10(b), is not ideally parabolic, but features fine sharp edges. However, we have already presented evidence, that in the case of nanostructured graded-index core fibers, fabricated using the stack-and-draw method and silicate glasses, diffusion between the glasses plays a role in shaping of the dispersion profile [26]. Diffusion is also expected to smooth the edges of individual glass rods and hence the effective refractive index distribution in the core of the discussed fiber. For this reason, higher accuracy between the ideal index gradient profile and the discretized index gradient profile can be expected in the case of a physically fabricated fiber. Therefore, we expect the dispersion of the fabricated fiber structure to match the ideal structure closer, than what is shown in Fig. 12.
5. Conclusions
We proposed a novel approach to the development of graded-index core chalcogenide fibers, which could overcome the present limitations of vapor deposition technologies, commonly employed in fabrication of graded-index silica fibers. With the use of the effective medium theory we designed a nanostructured core fiber with a parabolic index gradient. The proposed method allows to fabricate chalcogenide fibers with various types of graded-index core profiles, regardless of the technological limitations of chalcogenide glass. The calculated average distribution of the refractive index experienced by the guided mode in the core is in good agreement with the target ideal distribution, and other gradient profiles are possible using the same approach.
We have shown, that the graded-index fiber in the chalcogenide glass platform potentially enables dispersion and mode area (nonlinearity) engineering flexibility not achievable with step-index chalcogenide glass fibers. Specifically, spectrally flattened to within ± 10 ps/nm/km, normal dispersion profiles were anticipated with numerical simulations within about 4-14 µm wavelengths for the parabolic and x4 profiles of refractive index gradient, predisposing such designs, among other, for applications in mid-infrared pulse-preserving supercontinuum generation. The parabolic graded-index core fiber design with normal dispersion and a 10 μm diameter core enabled an over one octave spanning numerical SC generation from 3.3 to 8.5 μm, determined in 20 dB dynamic range. This result has been obtained with a pump pulse, which had 6.3 μm central pump wavelength, 1 nJ input energy and 200 fs pulse duration. Such pump pulse conditions are readily available with existing optical parametric amplifier systems.
The combination of dispersion engineering with maintaining of an all-solid glass structure facilitating enhanced power handling, as compared to e.g. suspended core chalcogenide glass fibers, should favor the application the proposed graded-index fiber designs in nonlinear frequency conversion applications. The recently reported new means of control of the multimode operation of silica graded-index fibers, based on nonlinear mode clean-up [38,39] could easily be extended onto the chalcogenide glass graded-index platform, using our approach, as well. This could potentially open up intriguing possibilities e.g. in mid-infrared imaging with unprecedented information handling capacity owning to the large effective area obtainable in this type of fibers.
Funding
National Science Centre in Poland (PRELUDIUM 2014/13/N/ST7/01931); Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund (TEAM TECH/2016-1/1).
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