1. Introduction

The utility of optical fibers with high numerical apertures (NA) has a long history in a variety of applications, which typically require the capture of light from broad area or wide angle sources. ‘Air-clad’ fibers have historically provided a valuable alternative for achieving high NAs, which are named so due to the discontinuous jacket of air which provides a large refractive index contrast with the core material. In particular, these fibres have recently received substantial interest for application with double-clad fibre lasers containing a doped inner-core [1]–[5]. The practical requirement for insensitivity of the fiber to its environment necessitates a solid outer jacket. Numerous fiber designs involving variations on the bridges which support the core in its air cladding have tried to balance fabrication constraints, mechanical stability and optical performance. While the mechanical properties of thick supporting bridges are desirable, these bridges provide core light with an avenue for leakage. In this respect, recent designs of microstructured fiber which possess bridges of sub-wavelength thickness have proven to be a substantial improvement both in terms of quality of fabrication and optical performance. Such fibers, with NAs in the range of 0.6 to 0.9 [1]–[5] have significantly exceeded the NAs obtainable in conventional doped or hard clad fibers [6]. To our knowledge however, few studies on the dependence of optical performance on bridge thickness and geometry, or fiber length have been conducted.

Recently a method has been demonstrated for calculating the far field transmission distribution of highly multimode microstructured fibers [7]. The distributions are determined from the confinement losses of leaky modes supported by the waveguide. The correctness and accuracy of the method has been established for simple structures through comparisons with other numerical methods, as well as for more complicated structures showing excellent agreement when compared with experimental measurements. We adopt this leaky mode technique over a number of alternatives for the following reasons: While conventional multimode fibers have accurately been modelled by the ray method [6, 8, 9], it is incapable of correctly describing the scattering of light off sub-wavelength features or the influences of Frustrated Total Internal Reflection (FTRI). A local-plane-wave approach can supplement this method by solving for the reflection matrices of an incident plane wave off a ‘flattened’ microstructured array. The overall fiber transmission for a given angle is then determined by the total number of reflections along the fiber length. While this approach has the merit of simple conceptual understanding, we believe it to be no less computationally demanding than the method used here.

Alternatively, a more heuristic method for estimating NA has been proposed which assumes a form of local mode coupling. That is, the coupling between core modes and fictitious but transient modes localized in the microstructured bridges (or cladding region). Such coupling is assumed to be most favorable when their effective indices are approximately equal. Once these bridge modes are excited they quickly propagate power into the outer jacket. The fundamental bridge mode, with effective index n _b, then provides the cut-off condition for guidance in the core, rather than the cladding index as in conventional fibers. The NA is subsequently defined by $\widetilde{NA}={\left(n_{\mathrm{co}}^2-n_{\text{b}}^2\right)}^{\frac{1}{2}}$. This simplified technique has been shown to provide good agreement with experimental measurements of NA on similar fibers [10]. Furthermore, the underpinning concepts of this approach have been affirmed by a deeper investigation of confinement losses in the fibers [7]. The limitations of this approach however, are evident for structures that have no obvious choice for a localized bridge mode or where the dependence of NA on bridge or fiber length is important. On the other hand, the question of accuracy of this approach for microstructured claddings with multiple layers is less evident. We show here that such a simplifying approximation must be made with a deeper insight into the physical processes by which confinement loss occurs. We demonstrate that the simplistic use of the most fundamental cladding or bridge mode will generally underestimate the NA of fibers with multiple layers. A more suitable expression is provided here.

2. Outline of method

In microstructured and air-clad fibers, propagation is often via leaky modes. When the core diameter D is large in comparison to the wavelength D≫λ, the real part of the mode propagation constant β can be used to specify an internal angle, as depicted in the schematic of Fig. 1. The confinement losses of these modes can be used to determine the far field transmission (power per unit solid angle) in the way outlined in [7].

 

Fig. 1. Cross section along optical fiber axis. The core refractive index, n _co=1.45, is assumed equal to the index of the infinite jacket. Both are lossless.

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Consider an internal source, or one in contact with the end-face of the fiber, with intensity distribution I(θ _in) for each polarization. The source is assumed to be small in comparison to the fiber core and centred so as to excite only near-meridional rays. The transmission within the fiber after a given length L is then

where

is the averaged transmission via the M leaky modes in the range $n_{\text{eff}}^{\mathrm{r}}$(θ _in)±Δ$n_{\text{eff}}^{\mathrm{r}}$/2 and where k=2π/λ.

Similarly, the far field intensity distribution in air (power per unit solid angle) is then given by

where F _TE, F _TM are the Fresnel reflection coefficients leaving the fiber and Snell’s law relates the internal and external angles by n _co sinθ _in=sinθ.

The simple interpretation of Eq. (3) is that far-field transmission for an incident angle is evaluated by averaging the transmission via modes that are excited in a narrow range about that angle. Such averaging is necessary since confinement losses are strongly dependent on the polarization of the mode (or mode class) at large θ _in.

The nominal numerical aperture (NA) is generally defined for optical fibers as

where θ _max is the maximum angle at which a meridional ray entering the fiber is transmitted. It is measured from the half-angle at which the far-field angular intensity has decreased to 5% of its maximum value. Numerically, the angle corresponding to 5% transmission as predicted by Eq. (3) can be used to determine the NA of microstructured fibers. However, while the transmission distributions of many fibers are simple such that the NA provides a good measure of light acceptance, the transmission distributions of more complicated fibers are not. The fibers with multiple layers presented below are an example, as they display more than one sharp feature in transmission. In such situations a more representative measure of light acceptance is the capture efficiency.

The meridional capture efficiency ε is defined as the fraction of emitted photons in forward propagation that remain after a length L. It is given by

where P_o =2π I_o for an isotropic source (I(θ _in)≡I_o ) or P_o =π I_o for a Lambertian source (I(θ _in)=I_o cosθ _in). Clearly the capture efficiency is a length dependent quantity. If the materials comprising the waveguide are assumed lossless, then ε provides an estimate for the aggregate confinement loss in a multimode fiber for a specified length.

We can obtain an approximate, yet simple relation between capture efficiency and NA for an internal source, if we neglect the attenuation due to Fresnel reflection leaving the fiber. Under the simplifying assumption that internal transmission obeys a unit-step-function U which is independent of polarization, ie. T _in(θ _in)≃2U(θ _in-${\theta }_{\text{in}}^{\text{max}}$), one can easily show using Eq. (5) that

for Lambtertian and isotropic sources respectively. Since we have omitted any attenuation due to Fresnel reflection, these expressions are only valid for NA≳1. We emphasise that the capture efficiency is sensitive to the choice of angular intensity distribution of the source, while NA is not, particularly when the fiber transmission functions has a distinct cut-off analogous to a step function. This allows us to approximately relate the capture efficiency from these two sources by the useful expression

3. Implementation

The class of structures investigated here is shown schematically in Fig 2. The key dimensions are defined in the caption. We emphasise that all bridges are of equal thickness and width. The fixed parameters were: core diameter D _o=150µm, core refractive index n _co=1.45 and λ=1.0µm. Furthermore, we assume an isotropic source I(θ _in)≡1 for all subsequent calculations of far field transmission and capture efficiency.

 

Fig. 2. Schematic illustrating the class of air-clad fiber geometry studied here. All bridges have equal thickness δ and width w. N denotes the number of bridges in each layer while R denotes the number of layers.

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To solve for the leaky modes in our microstructured fibers, we use the Adjustable Boundary Condition (ABC) method which has been recently developed [11]. The unique treatment of boundary conditions in this method correctly determines the outward radiating fields of leaky modes without difficult or manual searches in the complex n _eff plane. This high degree of automation, robustness and versatility was crucial to sampling in excess of 250,000 modes that have contributed to this paper. While the ABC method can be implemented with most numerical mode solvers, the present calculations were conducted using a small variation on the numerical scheme initially suggested. A radial finite difference scheme has replaced the basis function expansion, while the azimuthal Fourier expansion is retained. This alternative was adopted after it was found to vastly improve computation speed.

The radial and angular resolutions employed up to 4000 radial nodes and 20×N azimuthal Fourier components respectively, where N denotes the number of bridges in each layer. In this way we fully exploit the waveguide symmetry for the benefit of computational speed by including only Fourier components that are multiples of N. For each fiber structure the angular transmissions of Eq. (2) were obtained from 1600 near-meridional leaky modes in the interval 1.0≤$n_{\text{eff}}^{\mathrm{r}}$≤1.45. They were equally sampled from TE_0n, TM_0n, HE/EH_1n, HE/EH_2n and HE/EH_3n like mode classes, noting that the latter 3 are doubly-degenerate. All are meridional or near-meridional as is required for the calculation of NA. We emphasise that the HE and EH mode classes play a dominant role in power propagation in real fiber systems and must be included in the analysis for realistic results [7].

4. Dependence of capture efficiency and NA

The microstructured fibers considered here posses a substantial number of independent parameters that specify a particular fiber. In order to reduce the parameter space under consideration while still providing a systematic study, we have already prescribed that all bridge thicknesses and widths are equal. Since the refractive index contrast between the host material and air is very large in comparison to its wavelength dependence, we assume wavelength-scale equivalence [12] to obtain this most general dependence of capture efficiency ε(L/D,D/λ,w/λ,δ/λ,N,R). We reduce this parameter space by first noting that these multimode fibers necessarily lie in the extreme of D/λ≫1, where ε is weakly dependent on this variable. In addition it has been shown that values of N between 50 and 100 provide no discernable change in the calculated NA in single layer fibers [7]. This has been checked and confirmed for the multiple layer structures presented here, and we set N=100 for all calculations. Finally, in order to discount the influence of confinement loss due to frustrated total internal reflection (optical tunnelling) we impose the condition that w/λ≫1 on the air holes. Our choice of w/λ=8 is more than adequate for this purpose at all practical lengths and wavelengths [7]. Thus the dependence of capture efficiency is reduced to ε(L/D,δ/λ,R). By identical arguments, the dependence of numerical aperture is also NA (L/D,δ/λ,R). This parameter space in explored in the following sections.

5. Light acceptance of multiple layer air-clad fibers

It is rewarding to firstly investigate the typical characteristics of confinement loss in these fibers and determine their physical origins. The aim is to derive simple heuristic expressions for capture efficiency and NA that can be directly compared with full numerical simulation.

It has previously been shown that the condition for high confinement loss in a single layer fiber is $n_{\text{eff}}^{\mathrm{r}}$≳$n_{\mathrm{b}}^{\text{TE}}$, where $n_{\mathrm{b}}^{\text{TE}}$ is the effective index of the fundamental TE mode in a slab waveguide of equivalent thickness to the bridge [7, 10]. It is referred to as the TE local bridge mode, or just TE bridge mode for convenience, but we emphasise that such modes are not actual modes of the waveguide. The dispersion relations for these modes are available in many standard texts [13]. When satisfying this inequality, core light can efficiently couple to at least this local bridge mode, which quickly radiates power into the outer jacket, depending on the radial component of propagation in the bridge. This leads to one heuristic expression for numerical aperture

which has been proven to accurately estimate the NA of single layer structures. Using Eq. (7), the capture efficiency for an isotropic source can be estimated by

which is also suitable for single layer fibers.

Our attention so far has focused solely on the most fundamental local mode of the bridge. However, close in effective index is the second mode supported by a slab waveguide $n_{\mathrm{b}}^{\text{TM}}$ which is TM with respect to the slab. We had shown that it is only the TE-like core modes that do not strongly couple power into the TE bridge modes, since their polarizations are largely orthogonal. Only when the condition $n_{\text{eff}}^{\mathrm{r}}$≳$n_{\mathrm{b}}^{\text{TM}}$ was satisfied did the confinement loss of TE-like core modes sharply increase. However, since the TE-like mode class constitute a small fraction of the power carrying modes, the influence of this TM bridge mode on the NA was shown to be insignificant for single layer fibers. In stark contrast, this mode plays a very important role in determining both NA and capture efficiency of multiple layer fibers. In anticipation we have defined an alternative expression for the heuristic NA and capture efficiency similar to Eqs. (9) and (10)

The usefulness of these expressions are first evident in Fig. 3, which shows how typical farfield angular transmission distributions, Eq. (3), behave as additional layers are added to the fiber. Transmission in the window ${\theta }_{\mathrm{b}}^{\text{TE}}$<θ<${\theta }_{\mathrm{b}}^{\text{TM}}$ is greatly enhanced by the use of 3 or 4 rings, for which a new sharp cut-off is formed at the new angle determined by the TM bridge mode.

 

Fig. 3. Example external angular transmission distribution used to determine NA, showing dependence on the number of rings. Increasing the number of rings results in a shift in the transition from near ${\theta }_{\mathrm{b}}^{\text{TE}}$ to near ${\theta }_{\mathrm{b}}^{\text{TM}}$, where ${\theta }_{\mathrm{b}}^{\text{TE}/\text{TM}}$=sin^-1(N${\mathrm{A}}_{\mathrm{b}}^{\text{TE}/\text{TM}}$). The values δ/λ=0.3, N=100, w/λ=8 and L/D=20m/D _o were used in the calculation.

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A deeper insight is gained from Figs. 4 and 5, which illustrate the evolution of loss for the four major mode classes. In the range $n_{\text{eff}}^{\mathrm{r}}$>$n_{\mathrm{b}}^{\text{TE}}$, the bridges support no modes and core light is tightly confined regardless of polarization. This is demonstrated literally in Fig. 5 I and II. As previously discussed for $n_{\text{eff}}^{\mathrm{r}}$<$n_{\mathrm{b}}^{\text{TE}}$, the behaviour of light at the first layer of bridges has a simple explanation; TM-like core modes preferentially couple light into the TE bridge mode when it is supported by the bridge, since their polarizations are largely parallel. As do HE and EH modes, since their hybrid nature allows relatively efficient coupling between core and bridge modes regardless of polarization. TE-like core modes, do not couple efficiently to the fundamental TE bridge mode, but do couple to the TM bridge mode. However, in a multiple layer cladding, the radial bridges terminate at a 90 degree ‘T’ intersection of bridges of equal thickness. At these intersections the orientation of the TE and TM polarizations swap with the orientation of the bridge.

In the window $n_{\mathrm{b}}^{\text{TM}}$<$n_{\text{eff}}^{\mathrm{r}}$<$n_{\mathrm{b}}^{\text{TE}}$ core light may only couple to a TE mode in the radial bridges which are incapable of efficiently coupling to a TE mode in the azimuthal bridges since their polarizations are largely orthogonal. This is clearly demonstrated in Fig. 5 III and IV, where the cladding penetration of the TE and TM-like core modes is different, but their losses are relatively close. In fact, TM-like core modes are found to abruptly terminate at the end of the first radial bridges. As the number of layers and the number of intersections traversed by light escaping to the outer jacket increases, the total loss reduces for all mode classes that are capable of coupling to the TE bridge mode at the first layer.

For $n_{\text{eff}}^{\mathrm{r}}$≳$n_{\mathrm{b}}^{\text{TM}}$ a new knee in the loss curve is formed for all mode classes. In this range, power is coupled into both TE and TM modes at the first radial bridges. At each subsequent intersection, TE or TM modes in radial bridges are then able to preferentially couple to TM or TE modes respectively in the azimuthal bridges. Thus core light of all mode classes can penetrate deep into the microstructured cladding and radiate power to the outer jacket. This is also clearly shown in Fig. 5 V and VI, where due to the full penetration of light throughout the cladding the losses of TE and TM-like core modes become extreme.

A number of sharp peaks and minor transitions are visible in both these loss curves. They have been attributed to various resonances associated with the bridges such as the choice of w/λ or N. Specifically one can match most of these minor features, such as that at $n_{\text{eff}}^{\mathrm{r}}$≃1.16, with radial (standing wave) resonances in the bridges. They typically occur at $n_{\text{eff}}^{\mathrm{r}}$=$n_{\mathrm{b}}^{\text{TE}/\text{TM}}$-(p/4)λ/w, for p=1,2,3…. Further analysis is complex, without providing additional value to the present analysis.

 

Fig. 4. Total loss, -10log[T(θ)], for three mode classes clearly showing a shift in the loss knee for TE and HE+EH mode classes as the number of rings is increased. The vertical lines indicate $n_{\mathrm{b}}^{\text{TE}}$ and $n_{\mathrm{b}}^{\text{TM}}$, which are the effective indices of the TE fundamental and TM second order bridge modes. They are calculated for the equivalent slab waveguide of thickness 0.3 and the polarizations are prescribed with respect to the orientation of radial bridges. The values δ/λ=0.3, N=100, w/λ=8 and L/D=20m/D _o were used in the calculation.

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What has so far been revealed about the physical processes, suggests that while Eqs. (9) and (10) are applicable for single layer fibers, Eqs. (11) and (12) are more appropriate for multiple layered fibers. These expressions are directly compared against full numerical calculations for a wide range of δ/λ in Figs. 6 and 7. They serve as a convenient reference for a common practical fiber length and are expected to show robust agreement with a variety of air-clad fiber designs. The charts show that both NA and ε are well approximated by N${\mathrm{A}}_{\mathrm{b}}^{\text{TE}}$ and ${\epsilon }_{\mathrm{b}}^{\text{TE}}$ for single layer fibers, but become increasingly better represented by N${\mathrm{A}}_{\mathrm{b}}^{\text{TM}}$ and ${\epsilon }_{\mathrm{b}}^{\text{TM}}$ as more layers are added. The heuristic expressions for NA reach a maximum value of ($n_{\text{co}}^2$-1)^1/2, which is an un-physical result caused by neglecting the Fresnel reflections upon leaving the fiber to air. The insets display the potential improvement that can be achieved through the additional of multiple layers. They are simply estimations obtained from the heuristic models, but predict significant improvements in both the NA and the capture efficiency, reaching a maximum of ~20% and ~50% respectively. It is important for practical purposes to note that for high NAs or capture efficiencies exceeding 0.9 or 0.25 respectively, the addition of multiple layers allows one to more than double the bridge thickness and sill maintain this high performance.

 

Fig. 5. Example intensity plots of sample TE and TM-like modes in regions I, II ($n_{\text{eff}}^{\mathrm{r}}$≃1.3>$n_{\mathrm{b}}^{\text{TE}}$), III, IV ($n_{\text{eff}}^{\mathrm{r}}$≃1.172 between $n_{\mathrm{b}}^{\text{TE}}$ and $n_{\mathrm{b}}^{\text{TM}}$), and V, VI ($n_{\text{eff}}^{\mathrm{r}}$≃1.104<$n_{\mathrm{b}}^{\text{TM}}$). The extent of light penetration is shown to correlate with confinement loss which is similarly strongly polarization dependent.

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Fig. 6. Numerical aperture as a function of δ/λ and number of layers, full numerical results (data points) are compared to the heuristic estimates. Largely independent of source intensity distribution and weakly length dependent. L/D=10m/D_o . Inset: Percentage increase in NA by adding multiple layers, 100×(N${\mathrm{A}}_{\mathrm{b}}^{\text{TM}}$-N${\mathrm{A}}_{\mathrm{b}}^{\text{TE}}$)/N${\mathrm{A}}_{\mathrm{b}}^{\text{TE}}$.

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Finally Fig. 8 illustrates the dependence of capture efficiency on fiber length L/D, number of layers and bridge thickness δ/λ. The results are presented in terms of relative capture efficiency

in order to simplify the illustration. The small scale ripples in the curves are due to numerical error as well as actual fiber properties associated with radial and azimuthal bridge resonances. Evidently, the addition of multiple layers results in the trend ε→${\epsilon }_{\mathrm{b}}^{\text{TM}}$.

The aggregate confinement loss for an isotropic source in these fibers can be directly determined from Fig. 8. To estimate the fraction of power at a length L ₁ which is lost between during propagation between lengths L ₁ and L ₂, one would simply use |Δε (L ₂/D)-Δε(L ₁/D)|/ε(L ₁/D).

6. Conclusions

We have presented the calculated numerical apertures and capture efficiencies for a class of multimode microstructured fibres with thin supporting bridges forming an air-cladding. The calculations are performed using an original and accurate method employing leaky modes supported by the fiber. The findings build upon prior knowledge of the light acceptance properties of single layer structures, which have shown that exceptionally high NA can only be achieved for bridge thicknesses much smaller than the wavelength. This is due to the conditions that allow core light to efficiently couple to the fundamental TE bridge mode, which then radially propagates power into the outer jacket. Thus for single layer structures, the numerical aperture and capture efficiency is best approximated by N${\mathrm{A}}_{\mathrm{b}}^{\text{TE}}$ and ${\epsilon }_{\mathrm{b}}^{\text{TE}}$ respectively. The situation for multiple layer structures however, is significantly different. Here the condition for light acceptance is largely dictated by the second order TM bridge mode. This is understood in terms of the efficient transmission of light through numerous bridge ‘T’ intersections in the microstructured cladding. Since the orientation of the local polarization of TE and TM bridge modes swap between radial and horizonal bridges, both bridge modes are needed for light to penetrate deep into the cladding. The alternative heuristic expressions for both NA and capture efficiency are then N${\mathrm{A}}_{\mathrm{b}}^{\text{TM}}$ and ${\epsilon }_{\mathrm{b}}^{\text{TM}}$. When compared with full numerical calculations, they show good agreement. Investigations of different structural geometries will be the focus of future work.

 

Fig. 7. Capture efficiency as a function of δ/λ and number of layers for an isotropic source. L/D=10m/D_o . Inset: Percentage increase in capture efficiency by adding multiple layers, 100×(${\epsilon }_{\mathrm{b}}^{\text{TM}}$-${\epsilon }_{\mathrm{b}}^{\text{TE}}$)/${\epsilon }_{\mathrm{b}}^{\text{TE}}$.

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Fig. 8. Relative capture efficiency as a function of δ/λ, L/D and number of rings.

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