## Abstract

Numerical calculations predict that particular birefringent photonic crystal fiber designs exhibit slightly better performance in a coiled configuration than non-birefringent step-index fiber designs with respect to higher order mode suppression for the realization of large mode area effectively single transverse mode fibers. The passive losses of the fundamental and first few higher order modes of a birefringent photonic crystal fiber design with a 41*µ*m diameter core incorporating stress applying parts (SAP) were calculated using an integrated electromechanical finite element method. Minimum higher order mode losses of up to 5.5 dB/m were predicted for fundamentalmode losses of only 0.0014 dB/m. The bend performance of this PCF design was predicted to be relatively insensitive to manufacturing tolerances with respect to air hole size and device assembly tolerances with respect to coiling diameter based on the calculated dependence of the mode losses on these parameters. The positions and refractive index of the SAP render the numerical aperture of the core anisotropic allowing further tailoring of the bend performance by adjusting the angle between the coiling plane of the fiber and the orientation of the SAP within the cladding. Fundamental and higher-order mode losses are calculated for step-index fiber (SIF) designs with a 40*µ*m diameter core for comparison. The step-index fiber designs were predicted to exhibit slightly inferior bend loss mode discrimination and higher sensitivity to packaging configuration compared to the photonic crystal fiber designs presented.

© 2008 Optical Society of America

## Errata

Benjamin Ward, "Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture: errata," Opt. Express**20**, 7966-7972 (2012)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-7-7966

## 1. Introduction

Output power available from narrow-linewidth, high-average-power, single-transverse-mode fiber sources has increased dramatically over the past decade [1, 2, 3, 4, 5]. The primary impediment to power scaling of these devices is the onset of parasitic nonlinear processes due to the high optical intensity within the fiber core. The development of effective techniques for obtaining output dominated by the fundamental transverse mode for increasingly large core sizes has been one of the main approaches to raising nonlinear thresholds in single-transverse-mode fiber lasers [6, 7, 8, 9, 10]. For any particular application, a compromise between single-modedness and maximum power capacity for a step-index fiber (SIF) can be achieved by tailoring the core size and numerical aperture. As the core size is increased and the numerical aperture decreased to suppress the higher-order mode content, fundamental mode loss exhibits runaway behavior due to index matching between the bending-altered fiber cladding and the core [11].

The most demanding applications requiring single-frequency, single-polarization, multikilowatt continuous wave output, however, call for further improvements in both core size, fundamental mode confinement and modal purity. The bend-induced index variation across the fiber cladding is anisotropic and can be modeled as a unidirectional gradient [11]. This implies that if the fiber cladding is anisotropic in its index of refraction, bend-induced fundamental mode loss can be reduced while maintaining sufficient higher-order mode suppression through a suitably small effective index step between higher-order core-guided modes and the cladding modes. The stack-and-draw technique for fabrication of photonic crystal fibers (PCF) may be adapted in a straightforward manner to implement this approach.

This scheme imposes the additional requirement for practical devices that the fiber be coiled in a particular orientation relative to the anisotropy in the numerical aperture. Standard telecommunications-grade optical fiber has very little torsional stiffness and is therefore difficult to coil consistently in a particular orientation. Large mode area photonic crystal fiber, having a typical outer glass diameter of 500*µ*m or more has a large torsional stiffness and robustly maintains its orientation.

One method of introducing the desired guiding anisotropy is to incorporate regions of low refractive index into the cladding. Borosilicate glass commonly used in stress-applying parts (SAP) to create birefringence in PCF has a suitably reduced index of refraction [19]. Furthermore, birefringent PCF incorporating stress applying parts into the cladding naturally coil such that the SAP are in the coiling plane.

Numerical analysis of fiber designs enables optimization of fiber parameters without multiple design, fabrication and characterization iterations. One key feature required of a numerical model is the incorporation of stress induced refractive index variations due to both bending of the fiber and the difference in coefficients of thermal expansion between different components of the pre-form. Losses for each of the guided modes must be calculated in order to evaluate higher-order mode suppression. A fully-vectorial finite element method employing curvilinear elements, 3-dimensional mechanical/thermal stress analysis, and perfectly-matched layer boundary conditions on a structured symmetric mesh forms the basis for the analysis presented here. In this implementation, the structural and electromagnetic calculations are performed on the same structured mesh that conforms to a *C*
_{6v} symmetry. Although the finite element method for structural problems is well documented [12], a brief discussion is included here to facilitate the explanation of the details of its application to coiled optical fibers.

## 2. Structural model of the fiber

The fiber configuration discussed here is shown in Figure 1. The silicate glass comprising the fiber is approximated as a linear isotropic material characterized by its Young’s modulus and Poisson ratio. The first objective is to determine the values of the stress tensor components σ* _{ij}* within the fiber. The SAP in the cladding cause transverse stress loading within the fiber as it cools from the softening temperature during the drawing process. Bending of the fiber introduces a longitudinal strain distribution throughout the fiber cross-section in order to satisfy displacement boundary conditions for the fiber (e. g. the cladding boundary on the outside of a bend is longer than that on the inside.) The standard prescription for straight fibers is to employ the plane strain approximation so that deformations only occur in the plane normal to the electromagnetic propagation axis of the fiber with no longitudinal strain along this axis. If the bend is caused by a uniform constant-diameter coiling of the fiber, symmetry dictates that the longitudinal strain assumes a known distribution. The appropriate structuralmodel therefore incorporates fixed longitudinal strain variations which serve as an input to the determination of the strain in the plane perpendicular to the fiber axis. In order to determine the plane strain the elastic energy functional

must be minimized where **r** is the two dimensional position within the fiber cross-section Ω, **x**(*r*) is the three-component displacement field, *ε*(*r*) is the strain field, **C** is the elasticity matrix, and **R**
* _{th}* is the local thermal load vector [12].

At this point, the portion of the elastic energy due to longitudinal strain parallel to the fiber axis may be set by symmetry depending on the direction in which the fiber is bent. Establishing an *x,y* coordinate system in the plane perpendicular to the fiber axis, the longitudinal strain in the *z* direction may be written

where *r _{x}* and

*r*are the bend radii in the

_{y}*x*and

*y*directions respectively. The terms in (1) which depend on

*ε*may then be separated from the rest to yield

_{z}where the subscript *t* indicates the transverse component of the strain field. The term incorporating **C _{zi}** is now linear in the transverse displacements and therefore takes the form of a force in the elastic energy functional

with

In the finite element approach, the fiber cross-section is broken up into M elements defined on a lattice of nodes on which the displacements are defined as the elements of a global displacement vector. The elasticity matrix is defined to be uniform on each element and the load vector is defined as a force on each node. The strain within each element is evaluated using the shape functions *N* which interpolate the values of the displacements at the nodes on the boundary of the element throughout the interior of that element. Summing the elastic energy over all of the elements and minimizing the global elastic energy results a linear equation that may be solved to determine the global transverse displacement vector **x**
* _{t}*.

The displacement vector **x _{t}** is used to calculate the transverse strain field. This is combined with the longitudinal strain field to yield the full three-dimensional strain field within the fiber. The strain field is then combined with the stiffness matrix to yield the full three-dimensional stress-tensor.

One important characteristic of finite element calculation is the meshing scheme. A structured symmetric mesh made up of quadratic curvilinear Lagrangian elements is used in this particular implementation [13]. The meshing scheme used here is depicted in Figure 2. Just as rods and capillaries are stacked in a close-packed triangular lattice to create the PCF preform, the computationalmesh takes the form of a close-packed triangular lattice of space-filling hexagons. Each hexagonal cell consists of 54 elements. Two types of regions are employed. If the cell represents a capillary, the center 6 elements are curvilinear and form the air-hole. If the cell represents a solid rod (core material, stress rod, or bulk fused-silica), all elements in that cell are triangular and approximately equilateral. One important characteristic of such a meshing scheme is that the average element size is uniform across the entire fiber cross-section. This eliminates the dependence of the solutions on a priori determinations regarding their character. Plots of the stress distributions of the fiber depicted in Figure 1 are shown in Figure 3. It is apparent that even for a relatively tight coiling diameter of 16.8 cm, the SAP dominate the stress anisotropy and by extension the dielectric anisotropy as well.

## 3. Optical model of the fiber

The computational meshing scheme employed is depicted in Figure 2. The stress tensor within each element is used to determine the refractive index tensor within that element in the manner described in [13] for the coiled fiber. A conformal mapping [11] is then applied to describe the fields within the coiled fiber as though they were propagating in a straight fiber with a suitably modified refractive index tensor. Calculating the bend losses requires the use of appropriate conditions on the computational boundaries. Perfectly matched layer boundary conditions [17, 18] are applied here to minimize reflections and allow modal losses to be obtained from the imaginary part of the propagation constant [20]. The refractive index and magnetic permittivity tensor distributions in the boundary region should have the same symmetry as the rest of the mesh. This requires six different regions, each with a different orientation. This is accomplished by applying a rotation transformation to a region oriented in the purely horizontal or vertical direction. The PML boundary condition is implemented by a complex stretching transformation of the refractive index tensor and the magnetic permeability tensor. Thus the transformation employed here is a combination of stretching and rotational transformations.

Assuming harmonic time variation of the fields, the contribution to the electromagnetic action from a given element *i* is:

where Ω* _{i}* is the extent of the element within the fiber cross-section,

*H⃗*is the magnetic field which is interpolated within the element from values at the nodes bounding the element, and

*ω*is the temporal frequency of the wave.

The effective permeablilty tensor *$\tilde{\epsilon}$ _{i}* is constant on each element and on the interior of the computational area is obtained by calculating the photoelastic correction to the bulk permeability of fused silica based on the stress distribution imposed by the SAP and bending [13] and then applying a conformal transformation to map the fiber to a straight fiber with the correct refractive index variation

$${\epsilon}_{i,22}={\epsilon}_{0}\left({n}_{0}^{2}-2\left({C}_{2}{\sigma}_{1}+{C}_{1}{\sigma}_{2}+{C}_{2}{\sigma}_{3}\right)\right)$$

$${\epsilon}_{i,33}={\epsilon}_{0}\left({n}_{0}^{2}-2\left({C}_{2}{\sigma}_{1}+{C}_{2}{\sigma}_{2}+{C}_{1}{\sigma}_{3}\right)\right)$$

$$\phantom{\rule{.6em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}{\epsilon}_{i,\mathrm{jk}}=0,j\ne k$$

where *T _{i}* is the rotation transformation from the local principal stress axes in element

*i*to the global coordinate axes

*x,y,r⃗*is the location of the center of element

_{i}*i*relative to the center of the fiber cross-section,

*R⃗*is the vector from the center of the fiber cross-section to the center of the fiber coil,

*ε*

_{0}is the permeability of free space,

*n*

_{0}is the bulk refractive index of the fiber material,

*C*are the photoelastic constants, and

_{i}*σ*are the principle stresses.

_{i}In the PML regions, an additional transformation of the refractive index and magnetic permittivity is required and is given by [17].

where

The stretching variables *s _{i}* depend on the position and orientation of the layer relative to the center of the computational region. For layers oriented along the global coordinate

*x*, the PML transformation is represented by a diagonal matrix with the values

*s*=1-

_{x}*αd*

^{2}

_{i}

*j*,

*s*,

_{y}*s*=1 where

_{z}*d*is the depth within the layer of element

_{i}*i*and

*α*is the stretching parameter which is effectively the strength of the absorbing boundary condition and is proportional to the local conductivity of the medium. The parameter

*α*is unphysical in the sense that its value is not set by a property of the fiber, therefore the results of the calculations should be independent of its value. In order to preserve the symmetry of the computational mesh, 6 PML regions arranged as shown in Figure 1 are employed. The transformation matrices for these regions are obtained from this single region through a rotation transformation

where *S _{ix}* is the transformationmatrix for the element in the primary PML region with the same relative position within the layer as element

*i*. Thus, the final forms used for the refractive index tensor and magnetic permittivity in the finite element calculations are

where *µ⃡ _{i}*=

*µ*

_{0}

*I⃡*and all transformations

*S*are the identity transformation for elements that are not contained within the PML regions. Implementing the details of the finite element scheme as described in [13] the eigenproblem

is obtained where *K* and *M* are square sparse matrices of dimension 3*N* where *N* is the number of nodes in the mesh, *ϕ* is the vector defining the magnetic field vector *H* on the nodes, and *β* is the propagation constant of the fields along the fiber axis. This eigenproblem is then solved to yield the vector field distributions and complex propagation constants of the modes. Appendix A briefly describes the software implementation of this method. The imaginary part of the propagation constant determines the propagation loss through the relationship

The primary performancemetric analyzed here is the achievable higher ordermode propagation loss for a suitably negligible fundamental mode loss.

## 4. Numerical results

The fiber structure shown in Figure 1 is envisioned here as a component of a fiber laser or amplifier system thatmay be packaged in a standard rack-mount configurationwith a coiling diameter under 19″ or 48cm. Applications for such a system will also typically require a delivery fiber that may comprise a substantial fraction of the fiber length that is not part of the primary coil. All calculations were performed at a wavelength of λ=1064nm which is common for devices employing Ytterbium-doped fiber. The photonic crystal lattice spacing was also kept uniform at a value of 12.0*µ*m yielding a flat-to-flat core diameter of 41*µ*m. The Young’s modulus of borosilicate glass comprising the SAP is typically 60–64 GPa whereas that of fused silica is 73 GPa or approximately 20% higher. Thus the fiber naturally coils perpendicular to the axis along which the SAP are located or along the *x* axis defined here. Figure 4 shows the propagation loss of the first four guided modes of the fiber as a function of the hole diamter to lattice pitch ratio *d*/Λ for a coil diameter of 47.6 cm in the *x* direction. As this is a birefringent fiber, the two LP_{01} modes have different propagation constants. The one with the greater (lesser) propagation constant is the “slow” (“fast”) mode. The fast mode has a lesser effective index and experiences higher propagation losses. Also plotted are the losses for the LP_{11o} and LP_{11e} modes.

For high-efficiency devices, fundamental mode loss should be kept as small as possible. For many applications a loss of less than 1% or 0.04 dB is adequate. Typical Ytterbium dopant concentrations provide a multi-mode pump absorption of approximately 3–4 dB/m with a 400 *μ*m diameter pump cladding yielding device lengths of approximately 4–5 meters of active fiber for a single-end-pumped configuration. This means that the fundamental mode loss should be kept below approximately 8×10^{-3} dB/m. This requirement may be relaxed somewhat depending on variations in signal power along the length of the fiber. Figure 4 reveals a fairly slow variation in propagation losses with air hole size. The optimal hole size appears to be in the range of *d*/Λ=0.11-0.12 which yields a minimum higher-order mode loss of 1 dB/m.

If the fiber is coiled in the *y* direction, the modal losses are much higher for a given air hole size as shown in Figure 6. This is due to the SAP which assist the air hole lattice in confining the mode when the fiber is coiled in the *x* direction. In these calculations, the SAP have a refractive index that is 0.006 below that of pure fused silica, a value that is currently employed in manufactured birefringent photonic crystal fibers. This effect is so pronounced that even for *d*/Λ=0.17, the fundamentalmode loss is 8.2×10^{-2}, a factor of 10 larger than the target design value. If the air hole size is further increased in order to reduce fundamental mode losses, the higher order mode losses when the fiber is coiled in the *x* direction become negligible leading to poor mode discrimination. The best option appears to be to maintain the smaller air hole size and take care to coil the fiber in its natural orientation in the plane parallel to the SAP. A previously reported fiber design similar to the one considered here [19] employed the parameters Λ=12.3*µm* and *d*/Λ=0.12 which are very close to the optimum values determined here for coiling in the *x* direction. In the proposed scenario, the sensitivity to slight misalignment of the coiling direction becomes important. Figure 8 shows the variation in mode losses as the coiling plane is changed relative to the SAP plane. For *d*/Λ=0.12, the coiling plane may be misaligned by as much as 11° while keeping fundamental mode loss below 1×10^{-2} dB/m.

Integration of this fiber into various laser or amplifier systems would typically require a length of delivery fiber to extend from the main coil. Some applications may also be less sensitive to packaging dimensions therefore the characteristics of the fiber at different coiling radii are of interest. Figures 9 and 10 show the effect of coiling at different radii in both the *x* and *y* directions on the modal losses. Perhaps the most important quality of the fiber in this regard is the relatively gentle variance of bend losses with coiling diameter. As the diameter is varied within the range of 20–28 cm, the higher-order mode loss remains above 1 dB/m while the fundamental mode loss drops from 0.007 dB/m to 0.0006 dB/m. This means that practically speaking, a coiling diameter variation of a few cm in this range is unlikely to have a drastic impact on performance. This relaxes the requirement for precise design and assembly tolerances for devices incorporating fibers such as these.

Another characteristic worth noting is the performance of the fiber with *d*/Λ=0.12 when coiled gently in the *y* direction. A higher order mode loss of 5.5 dB/m is achievable at a diameter of 95 cm with only a 0.0014 dB/m loss in the fundamental mode. If a 1 meter form factor is tolerable, this configuration should be able to achieve superior mode discrimination and efficiency. Alternatively, a larger diameter of curvature in the *y* direction may be incorporated into a tighter primary coil in the *x* direction.

The predicted performance of the PCF described here may be compared to the performance of step-index fiber with a comparable core size. Recently an improved bend loss formula applicable to multi-mode step-index fibers has been put forth and found to agree with experimental measurements and simulations alike [11]. One of the key consequences of this work is that previous calculations of bend loss discrimination between the fundamentalmode and higher-order modes of step-index fibers have been overestimated. The key aspects of this analysis include the effects of bend-induced stress, bend-induced mode field distortion, and bend-induced changes in the modal propagation constants. The computational method used to analyze the PCF was adapted to analyze the mode loss of a coiled step-index fiber with a comparablemode field area to the PCF considered here. This solution method is equivalent to an imaginary distance beam propagation method [20] in the limit that the propagation step size approaches zero. It incorporates stress-induced modification of the effective coiling diameter as well as bend-induced propagation constant and mode field distribution modifications. The first step-index fiber design considered here has a circular core with a diameter of 40*µ*m and a numerical aperture of 0.06 which is typical for large mode area step index fibers. The results are shown in Figure 11. No SAP are incorporated into this fiber design so only one of the polarizations of the fundamental mode is considered although there is some slight stress-induced birefringence due to bending. At a coiling diameter of 12 cm, the calculated stress-induced birefringence of this fiber is on the order of 10^{-7} supporting the necessity of SAP for ensuring sufficient birefringence for single-polarization operation. The calculations indicate that the optimal coiling diameter for this fiber is between 8 cm and 12 cm. A diameter of 8.4 cm yielded a predicted fundamental mode loss of 0.5 dB/m and a higher-order mode (LP_{11o})loss of 2 dB/m. At a somewhat larger diameter of 12.0 cm the fundamental and LP_{11o} mode losses were predicted to be 0.0016 dB/m and 0.019 dB/m respectively. Interpolating the higher-order mode loss to the *x*-coiled PCF value of 1 dB/m yields a fundamentalmode loss of 0.2 dB/m which for a device of several meters would impose a significant penalty on device efficiency. The much tighter optimum coiling diameter for the step-index fiber results in a severe distortion of the fundamental mode and reduction in the mode field area as shown in Figure 12.

The optimum coiling diameter for this fiber is much smaller than for the PCF. This raises the question as to whether a step-index fiber with a numerical aperture comparable to the PCF would exhibit similar performance. To address this question, a fiber design was analyzed with the same 40*µ*m core diameter but a smaller numerical aperture (0.035). This represents and index step of 5×10^{-4}. The reduced NA enables the SIF to behave more like the PCF with respect to bend-loss mode discrimination. This fiber appears to have an optimal coil diameter of 56cm at which the fundamental mode loss is around 0.006 dB/m with a LP_{11o} mode loss of about 2 dB/m. If the fiber is coiled down to the design target of a 19″ (48 cm) diameter, the fundamental mode loss increases to about 0.1 dB/m. The results are summarized in Table 1. The change in fundamental mode loss as coiling diameter is varied between 67cm and 37cm provides a measure of the sensitivity of the fiber to irregularities in the coil configuration. For the low NA SIF the loss decreases from about 2 dB/m to about 3×10^{-4} dB/m or about 5 orders of magnitude. The loss for the PCF coiled in the *y* direction decreases from about 60 dB/m to about 0.06 dB/m or about 3 orders of magnitude. When coiled in the *x* direction, the PCF loss decreases from about 4×10^{-2} dB/m to 2×10^{-5} dB/m, also a change of 3 orders of magnitude. These results show that the PCF is less sensitive to coiling diameter than the SIF and exhibits lower losses when coiled in the plane of the stress rods.

One potential drawback of the meshing approach employed here is that the chosen fiber parameters, particularly the air hole size, directly determine the sizes of the elements. The accuracy of approximations such as the plane wave expansion [14] and multipole method [15] can be checked by examining the changes in the results as the number of terms in a series is increased. The corresponding check for the finite element method is to increase the number of elements employed for a given area. The scheme presented here is at present only implemented for 54 elements per cell of the lattice. Alternatively, the behavior of a simple, un-coiled photonic crystal fiber determined by the finite element approach in the absence of stress contributions may be compared to results obtained with the multipole method [16]. This was carried out

for a PCF consisting of a core of 7 close-packed rods surrounded by 3 rings of air holes with uniform air hole size in the range *d*/Λ=0.5-0.8. The multipole method predicted losses of 46±9% of those predicted by the FEM over 7 different hole sizes. This represents differences in the complex part of the effective index of 1.1±1.0×10^{-8}. The real part of the effective indices differed by 5.9±1.4×10^{-7}. While the agreement could definitely be improved upon, the same numerical scheme was applied here to both PCF and SIF so the conclusions derived from comparing their bend losses should be valid.

The calculations presented here do not always produce smooth, monotonic curves. This indicates that special attention should be given to checking the running parameters. For these calculations, the choices of running parameters include the thickness of the boundary, the distance from the outermost air holes to the boundary, the absorption coefficient *α* characterizing the boundary region, and the approximate effective index of the modes to be calculated. All other inputs are set by the physical properties of the fiber. In order to assess the effect of the running parameters on the results, the results shown in Figure 10 which displayed the most oscillatory behavior, were calculated with boundary thicknesses of 35 and 55 *µm*, cladding to boundary distances of 10 and 20 *µm* and values of *α* of 0.02 and 0.1. The two results for propagation losses in dB/m were indistinguishable when plotted and agreed to within 5%. For the remainder of the calculations, the value α=0.1 was used, a thickness of 35 *µm*, and a boundary distance of 10 *µm* as depicted in Figure 1.

There are several reasons why this complex behavior might be expected. First, coupling between the fundamental mode and cladding modes has been shown to cause non-monotonic behavior in bending losses in photonic crystal fibers [21]. Furthermore, in multi-mode step index fibers, non-monotonic behavior has been obtained both by analytic and computational means. See for example Figure 9 of reference [11]. Another cause of oscillatory behavior may be slight reflections from the perfectly-matched layers [18] which play a role similar to that of reflections from the fiber cladding which have been shown to give oscillatory loss spectra [11]. Finally, of the four modes considered here (the fast and slow fundamental modes and the two LP_{01} modes), the fast fundamentalmode is unique in that it is the only mode polarized along the fast axis. The LP_{11} modes are both slow modes. Stress-induced birefringence can account for modes with different polarizations exhibiting oscillatory loss behavior with peaks and valleys at different points in the parameter space as the stress-induced birefringence causes the guided modes to couple to cladding modes with different effective indices. The simultaneous presence of all of these factors indicates that oscillatory behavior is not in itself cause to doubt the validity of the calculations. Rather, they should be checked with other methods applicable to bend-loss analysis of large mode area fibers as well as experimental results. One possible future avenue of investigation is to use the mode field distributions calculated here to evaluate Equation 13 of reference [11] to provide an independent value of the modal propagation losses.

## 5. Discussion

One way to conceptualize the behavior of photonic crystal fibers is to consider the average refractive index within the cladding. This picture implies that the bend performances of photonic crystal and step index fibers should be the same. This would seem to limit the advantages of photonic crystal fiber designs to providing a way tomore precisely control the effective cladding index through the air hole size and lattice pitch. The calculations presented here suggest that this picture is incomplete. Comparing the effective refractive index of the cladding of each type of fiber provides some insight into the their respective behaviors. Even when effectively raised through coiling, the index of refraction within the air-holes on the outside of the coil is still below the effective index of any mode guided in the core due to the large index step between glass and air. This means that the intensity profile of cladding modes in PCF must distort to avoid the air holes. In SIF, there is no localized region beyond the caustic boundary [11] with an index of refraction below the effective index of the guided mode and so the cladding modes remain undistorted by localized index variations. In PCF, this distortion lowers the effective index of the cladding mode. The different manner in which the effective indices of the cladding modes are determined in PCF and SIF lead to different cladding mode spectra which can differentiate the losses of guided modes in the two types of fibers. In particular PCF can exhibit stronger coupling of higher order modes to cladding modes. An intuitive way to express this property is to say that the intensity lobes characterizing the higher order core modes “fit” within the interhole spaces within the photonic crystal cladding and can therefore leak through the cladding out of the core. Finally, for the PCF designs considered here, the SAP render the core numerical aperture effectively anisotropic causing tigher confinement of the guided modes when the fiber is coiled in the plane of the SAP. This allows the air-holes to be smaller for a given maximum fundamental mode bend loss specification which allows improved bend-loss discrimination of the fundamental mode. A similar enhancement mechanism will likely be exhibited in SIF in-corporating reduced-index SAP as well although this has not yet been investigated.

Photonic crystal fibers of the type considered here are only one class of microstructured optical fibers. The flexibility in tailoring the refractive index profile enables a wide spectrum approaches to design goals for different devices including high-power fiber lasers. This should prove a fruitful avenue for further development. High power devices featuring photonic crystal fiber splices and fiber Bragg gratings are currently available commercially for industrial applications. Consequently photonic crystal fibers comprise a viable approach to developing rugged durable all-fiber format lasers and amplifiers with improved power capacity and beam quality.

## Appendix A: Computational details

The mechanical and electromagnetic stiffness matrices and load vectors were assembled in parallel using MatlabMPI [22]. The equations determining the displacements were solved using PETSc [23]. The eigenvalue problems were solved using the SLEPc [24] implementation of the Krylov-Schur algorithm with a shift-invert spectral transformation.

## Acknowledgments

The author gratefully acknowledges financial support by the High Energy Laser Joint Technology Office through the Multidisciplinary Research Initiative Program and computational support by the Air Force Research Laboratory Major Shared Resource Center and the Department of Defense High Performance Computing Modernization Program User Productivity Enhancement and Technology Transfer initiative.

## References and links

**1. **Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. **30**, 459–461 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-5-459. [CrossRef]

**2. **M. Hildebrandt, M. Frede, P. Kwee, B. Willke, and D. Kracht, “Single-frequency master-oscillator photonic crystal fiber amplifier with 148 W output power,” Opt. Express **14**, 11071–11076 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11071. [CrossRef]

**3. **A. Liem, J. Limpert, H. Zellmer, and A. Tnnermann, “100-W single-frequency master-oscillator fiber power amplifier,” Opt. Lett. **28**, 1537–1539 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=ol-28-17-1537. [CrossRef]

**4. **S. Hfer, A. Liem, J. Limpert, H. Zellmer, A. Tnnermann, S. Unger, S. Jetschke, H. -R. Mller, and I. Freitag, “Single-frequency master-oscillator fiber power amplifier system emitting 20 Wof power,” Opt. Lett. **26**, 1326–1328 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=ol-26-17-1326. [CrossRef]

**5. **I. Zawischa, K. Plamann, C. Fallnich, H. Welling, H. Zellmer, and A. Tnnermann, “All-solid-state neodymiumbased single-frequency master-oscillator fiber power-amplifier system emitting 5.5 W of radiation at 1064 nm,” Opt. Lett. **24**, 469–471 (1999), http://www.opticsinfobase.org/abstract.cfm?URI=ol-24-7-469. [CrossRef]

**6. **J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. **25**, 442–444 (2000), http://www.opticsinfobase.org/abstract.cfm?URI=ol-25-7-442. [CrossRef]

**7. **J. Limpert, O. Schmidt, J. Rothhardt, F. Rser, T. Schreiber, A. Tnnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express **14**, 2715–2720 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-7-2715. [CrossRef]

**8. **T. -w Wu, L. Dong, and H. Winful, “Bend performance of leakage channel fibers,” Opt. Express **16**, 4278–4285 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-6-4278. [CrossRef]

**9. **W. S. Wong, X. Peng, J. M. McLaughlin, and L. Dong, “Breaking the limit of maximum effective area for robust single-mode propagation in optical fibers,” Opt. Lett. **30**, 2855–2857 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-21-2855. [CrossRef]

**10. **J. Fini, “Design of solid and microstructure fibers for suppression of higher-order modes,” Opt. Express **13**, 3477–3490 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-9-3477. [CrossRef]

**11. **R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. **43**, 899–909 (2007). [CrossRef]

**12. **Robert D. Cook, *Finite Element Modeling for Stress Analysis* (Wiley, New York, 1995).

**13. **B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Anisotropic Inhomogeneous Refractive Index Tensor,” IEEE J. Quantum Electron. **44**, 150–156 (2008). [CrossRef]

**14. **Kazuaki Sakoda, *Optical Properties of Photonic Crystals* (Springer, New York, 2001).

**15. **T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322–2330 (2002). [CrossRef]

**16. **B. T. Kuhlmey, University of Sydney Australian Research Counsel Centre of Excellence for Ultrahigh-bandwidth Devices for Optical Systems Microstructured Optical Fibre Utilties Software 2004.

**17. **F. L. Teixeira and W. C. Chew, “General Closed-Form PML Constitutive Tensors to Match Arbitrary Bianisotropic and Dispersive Linear Media,” IEEE Microwave Guid. Wave Lett. **8**, 223–225 (1998). [CrossRef]

**18. **J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, pp. 185–200, (1994). [CrossRef]

**19. **T. Schreiber, F. Rser, O. Schmidt, J. Limpert, R. Iliew, F. Lederer, A. Petersson, C. Jacobsen, K. Hansen, J. Broeng, and A. Tnnermann, “Stress-induced single-polarization single-transverse mode photonic crystal fiber with low nonlinearity,” Opt. Express **13**, 7621–7630 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-19-7621. [CrossRef]

**20. **K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. **38**, 927–933 (2002). [CrossRef]

**21. **J. Olszewski, M. Szpulak, and W. Urbanczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express **13**, 6015–6022 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-6015. [CrossRef]

**22. **Jeremy Kepner, “Parallel Programming with MatlabMPI,” in *Proceedings of the High Performance Embedded Computing (HPEC 2001) workshop*,(MIT Lincoln Laboratory, Lexington, MA, 2001).

**23. **Satish Balay, Kris Buschelman, Victor Eijkhout, William D. Gropp, Dinesh Kaushik, Matthew G. Knepley, Lois Curfman McInnes, Barry F. Smith, and Hong Zhang, *PETSc Users Manual* ANL-95/11 - Revision 2.1.5, Argonne National Laboratory, 2004.

**24. **Vicente Hernandez, Jose E. Roman, and Vicente Vidal, “SLEPc: A Scalable and Flexible Toolkit for the Solution of Eigenvalue Problems,” ACM Transactions on Mathematical Software, 31, 351–362 (2005). [CrossRef]