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Abstract
We test the development of a silica all-glass optical fiber with a highly birefringent large mode area (HB-LMA). In the fiber, the birefringence and single mode operation are independent of bending and results from the internal nanostructuring of the core, which makes the glass anisotropic. Taking into account technological limitations of the doped silica glasses, we optimized the HB-LMA fiber properties by appropriate selection of germanium and fluorine doping level of silica used in the fiber core and cladding. We demonstrated that the anisotropic glass can be successfully used as a core material in large core area fibres in C-band for polarization components of the fundamental mode. We obtained phase birefringence of 1.92 × 10−4 in the fiber with the core diameter of 30 µm and the effective mode area equal to 573 µm2 and 804 µm2, for x- and y-polarization, respectively. The same approach was applied to designing a single mode fiber with 40 µm core diameter and effective mode area over 1000 µm2, which supports only single polarization.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Due to the increase of output power offered by progress in fiber lasers [1], we have recently observed a trend toward developing optical fibers with a large mode area (LMA). The LMA fibers offer reduction of power density in the core, thus lower nonlinearities and higher damage thresholds, extremely small numerical aperture (NA) (i.e., equal to 0.034) [2], very large mode area (e.g., 3700 µm2) [3] and single mode (SM) regime of work [4]. However, the demands for fiber lasers or high power beam delivery systems as maintaining polarization, or single polarization guidance in LMA properties fibers are also highly desired [4].
All these criteria are difficult to match using conventional fiber designs and standard manufacturing techniques based on CVD (chemical vapour disposition) technology [5]. Recently, several approaches have been proposed for developing LMA fibers with truly SM performance, such as suppressing higher order modes in multi trench fibers [6], all-solid photonic bandgap fiber [7], or pixelated Bragg fibers (PBF) [8]. However these concepts do not offer robust single-mode operation for general purposes, since bending loses limits their performance. Moreover, only few concepts of polarization maintaining (PM) LMA fiber have been reported up to now [7–10]. The two main ways of PM in such fibers pertain to the geometrical induction of birefringence in non-symmetrical fiber cores [9], and to applying stress to the fiber [7,8,10]. In the first case, called form birefringence (FB), the simplest way to perturb the core symmetry is to introduce twofold rotational symmetry in the refractive-index profile by making the elliptical shape of the core [9]. In the second case, similarly to the conventional special fibers, the PM performance of LMA fibers has been achieved for stress induced birefringence, usually by stress-applying parts (SAP) placed in the cladding [8] or the core [7] of an all-solid fiber, or in the cladding of hole assisted fibers [4]. However, the stress-applying parts resulting in relatively high birefringence of (1–4) × 10−4 [11,12] have to be placed close to the fiber core. Increasing the stress brakes the symmetry of the mode distribution [10] or introduce higher order polarization mode [8]. In commercially available PM-LMA all-solid fibers [13] similar birefringence of (1–4) × 10−4 was achieved by very big SAPs (∼100 µm in diameter) located far from the core, thus mode is symmetric. This, however, imposes using large fiber diameter, e.g. within the range of few hundreds of micrometers. Other drawbacks of stress-induced birefringence are an increased sensitivity of the fiber to temperature [14] and pressure [15], which is undesired for e.g. laser systems. Also, commercially available PM-LMA are usually few-mode fibers, which need bending to obtain SM performance. SM-LMA fiber with single polarization was proposed by Aleshkina et al. [7], who obtained single-modedness at 1064 nm in a large mode area of 870 µm2, but the polarizing performance of this photonic bandgap fiber was bending dependent.
Overall, there is a lack of robust designs of fiber with large mode area for general purposes, which simultaneously support only the fundamental mode and maintain polarization. Large mode area and single-modedness together were deeply studied in the literature and many successful designs basing on different guiding regimes were showed [2,3,6–8,10,12,13]. However, the introduction of birefringence is limited in fact, to only single concept implementation of stress applying parts (Table 1). Most of the SM fibers were designed to work at operating wavelength (OW) around 1 µm reaching mode field diameters (MFD) in the range of 16.7–33 µm. This corresponds to the mode field areas of 216–870 µm2, respectively, which is appropriate for LMA fibers. The properties of design considered within this paper are also showed in Table 1 for comparison.
Table 1. Properties of SM–LMA Fibersa
Nevertheless, birefringence can be induced by the use of an alternative mechanism, whose origin is different than that of geometrically induced or stress induced birefringence in the case of PM fibers. Based on the second order Maxwell-Garnett effective medium theory [17], birefringence can be obtained through placing subwavelength elements in the fiber core, which allows to obtain an artificially anisotropic glass. This idea was first proposed by Waddie et al. [18], who analysed a bulk material composed of subwavelength layers of two types of silicate glasses (soft glasses) and numerically achieved artificial birefringence in C-band equal to 2.3 × 10−3. Further, a subwavelength structure in a lamellar-core fiber was reported by Wang et al. [19], who first achieved birefringence of 6.4 × 10−3 at the wavelength of 543 nm for pair of lead oxide glasses (soft glasses). Also the nanostructured fiber presented by Stepniewski et al. [20], made of lead oxide and borosilicate glasses had an artificially anisotropic rectangular core with the size of 6.2 × 2.7 µm. This work for the first time showed experimentally flat birefringence characteristics in a broad wavelength range. The obtained birefringence value was very high, since it reached a level of 10−3 in a wide spectral range. However, both of these fibers [19,20] were made of soft glasses, which limits their practical use due to incompatibility to standard telecommunication systems. Thus, to increase the applicability of such polarization maintaining fibres, attempts were to fabricate them of silica glasses. Recently, the PM performance of anisotropic core fiber made of silica glasses was analysed [21] and numerical studies proved that the proposed SM fiber achieved birefringence equal to 1.21 × 10−4 and was compatible with conventional SM fibers.
High birefringence can be defined as that at least in the range of 10−4. Also, due to the limits of the conventional approach for birefringence induction, PM fibers cannot be easily scaled to become highly birefringent large mode area fibers (HB-LMA). In this work we consider transferring the early concept of anisotropic core fibers [21] towards the HB-LMA fiber design. We present the numerical analysis on birefringence enhancement in an anisotropic core silica fiber, which is simultaneously characterized by a large mode area and the bending-independent, single mode performance in the C-band.
2. HB-LMA optical fiber design
We consider an optical fiber with solid silica cladding and anisotropic core composed of interleaved subwavelength layers of high and low refractive indices (Fig. 1). We have previously shown that these type of structures express high birefringence and can be used as core in optical fibers [21]. We aim to maximise phase birefringence, effective mode area and rigorously maintain single-mode performance at the wavelength of 1.55 µm. The analysis starts from 10 µm of the core diameter, which is usually considered a bottom limit for large mode area fibers, and is carried out up to 40 µm where the fiber supports only single polarization. We limit our analysis to technological constrains of CVD doping of silica glass and thermal matched of pure and doped silica-based glasses.
Fig. 1. Scheme of an all-solid optical fiber with anisotropic core composed of interleaved subwavelength layers of high and low refractive indices.
We defined two types of the layers in the core: the first type–made of high refractive index glass n1GeO2 and the second type–made of low refractive index glass n1F. The third type of glass was used for the cladding of the fiber and was defined as glass material with middle index of refraction n2. It is important to note that the core formed in this way is characterized by its effective properties. According to the Maxwell-Garnett model of effective medium [17], the size of a single element (here the thickness of the single layer) has to be smaller or comparable to λ/2π. However, as we have shown in the previous works [22,23], in quasi-2D structures, due to the presence of diffusion in the nanostructured fiber, the effective medium condition is already met for the size of a single element equal to λ/5 or even to λ/3. Based on our earlier studies [21–23] we limited the analysis to the fixed layer thickness of 1/5 of the operating wavelength. The entire analysis was conducted for the wavelength of 1.55 µm, thus the single layer thickness was equal to 0.3 µm. The core diameters considered were in the range of 10–40 µm and the cladding diameter was constant for all optical fiber designs, and equalled 125 µm.
The main objective of this work was to obtain high birefringence (HB) in the proposed fiber design, i.e., at least of the order of 10−4, highly demanded for PM fibers [11,12]. To achieve this, we consider three optical fiber structures schematically presented in Fig. 2, each related to material modification in the core, or in the cladding area: (F#1) the initial design, (F#2) structure to the improvement of birefringence, (F#3) structure to testing and optimalisation the guiding properties towards SM (single-mode regime) transmission. The structures are described below.
Fiber structure F#1. We consider the construction of the base fiber design similar to a concept presented in [21]. In this case the initial design is a fiber with silica cladding and the core layers made of germanium (Ge) doped silica (high refractive index n1GeO2) and pure silica layers (refractive index n2), pure silica glass is also considered as a cladding. The refractive index profile of the fiber is shown in Fig. 2(a) for the y axis (perpendicular to the layers) and describes the relations of absolute indices for used materials not effective ones. The results of the analysis are discussed in the Section 4 of the paper.
Fiber structure F#2. On this structure we study what maximum birefringence can be obtained. Preliminary studies [21] have shown that with the increase of the doping level in the Ge-doped layers (increasing the refractive index n1GeO2 of those layers), the birefringence of the fundamental mode also rises. Since the refractive index in the silica layers n2 is constant and for the n1GeO2 it increases, the contrast Δn between those refractive indices also grows. Unfortunately, due to technological limitations (which will be discussed in details in the next section), only a limited range of the Δn increase is possible by a higher Ge-doping of one type of layers. Therefore, to further increase the birefringence, in the proposed fiber structure F#2 we allowed for reducing the refractive index n1F and, consequently, for increasing the value of Δn in the core. This can be achieved with fluorine (F) doping of the pure silica layers. Figure 2(b) presents the scheme and material refractive index profile of the structure F#2. The results of the analysis are discussed in the Section 4 of the paper.
Fiber structure F#3. The assumption of the new fiber structure was mainly to optimize the modal properties. We analysed and optimized the structure F#2 in terms of multimode, single-mode and non-guiding regimes. To improve the guiding properties (i.e., to obtain SM regime of work), we consider introducing additional Ge-doping of the cladding material, thus increasing the refractive index n2, Fig. 2(c). This approach allows to decrease the numerical aperture (NA) of the fiber and reduces the V number (normalized frequency parameter). The results of the analysis for fiber structure F#3 are presented in the section 5 of the paper.
We divided the modelling of the considered structures into two steps. Steps 1 named ‘effective parameters’, is dedicated to obtaining the initial estimations of the material and geometrical parameters of the fibers structure based on their effective parameters. For this purpose we use Sellmeier equations for material dispersion, and equations for numerical aperture (NA), V-number derived for the optical fibers with cylindrical symmetry. We also use analytical expressions for birefringence estimation of the fiber with interleaved layers in the core. The description of this part of modelling is presented in Sections 4 and 5. Steps 2 named ‘numerical modelling’, contains numerical modelling of fibers based on solving Maxwell equations for fibers with determined geometrical and material parameters. We implemented the HB-LMA optical fiber structure (acc. with the scheme in Figs. 1 and 2) into the finite element method (FEM) (Comsol Multiphysics [24]). Those results are presented in Section 6.
3. Glass material selection and technological limits
We aimed to design the optical fiber based on commonly used glasses in telecommunication and laser techniques. Therefore the silica glass is a base of HB-LMA fiber design. To vary the refractive indices in the core layers and in the cladding we applied silica doped with different elements, as to obtain the glass with higher or lower refractive index than pure silica. To raise the refractive index, the silica can be doped with e.g. germanium or phosphorus [25]. For our design we selected germanium, as it allows to obtain higher refractive index for the same doping level than phosphorus and is commonly used as a core material in optical fibers. Lowering the refractive index of silica is also possible by using fluorine (F) or boron doping [25]. For our analyses we selected a F dopant, which introduces lower attenuation of glass than the boron [25].
The material dispersion of fused silica [Fig. 3(a)] is defined by Sellmeier’s coefficients available in several databases as in [25]. The refractive index of the silica doped with germanium dioxide can be estimated with use of formula proposed by Fleming et al. [26]:
Fig. 3. Material dispersion of pure silica, Ge-doped and F-doped silica for selected molar fractions (a) and (b) refractive index difference for doped silica glasses in relation to pure silica glass.
To describe the dispersion of F-doped silica glass we used interpolation formula proposed by Sunak et al. [27]:
In the analyses, we have also assumed technological limitations on the maximum doping levels allowed for each parts of the designed fiber. Since the HB-LMA optical fibers considered here are composed of three glasses, the glasses have to be matched in terms of, e.g., thermal expansion coefficients, viscosities and transition temperatures. It is a consequence of the thermal-chemical relations of the glasses used for fiber design. We note that the thermal expansion coefficient of Ge-doped glass (e.g., 8.5 mol.%) is approximately two times higher than the one for the pure silica [28], whereas it is a little smaller for F-doped silica [28]. At the same time, the viscosities for both doped glasses are smaller than for silica [28–30]. Taking into account the fibre-drawing process, the mismatch between the thermal expansion coefficients should be avoided, since it causes breaks and cracks in the fiber structure after drawing, due to the residual frozen stress between the glasses. All these have to be considered while establishing the maximum levels of Ge and F silica doping, as discussed below.
Considering Ge-doped silica glass, we limit in our study the doping levels up to 15 mol.%. This value is achievable thanks to using modified chemical vapour deposition (MCVD) technique devoted to optical fiber technology [5]. A higher molar fraction of GeO2 in silica rods is also possible to obtain, but the MCVD process is then more challenging. The internal stress in the rod raises with the doping level, and its distribution has to be carefully controlled to avoid total destruction of the rod related to the stress cracks, especially during and after the collapse step. What is more, with the increase of Ge-doping level the optical losses also grow, which is inconvenient for the optical fiber because its transmission performance degrades [31]. Finally, for fiber structure F#3, we also consider the Ge-doping of the cladding. In this case, for technological reasons we have limited the maximum Ge-doping of the cladding to the level of 2 mol.%. This is also because of the considerable cost of the doped silica.
As far as the F-doping is concerned, we assumed a limit for F-doped silica at the level of 6.1 mol.% (2 wt.%). Although the F-doping of the silica higher than 6.1 mol.% is possible to obtain, there is a trade-off between the preferred value of RI of glass and the, so called, ‘glass stability’. The fluorine atom replaces the oxygen in the Si-O-Si bonds in silica glass, destroying the bridge bond made by the oxygen, which leads to the weakening of the glass network [29]. In result, also the systematic decrease of viscosity is observed for higher F-doping levels. Fluorine also has an impact on the other physicochemical properties of the glass such as the transition temperature, density, etc. [29,30].
Summarizing, for the HB-LMA optical fiber design we selected pure silica glass, Ge-doped silica glass with maximum doping level of 15 mol.% for the core, 2 mol.% for the cladding and F-doped silica glass with maximum doping level of 6.1 mol.%. These limits were introduced in our simulations to consider only commercially accessible and technologically feasible glasses in form of glass rods and capillaries, which allow to stack the preform of HB-LMA fiber.
4. Enhancement of the phase birefringence of HB-LMA fiber
Our analysis concerns the first two structures considered, which aimed to improve birefringence by modifications introduced to the fiber core shown in the Section 2 and in Figs. 2(a) and 2(b): the initial structure design (F#1), and the fiber F#2 with increased refractive index contrast Δn.
Theoretical assumptions. Firstly, we consider a design of HB-LMA fiber with the core composed of interleaved layers of glass with high n1GeO2 and low n2 refractive indices [Fig. 2(a), fiber F#1]. We assume that the distance between the layers of the same material is Λ = 0.6 µm. This value was chosen based on our previous analyses taking into account the birefringence and the limitations of the material's effective properties [18,21]. The refractive index of cladding is defined as n2.
For the core we also define refractive index contrast Δn as follows:
where n1F = n2. To study an influence of material properties on guiding properties of the fibers with the varying core diameter, we introduce average refractive index of anisotropic core following equation after [17,32]:In the further analysis we assumed both fractions to be equal f1F = f1GeO2 = 0.5, since this condition corresponds to maximum phase birefringence [18]. The average refractive index of the core n1 [Eq. (4] is only a linear, rough approximation, but it allows to analyse the dependencies of the averaged RI on the doping levels of layers, as well as those of the core diameters. However, considering the periodic structure of a finite diameter core, we used a more detailed model in order to analytically determine the value of the fiber phase birefringence.
For an infinite layered structure, the effective RI for a polarization of the electrical field parallel (nX) and perpendicular (nY) to the layers are given by [17,32]:
If the size of the layered structure is limited only to the core region of diameter Φ, the effective refractive index $n_{X,Y}^{{\rm eff}}$ components of the fundamental mode can be calculated from the formula [33]:
where nX,Y is equal to nX or nY depending on the polarization components. The normalized propagation constant b is given by the approximate formula [33]: dependent on a normalized frequency V:Taking this into account, the birefringence of a fiber with a periodic core of finite diameter is given by the formula:
Example results, for the first two considered structures F#1 and F#2 [Figs. 2(a) and 2(b), Section 2), obtained from the introduced equations are presented in Fig. 4. Figure 4(a)-(c) shows the results for fiber F#1. Figures 4(d)–4(i) shows the results for fiber F#2. In Figs. 4(b), 4(e), and 4(h), the white dashed lines indicate the three core diameters selected for the further estimation of birefringence according to Eq. (11).
Fig. 4. Birefringence estimation for HB-LMA fiber based on the effective parameters. The first row (a)–(c) refers to F#1 structure. The second (d)–(f) and the third rows (g)–(i) refer to F#2 structure for different F-doping of the core, respectively for levels of 1 and 2 wt.%. The first column (a), (d), and (g) shows the refractive index contrast Δn in the core. The second column (b), (e), and (h) shows the map of the core refractive index n1. The last column (c), (f), and (i) shows the phase birefringence as the function of Ge-doping level in the core layers for three selected core diameters indicated by the white dashed lines in the second column.
Fiber structure F#1. Let us first consider the initial fiber design F#1. Figures 4(a)–4(c) (first row) shows the results of birefringence analysis for the fiber with a core made of pure silica and Ge-doped silica, for which the doping levels change from 0 mol.% to 15 mol.%. In Fig. 4(a), according to Eq. (3), the RI contrast of the core layers is plotted as the function of Ge-doping level of the silica glass. An almost linear growth of Δn is observed, reaching the value of about 22.5 × 10−3 for the assumed maximum Ge-doping level of 15 mol.% which corresponds to the numerical aperture NA = 0.18. Figure 4(b) shows the n1 (average refractive index of anisotropic core), according to Eq. (3), and maps its changes with the Ge-doping level and the core diameter Φ. We note that the fiber with 0 mol.% Ge-doping level does not guide the light, which cannot be clearly shown in the Figs. 4(b) and 4(c). The remaining fiber structures support at least one mode.
The phase birefringence (B) is calculated in function of Ge-doping level for the HB-LMA optical fiber designs with the core diameters of 15 µm, 25 µm and 35 µm [Fig. 4(c)]. It can be seen that the birefringence grows with the increase of the Ge-doping. Nevertheless, the values of the phase birefringence are relatively small, since the threshold of 10−4 (typical minimum birefringence demanded for PM fiber [11,12]) is achieved for the fibers with doping levels higher than 10 mol.%. Because in the analysis we limit the Ge-doping level of core layers to 15 mol.%, the maximum value of birefringence in HB-LMA fiber F#1 [Fig. 2(a)] made of pure silica and Ge-doped silica equals 2 × 10−4, almost regardless of the diameter of the core. For a fiber with the core composed of interleaved subwavelength layers, the fundamental mode includes two polarization components. The effective mode area Aeff is larger for the y-polarization as shown in Fig. 5(a). For example, for a 35 µm diameter fiber the effective mode areas equal 556 µm2 and 648 µm2 for the x- and y-polarization modes.
Fig. 5. Effective mode area for fibers (a) structure F#1, (b) structure F#2 for 1 wt.% F-doping and (c) structure F#2 for 1 wt.% F-doping.
Fiber structure F#2. Figures 4(d)–4(i) (second and third rows) shows the results of the analysis, for the modified HB-LMA fiber structure where F-doping was introduced [fiber F#2, Fig. 2(b)]. According to the results obtained for fiber structure F#1, we can conclude that birefringence also raises for the growing Ge-doping level, thus for the increasing contrast Δn between the layer’s glass materials. Since we cannot increase the Ge-doping any further, to increase Δn is to decrease the RI of the low refractive index layers in the core by means of F-doping.
We performed analysis for F-doping level of 1 wt.% (3.1 mol.%) and 2 wt.% (6.1 mol.%). The results for the 1 wt.% F-doping level are presented in Figs. 4(d)–4(f) (second row). In this case, refractive index contrast Δn in the core varies from 5 × 10−3 to 27 × 10−3 [Fig. 4(d)], which corresponds to NA = 0.16 for the maximum core Ge-doping of 15 mol.%. The map of n1 in Fig. 4(e) additionally shows the area in which HB-LMA fibers do not guide the light (not coloured area). This is because the average n1 value in the core is lower than the cladding refractive index n2, so the light cannot be guided. Thus, the birefringence plots in Fig. 4(f) only show the Ge-doping levels higher than 3.7 mol.%. In Fig. 4(f) the difference in birefringence depending on core diameter is more visible than in Fig. 4(c). Slightly smaller B values are obtained for the smallest diameter of the core. The threshold of the birefringence set at 10−4 is reached for the lower Ge-doping levels around 7 mol.%. The results in the maximal birefringence are 3 × 10−4 for the fiber structure with the F-doping of 1 wt.% in the low refractive index layers in the core and with Ge-doping of 15 mol.% in the high refractive index layers in the core. Increasing the contrast Δn also increases the difference between the effective mode areas Aeff for the x- and y-polarization components. For a 35 µm diameter fiber, the effective mode areas equal 539 µm2 and 674 µm2, respectively [Fig. 5(b)].
Next, we consider the F-doping of low refractive index layers in the core at level of 2 wt.% (6.1 mol.%) as presented in Figs. 4(g)–4(i), (third row). The RI contrast in the core increases significantly to the maximum of 30 × 10−3, which corresponds to NA = 0.14 for the maximum core Ge-doping. The range of the material parameters for which the fiber does not guide the light is also larger. The high refractive index layers in the core have to be doped with Ge at least at a level of 6.4 mol.% to raise n1 above refractive index of the cladding. The birefringence is now slightly higher for the fibers with larger diameter of the core, reaching B = 4 × 10−4 for maximum considered doping levels of the F and Ge in the core layers (Fig. 4(i)). The effective mode areas for the 35 µm diameter fiber are 522 µm2 and 697 µm2 for the x- and y-polarization modes [Fig. 5(c)].
Overall, we have shown that the contrast between the high and low refractive index layers in the anisotropic core determines the birefringence in the HB-LMA fiber. Maximizing the Ge and F doping level in the appropriate layers resulted in the significant increase of the phase birefringence (B = 4 × 10−4) with respect to the reference fiber structure F#1. Importantly, the birefringence increased with the diameter of the core.
5. Modal properties of HB-LMA fiber
In section 4 we focused on birefringence optimization, while the modal properties of the anisotropic core fibers were neglected. In this section the silica doping in the cladding is considered according to the fiber structure F#3 [Fig. 2(c), Section 2]. We analyse which HB-LMA fiber structures are non-guiding, which guide only the fundamental mode and which support more than one mode. Therefore, the cut-off wavelength is introduced [33]:
where Φ is core diameter, V-normalized frequency [Eq. (10)] (for single-mode operation it equals 2.405 in step-index fibers), and n2 stands for the cladding refractive index. Again, n1 is the averaged refractive index in the core calculated from Eq. (4).Similarly to section 4, we calculated the maps of n1 for all fibers indicating the areas of guidance (Fig. 6, first column). The single-mode (SM) performance of the fiber is indicated in blue, the multi-mode (MM) regime in red, whereas the white areas stand for the lack of guidance. The target HB-LMA fibers should guide only the fundamental mode. Thus, for the three selected core diameters (15, 25, and 35 µm), we show the birefringence calculated from the effective parameters (Fig. 6, second column) in a ranges of Ge-doping core which satisfy the condition for SM operation. For all doping levels, the widest range of the SM regime is obtained for small core diameters and it drastically decreases for larger cores. The highest birefringence values we achieved for the maximum F-doping level [Fig. 6(f)], and only there the phase birefringence exceeds the specified HB limit 1.0 × 10−4. As far as the SM regime is considered, the highest Ge-doping level is limited to 8.8 mol.%, 7.0 mol.% and 6.5 mol.%, respectively for core diameters of 15 µm, 25 µm and 35 µm. These parameters correspond to the maximum phase birefringence equal to 1.4 × 10−4, 1.1 × 10−4 and 1.0 × 10−4, respectively. In particular, the final value obtained for the fiber with the core diameter of 35 µm shows that the modification of the core doping only, does not allow to obtain very high birefringence in large mode fiber in SM regime within the material-technological limits assumed preliminarily.
Fig. 6. Birefringence estimation based on the ‘effective parameters’ for the HB-LMA fiber working in the SM regime. The first row (a) nd (b) refers to F#1 structure. The second (c) and (d) and the third rows (e) and (f) refer to F#2 structure for different F-doping of the core, respectively for the levels of 1 and 2 wt.%. The first column presents the cut-off wavelength maps (a), (c), and (e) for the selected range of core diameters. The second column presents phase birefringence (b), (d), and (f) as the function of Ge-doping level in the core layers for three selected core diameters indicated by the white dashed lines in the first column.
In order to further increase the birefringence in the SM LMA fibers additional material modification has to be introduced into the cladding. As shown in Section 4, the birefringence increases with the increasing contrast Δn of the interleaved layers in the core [Eq. (3)]. For a given F-doping, this can be achieved by increasing the Ge-doping in the high refractive index layers, which results in an increase of the average RI of the core n1 [Eq. (4)]. The limit of the SM operating regimes can be derived from Eq. (12) and transformed into the form:
where c = (πΦ/Vλ)2. To maintain SM performance of the fiber, the increasing average RI of the core n1 has to be compensated by a corresponding increasing RI of the cladding n2. This can be achieved by Ge doping of the silica cladding [see Fig. 2(c), fiber structure F#3].Fiber structure F#3. The results of the analysis for fiber structure F#3 are presented in Fig. 7. The first column shows the required Ge-doping in both the core and cladding, and F-doping in the core to guarantee SM operation according Eq. (12) as the function of the core diameter. Similarly to the earlier analysis (structure F#1 and structure F#2), we consider three cases of the core layers doped with F at a level of 0 wt.%, 1 wt.% and 2 wt.%. Colours stand for the Ge-doping level in the cladding which guarantees operation in SM mode. The white dashed line indicates the limit set for Ge-doping level in the cladding, which is 2 mol.%. This line shows that for large core diameters the SM regime imposes lower Ge-doping levels in the core. For example, in the absence of F-doping at the core [Fig. 7(a)], for the core diameter of 40 µm only 5 mol.% of Ge-doping in the core is needed. Thus, the resulting birefringence depicted in Fig. 7(b) slightly drops for larger core diameters. Maximum B value of 0.72 × 10−4 was achieved for the fiber with Φ = 10 µm.
Fig. 7. Birefringence estimation for the HB-LMA fiber with Ge-doped cladding (F#3 structure) working in the SM regime. The first column (a), (c), and (e) shows the required doping of both core and cladding to guarantee SM operation. The white dashed lines indicate 2 mol.% limit in cladding doping level. The second column (b), (d), and (f) presents the Ge-doping level in the core (red line) and phase birefringence (black line) as the function of core diameter based on the ‘effective parameters’. The rows present the cases for different F-doping in the core, respectively for levels of 0, 1 and 2 wt.%.
A similar behavior of birefringence was obtained for 1 and 2 wt.% of F doping in the core layers [Figs. 7(c) and 7(d)]: the maximal value was obtained for the smallest core diameter of 10 µm B = 1.76 × 10−4 for 1 wt.% of F, and for the 11 µm core diameter B = 2.76 × 10−4 for 2 wt.% of F. Importantly, although B decreased for larger diameters, it was improved in comparison to the designs without the Ge-dopant in the cladding [structure F#2, Fig. 6(f)]. Relative to the F#2 structure, B almost doubled and equalled 1.75 × 10−4 for the fiber with 35 µm in diameter.
Because the correlation between the cladding n2 and the core n1 refractive indices was affected by the doping level, operation in the SM regime leads to an inversely proportional dependence of the NA on the core diameter. Using Eq. (13) and the definition of numerical aperture for a step-index fiber NA = (n1-n2)1/2, we can see that NA = Vλ/πΦ, which gives NA = 0.02 for a 30 µm diameter fiber. This is a very low value and such small numerical apertures are difficult to obtain for standard step-index fibers made by CVD methods, since the doping level has to be extremely small to achieve the refractive index precisely controlled at the level of 10−4 [2]. However, such values are possible to obtain thanks to applying the nanostructuring [20,22] in the fiber core.
6. Numerical modelling
The analysis in the previous sections confirmed that improving the birefringence in SM regime of HB-LMA fiber can be achieved by Ge-doping of the cladding in the fiber structure F#3. However, in that analysis the SM condition was introduced with use of Eq. (12), which is correct only for step-index fibers with cylindrical symmetry. Similarly, in the birefringence calculations in Sections 4 and 5 were based on the effective parameters and approximate formulas in Eqs. (4–11). Since our HB-LMA fiber has an anisotropic core, a detailed analysis was performed using the finite element method.
In Fig. 8 we compared the results for the SM condition based on Eqs. (11) and (12) with the FEM simulation for the fibers with 2 mol.% of Ge-doping in the cladding and 6.1 mol.% (2 wt.%) of F-doping in the core layers. The results based on the equations are named ‘effective parameters’ and the FEM simulation is named ‘numerical modelling’. The Ge-doping in the core layers was selected as to fulfil the SM condition for the ‘numerical modelling’. In the left column of Fig. 8 we compared the birefringence values (black series) and the corresponding Ge-doping level in the core layers (red series). It can be seen that the birefringence obtained from the ‘numerical modelling’ is different than that from ‘effective parameters’, e.g., higher for the core diameters larger than 15 µm. What is more, in the cores with diameters larger than 30 µm the birefringence cannot be estimated even for the lower Ge-doping. This is related to the cut-off of the y-polarization mode which occurs in the anisotropic core of the LMA fiber because the effective refractive index of the y-polarization mode converges to the cladding RI.
Fig. 8. A comparison of the analyses based on the ‘effective parameters’ (first row) and the results of ‘numerical modelling’ (second row). The first column shows the optimal Ge-doping (red series) in the core of the HB-LMA fibers with assumed SM condition and the corresponding birefringence (black series) in a function of core diameter. Effective mode areas b) and d), calculated for fibers with parameters from a) and c), respectively. Insets show the normalized electric field amplitude for both polarization modes for the selected diameters of the cores indicated with the black dashed lines.
We then studied the effective mode area Aeff parameter, since our proposed fiber was designed as to have a large mode area. The right column in Fig. 8 presents Aeff estimated for all considered core diameters. We show the results separately for both polarization modes LP01x and LP01y. For the ‘effective parameters’ Aeff does not change much between the polarization modes up to the core diameters of 20 µm, and area is noticeably larger for the y-polarization mode [see Fig. 8(b)]. However, both polarization modes are supported even for the largest considered core diameter Φ = 40 µm, for which the maximum values of Aeff are respectively 821 µm2 and 1168 µm2, for the x- and y-polarization modes. The situation changes remarkably for the ‘numerical modelling’ SM condition [Fig. 8(d)]. The polarization mode y has a much larger area than the one for x, e.g., for the core diameter of 30 µm Aeff respectively equal 573 µm2 and 804 µm2 for the x- and y-polarization mode. For diameters higher than 30 µm the effective refractive index of the fundamental mode in the y direction is too small to ensure guidance of the y-polarization mode, thus neither the birefringence nor the effective mode area for y-polarization can be estimated. The mode area of the x-polarization mode grows with the core diameter reaching 1022 µm2 for Φ = 40 µm.
Concluding, the numerical analysis based on FEM, revealed the unique performance of our HB-LMA fiber. We achieved fibers with birefringence at least at a level of 10−4, which support both polarization modes with different mode areas up to 30 µm of the core diameter. For larger core diameters, we obtained HB-LMA fibers which support only one polarization mode and can be called single polarization fibers [34]. The material properties, as well as the obtained birefringence and effective mode area values for the optimized HB-LMA fibers are summarized in Table 3.
Table 3. Optimized Single Mode HB-LMA Fibers: Composition, Birefringence and Effective Mode Area
Preliminary numerical studies on the influence of bending on HB-LMA fibers have shown that they do not significantly affect birefringence [Fig. 9(a)]. However, bending affects the propagation losses significantly only when the bending radius is less than 10 mm [Fig. 9(b)]. Moreover, in the presented fiber it is possible to embed additional loss limiting structures such as trenches, because due to the lack of stress zones, it has a lot of space in the cladding.
Fig. 9. Influence of bending on (a) phase birefringence and (b) losses. Insets show modes for a bending radius of 10 mm. For better visibility, the modes views show the core area and not alternating layers.
7. Conclusions
We have verified numerically a possibility of development a large mode area polarization maintaining silica fibers for C-band using anisotropic glass approach where birefringence is induced only by internal nanostructure of a core without any external stress areas or fiber cladding structuring. In this work we consider a development of fiber core of interleaved subwavelength layers of high and low refractive index glasses. For this purpose we use silica and silica doped glasses with current technological limits of doping level offered by standard MCVD technology to ensure high quality optical glass and compatibility with current silica based fiber systems. Therefore we assumed a maximum doping level with germanium of 15 mol.% and fluorine of 6.1 mol.%. In addition, we apply for some design a Ge-doped silica at maximum level of 2 mol.% in the fiber cladding to ensure a single mode operation.
All proposed solutions work entirely in the single mode regime without the need to induce bending or any resonance structures in the cladding. Moreover the proposed solutions are easy to implement in practice in the all-fiber systems. This is because the fibers are straightforward for standard splicing due to their all-solid structure (no air-holes in the core or cladding) and standard external dimension of the fiber (no large stress zones).
We show that the birefringence a level of 2 × 10−4 can be achieved for Fiber #1 with the core diameter up to 35 µm, NA = 0.18 and the effective mode areas of 556 µm2 and 648 µm2 for the x- and y-polarization modes. In this case, the core is composed of interleaved layers of pure silica glass and 15 mol.% Ge-doped silica glass with the thickness of 300 nm each. A further increase of the birefringence is possible in Fiber #2 structure, where the un-doped low refractive index layers in the core are replaced by the F-doped silica. A single mode fiber maintaining polarization (B = 4 × 10−4) can be achieved for the core composed of 300 nm thick interleaved layers of 15 mol.% Ge-doped silica glass and 6.1 mol.% of F-doped silica. Again, it has been demonstrated that, for the core diameter of 35 µm, the effective mode areas are 522 µm2 and 697 µm2 for the x- and y-polarization modes.
Finally, the single mode LMA polarization maintaining fiber with the core diameter of 30 µm, and the effective mode areas of 573 µm2 and 804 µm2 for the x- and y-polarization modes can be obtained if the fiber cladding is composed of 2 mol.% Ge-doped silica to increase its refractive index and ensure single mode operation. In this case, the phase birefringence is estimated as B = 1.92 × 10−4 for the 1550 nm wavelength. A further increase of the core diameter causes that only the x-polarization component of the fundamental mode is guided. This property can be used to develop polarizing fibers with a large mode area over 766 µm2.
The proposed solution offers a higher effective mode area than previously proposed LMA polarization maintain fibers (Table 1). The main strength of our design is use of the artificially anisotropic core glass, which allows for inducing the birefringence with no need to use any stress-applying parts, as in most other solutions so far. With no stress applied, the fiber diameter can be scaled independently, e.g., can be adapted to the LMA active fibers for pump delivery, or to telecom components for the standard 125 µm diameter. The presented results are limited to assumed level of doping of silica glass. Further increase of effective mode area and birefringence of single mode fibers with anisotropic core are possible if higher doping level of silica is accepted.
The presented results are based only on theoretical considerations and computer simulations. However, importantly the birefringent properties of a fiber with an anisotropic core composed of interleaved subwavelength layers of high and low refractive indices have also been confirmed experimentally [20].
Funding
Fundacja na rzecz Nauki Polskiej (POIR.04.04.00-00-1C74/16).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are based on numerical data obtained from the formulas presented in this paper therefore are not additionally published, but may be obtained from the authors upon reasonable request.
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