Design and tolerance analysis of a low bending loss hole-assisted fiber using statistical design methodology

doi:10.1364/OE.16.005061

Design and tolerance analysis of a low bending loss hole-assisted fiber using statistical design methodology

Jurgen Van Erps, Christof Debaes, Tomasz Nasilowski, Jan Watté, Jan Wojcik, and Hugo Thienpont »View Author Affiliations

Optics Express, Vol. 16, Issue 7, pp. 5061-5074 (2008)
http://dx.doi.org/10.1364/OE.16.005061

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▸ Article Outline
– Abstract
1. Introduction
2. Modeling and analyzing conventional single-mode fiber SMF-28
3. Design of low bending loss hole-assisted fibers
4. Detailed optical simulation of the resulting hole-assisted fiber designs
5. Conclusion
– References and links

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Abstract

We present the design of a low bending loss hole-assisted fiber for a 180°-bend fiber socket application, including a tolerance analysis for manufacturability. To this aim, we make use of statistical design methodology, combined with a fully vectorial mode solver. Two resulting designs are presented and their performance in terms of bending loss, coupling loss to Corning SMF-28 standard telecom fiber, and cut-off wavelength is calculated.

1. Introduction

Although present-day service providers bridge the last kilometers in the telecom network using copper wires, the increasing number of subscribers and the potential growth of the bandwidth demand in the coming years require a shift to all-optical full service access networks [1

D. B. Payne and R. P. Davey, “The future of fibre access systems?,” B T Technol. J. 20, 104–114 (2002). [CrossRef]

]. One of the drivers to allow for low cost fiber-to-the-home and fiber-in-the-home deployments is a low bending loss fiber that can be handled as though it were a metal cable for ease of handling. Indeed, bend diameter limitations of standard single-mode fiber is becoming a problem for access-type wiring, including home wiring, which is likely to be curved in small curvatures at the corners of walls and in ducts [2

K. Himeno, S. Matsuo, N. Guan, and A. Wada, “Low-bending-loss single-mode fibers for fiber-to-the-home,” J. Lightwave Technol. 23, 3494–3499 (2005). [CrossRef]

The simplest way to reduce bending loss of a fiber is to increase the refractive index difference between the core and the cladding in the step-index profile. When doing this, it is necessary to decrease the core size accordingly to keep the fibers operating in the single-mode regime. Fibers of this type are commercially available but suffer from a major drawback as a consequence of this way of bend loss reduction: the Mode Field Diameter (MFD) of these fibers shows a large mismatch with the MFD of Corning SMF-28 standard telecom single-mode fiber. This in turn results in unacceptably large losses when butt-coupling or splicing these specialty fibers to SMF-28. To tackle the trade-off between bending and splice loss in a single-mode fiber, fibers with a trench-index profile have been proposed [2

K. Himeno, S. Matsuo, N. Guan, and A. Wada, “Low-bending-loss single-mode fibers for fiber-to-the-home,” J. Lightwave Technol. 23, 3494–3499 (2005). [CrossRef]

]. In recent years, a number of hole-assisted fibers, in which a number of air holes is distributed around the core to confine the light to the core region when the fiber is exposed to a macro-bend, have been proposed [2]–[6]

K. Himeno, S. Matsuo, N. Guan, and A. Wada, “Low-bending-loss single-mode fibers for fiber-to-the-home,” J. Lightwave Technol. 23, 3494–3499 (2005). [CrossRef]

. For use in Fiber-To-The-Home (FTTH) network deployments, the hole-assisted low bending loss fibers should exhibit the following properties: minimal coupling loss when butt-coupled to SMF-28 (i.e. have matching MFD’s), cut-off wavelength well below 1310nm and last but not least minimal bending loss for small bending radii.

Statistical design methodology is well-known and widely used for experimentation or for process optimization. The planning and conducting of experiments along with the analysis of the resulting data to draw valid and objective conclusions, is often referred to as “Design of experiments” [7

D. C. Montgomery, Design and Analysis of Experiments, 5th ed., (John Wiley & Sons, New York, 2001).

]. However, these methods can also prove to be powerful in the design and simulation of systems when simulation time becomes an issue [8

T. J. Santner, B. J Williams, and W. I. Notz, The Design and Analysis of Computer Experiment (Springer-Verlag, 2003).

]. In this paper, we present the use of statistical design methodology and tools for the design and optimization of a hole-assisted fiber fulfilling the threefold requirements mentioned earlier. The specific application for which we designed a low bending loss fiber is a novel 180 °-bend fiber socket [9

J. Van Erps, et al., “Mass manufacturable 180°-bend single mode fiber socket using hole-assisted low bending loss fiber,” IEEE Photon. Technol. Lett. 20, 187–189 (2008). [CrossRef]

]. This 180°-bend fiber socket targets the connection of two side-by-side positioned standard single-mode fibers by means of a looped fiber, bent on a radius of 5mm, as shown in Fig. 1. Compared to other hole-assisted low bending loss fiber designs, an efficient coupling to SMF-28 is of utmost importance in this application.

Fig. 1. Schematic representation of a 180°-bend fiber socket using low bending loss hole-assisted fiber bent on a 5mm radius to interconnect two SMF-28 fibers.

Finally, the statistical analysis enables us to perform an extensive tolerance analysis for manufacturability of the designed hole-assisted fibers by making use of Monte-Carlo simulations. The optical simulations are performed using Lumerical MODE Solutions [10

Lumerical MODE Solutions™, http://www.lumerical.com/mode.php.

], a fully vectorial mode solver based on a finite difference engine. The hole-assisted fiber designs resulting from the statistical design are fully analyzed and their bending loss performance is compared to conventional single mode fiber SMF-28.

2. Modeling and analyzing conventional single-mode fiber SMF-28

Since we are interested in the coupling loss to SMF-28, we start with the calculation of the fundamental mode of this standard telecom single-mode fiber at 1310nm and at 1550nm. The implementation in MODE Solutions is done with the following parameters. The radius of the GeO2-doped core is 4.15µm and its index of refraction as a function of wavelength is calculated using [11

H. R. D. Sunak and S. P. Bastien, “Refractive index and material dispersion of doped silica in the 0.6–1.8µm wavelength region,” IEEE Photon. Technol. Lett. 1, 142–145 (1989). [CrossRef]

\frac{n^{2} - 1}{n^{2} + 2} = \sum_{i = 1}^{3} \frac{(A_{i} + B_{i} f) λ^{2}}{λ^{2} - z_{i}^{2}}

(1)

where A ₁=0.2045154578; A ₂=0.06451676258; A ₃=0.1311583151; B ₁=-0.1011783769; B ₂=0.1778934999; B ₃=-0.1064179581; z ₁=0.06130807320×10^-6 µm; z ₂=0.1108859848×10^-6 µm; z ₃=8.964441861×10^-6 µm and f=0.033, the molar fraction of the dopant. The cladding, having a diameter of 125µm, is modeled as Corning fused silica with the Sellmeier dispersion equation coefficients found in [13

Corning HPFS® Standard Grade, http://www.corning.com/docs/specialtymaterials/pisheets/H0607_hpfs_Standard_ProductSheet.pdf.

We use a simulation area of 50µm×50µm with a grid spacing of d_x =d_y =0.25µm for this and all subsequent simulations, unless specified otherwise. We use Perfectly Matched Layer (PML) boundary conditions for the horizontal boundaries of the simulation area [14

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computational Phys. 114, 185–200 (1994). [CrossRef]

]. The vertical boundaries are delimited by a PML on one side and a symmetric boundary on the other side, as illustrated in Fig. 2. Notice that by this choice of simulation area, we use the simplified model of a core surrounded by an infinite cladding. The fundamental modes of this fiber and their respective waveguide (confinement) loss at respectively 1310nm and 1550nm are stored as reference modes for later coupling loss calculations. Notice that only the waveguide confinement loss is calculated, whereas no material or scattering loss has been taken into account in the simulations. For comparison purposes later on, we simulate the bending characteristics of the SMF-28 by sweeping the bending radius from 17mm down to 4mm, for a bend in the horizontal plane (see Fig. 2), at 1310nm and at 1550nm wavelength. MODE Solutions calculates the modes in a bent waveguide by solving Maxwell’s equations in a cylindrical coordinate system and measures the radiative losses caused by the bend by using PML boundary conditions to absorb the radiation from the bent waveguide. The results are shown in Fig. 3 (and Fig. 9), where the bending loss is calculated for the fundamental mode of the fiber. This Fig. shows that the bending loss of SMF-28 increases rapidly for decreasing bend radii. To prove the validity of our simulations, we have added to this graph the bend loss for SMF-28 single-mode fiber using the analytical method proposed by Faustini [12

L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15, 671–679 (1997). [CrossRef]

]. An excellent agreement between the analytical results and the simulated values can be clearly observed. If an extra coating layer had been taken into account (rather than using an infinite cladding), oscillations would appear in the bending loss curve.

Fig. 2. Definition of the simulation area of 50µm×50µm and the boundary conditions of SMF-28 in MODE Solutions.

Fig. 3. Bending loss of SMF-28 as a function of bending radius at 1310nm and at 1550nm.

Fig. 4. Low bending loss hole-assisted fiber, design Type A (left) and Type B (right). The GeO₂-doped core, indicated in light gray, has a radius c (not indicated). The white zones represent air holes, the gray zones represent Corning fused silica.

3. Design of low bending loss hole-assisted fibers

In this section, we will design a hole-assisted fiber with low bending losses for small bending radii and minimal coupling loss when butt-coupled to SMF-28. As we will see, a good way to mitigate the bending loss is to completely surround the GeO₂-doped core with an air gap to confine the light to the core region. To work towards the manufacturability of such a fiber, we investigate various ways to connect the inner cladding to the outer cladding, in a similar way as for the suspended core fiber presented in [15

A. S. Webb, F. Poletti, D. J. Richardson, and J. K. Sahu, “Suspended-core holey fiber for evanescent-field sensing,” Opt. Eng. 46, 010503 (2007). [CrossRef]

]. Two different connection schemes result in two designs: in design type A, we use a straight bridge between the inner and the outer cladding, whereas in design type B, capillaries are linking the inner and outer cladding, as illustrated in Fig. 4.

3.1. Hole-assisted fiber Type A

The parameters of the first design, as indicated in Fig. 4, are the core radius c, the inner cladding radius r, the bridge thickness t and the hole width w. Using MODE Solutions, we calculate the modes of the straight fiber and of the bent fiber for a bending radius of 5mm, both at 1310nm and at 1550nmwavelength. The bending radius of 5mmwas chosen in view of the special 180 °-bend fiber socket for which the low bending loss hole-assisted fiber was originally designed [9

J. Van Erps, et al., “Mass manufacturable 180°-bend single mode fiber socket using hole-assisted low bending loss fiber,” IEEE Photon. Technol. Lett. 20, 187–189 (2008). [CrossRef]

]. The coupling loss to SMF-28 is given by the overlap integral η between the fundamental mode of the straight hole-assisted fiber (E⃗,H⃗) and the fundamental mode of the SMF-28 (E⃗_SMF ,H⃗_SMF ), found in section 2 [16

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

η = ℜ [\frac{(\int \vec{E} \times {\vec{H}}_{S M F}^{*} \cdot d \vec{S}) (\int {\vec{E}}_{SMF} \times {\vec{H}}^{*} \cdot d \vec{S})}{\int \vec{E} \times {\vec{H}}^{*} \cdot d \vec{S}}] \frac{1}{ℜ [{\vec{E}}_{SMF} \times {\vec{H}}_{SMF}^{*} \cdot d \vec{S}]}

(2)

where ℜ indicates the real part. The bending loss can be calculated in two ways. Either we track the waveguide loss of the fundamental mode of the bent fiber, or we calculate the bending loss by taking into account all the modes present in the bent fiber, in which case the expression for the bending loss BL becomes:

B L = \sum_{i = 1}^{n} L_{i} η_{i}

(3)

where n is the number of modes ϕ_i (E⃗, H⃗) found by MODE Solutions for the bent hole-assisted fiber and L_i is the waveguide loss of each mode ϕ_i . η_i is the overlap of each mode ϕ_i of the bent fiber with the fundamental mode ζ (E⃗, H⃗) of the straight hole-assisted fiber. This second method will result in an overestimation of the bending loss, since it assumes an abrupt transition between a straight and a bent fiber. However, if the transition occurs smoothly, i.e. if we ensure that the rate of change of the bend radius is small enough, the propagation along the waveguide will be approximately adiabatic [17

J. D. Love and C. Durniak, “Bend loss, tapering, and cladding-mode coupling in single-mode fibers,” IEEE Photon. Technol. Lett. . 19 1257–1259 (2007). [CrossRef]

], in which case calculation of the bending loss of the fundamental mode only would suffice. Moreover, since the higher-order modes in the bent fiber are closer to the inner cladding–air interface, they will be more susceptible to scattering loss. Therefore, we decide for the statistical design that follows to define the bending loss as the waveguide loss of the fundamental mode of the bent fiber, and suppose an adiabatic coupling between the modes of the straight and the bent hole-assisted fiber. In section 4, where we analyse the performace of the resulting optimal fiber designs, we compare both methods of bending loss calculation. Because we designed these low bending loss fibers originally with a specific 180°-bend fiber socket application in mind as mentioned earlier [9

J. Van Erps, et al., “Mass manufacturable 180°-bend single mode fiber socket using hole-assisted low bending loss fiber,” IEEE Photon. Technol. Lett. 20, 187–189 (2008). [CrossRef]

], we also define a total link loss parameter. This is defined as follows: it comprises twice the coupling loss to SMF-28 and the bending loss corresponding to one and a half turn (540 °) of 5mm radius (see Fig. 1). The extra turn is added in view of providing some fiber overlength for cleaving purposes. In the statistical design that follows, we target a minimization of this total link loss at 1310nm and at 1550nm.

The approach which is commonly used in optical simulations relies on a sensitivity analysis of one factor at a time, in which each factor is successively varied over its chosen range with the other factors held constant at a so-called baseline level. The main disadvantage of this strategy is that it fails to consider any possible interaction between the factors. The correct approach to dealing with several factors is therefore to conduct a factorial design, in which factors are varied together instead of one at a time [7

D. C. Montgomery, Design and Analysis of Experiments, 5th ed., (John Wiley & Sons, New York, 2001).

]. The effect of a factor is defined to be the change in response produced by a change in the level of the factor. If the number of factors becomes too large to investigate all combinations of factor levels, only a subset of the runs can be made by using a so-called fractional factorial design. A major use of fractional factorials is in screening designs, in which many factors are considered and the objective is to identify those factors (if any) that have large effects. In this work, we will adopt Plackett-Burman designs, which are defined as two-level fractional factorial designs for studying up to k≤N-1 factors in N runs, where N is a multiple of 4 [18

R. L. Plackett and J. P. Burman, “The design of multifactorial experiments,” Biometrika 33, 305–325 (1946). [CrossRef]

]. Plackett-Burman designs are very efficient screening designs when only main effects are of interest.

The factors that are identified as important during the screening design are then investigated more thoroughly in subsequent designs. To this end, we will use response surface methodology, in which the goal is to minimize the response influenced by several variables. A response surface is generated by fitting the relationship between the response y (in our case the total link loss), and the independent variables x to a second-order model [7

D. C. Montgomery, Design and Analysis of Experiments, 5th ed., (John Wiley & Sons, New York, 2001).

y = β_{0} + \sum_{i = 1}^{k} β_{i} x_{i} + \sum_{i = 1}^{k} β_{i i} x_{i}^{2} + \sum_{j = 1}^{k} \sum_{i < j}^{k} β_{i j} x_{i} x_{j} + ε

(4)

Of course, it is unlikely that this polynomial model will be a reasonable approximation of the true relationship over the entire space of the independent variables, but for a relatively small region, they usually work quite well.We will use a Box-Behnken design for fitting this response surface [19

G. E. P. Box and D.W. Behnken, “Some new three level designs for the study of quantitative variables,” Technometrics 2, 455–476 (1960). [CrossRef]

]. This design is formed by combining full factorial designs (requiring 2 ^k runs for k factors) with incomplete block designs and are very efficient in terms of the number of required runs.

Fig. 5. Pareto charts resulting from the Plackett-Burman design analysis, showing that the important factors at 1310nm are the inner cladding radius r, hole width w and bridge thickness t. At 1550nm, the core radius c is also significant.

Our design strategy is thus as follows: first, we perform a Plackett-Burman fractional factorial screening design to determine which are the significant parameters in our system and then we make a refined surface response Box-Behnken analysis with the selected critical parameters. From the resulting surface response fitting coefficients, we can determine the optimal hole-assisted fiber design and in a final step perform Monte-Carlo simulations to determine the tolerance for manufacturability of each parameter.

3.1.1. Plackett-Burman screening analysis

In Table 1 we show the Plackett-Burman design table, including the lower and upper level values assigned to each of the four parameters of our system (c, r, w and t, as defined above). The factor levels were chosen from expected technological fabrication constraints and initial parameter sweeps. No blocking or randomization of run order was used here, since it does not make any sense when using the statistical design tools for optimization via simulations, as opposed to the case of physical realizations in real design of experiments [7

D. C. Montgomery, Design and Analysis of Experiments, 5th ed., (John Wiley & Sons, New York, 2001).

]. The total link loss values at 1310nm and at 1550nm result from simulations with MODE Solutions, and are used separately as the response in the statistical analysis. The latter was carried out with the help of Minitab 14 Statistical Software (by Minitab Inc., [20

Minitab Statistical Software, http://www.minitab.com/products/minitab/.

]). Effect plots (called Pareto charts) allow to look at both the magnitude and the importance of an effect. The Pareto charts resulting from the analysis of the data in Table 1 are shown in Fig. 5, where the reference line corresponds to a significance level α=0.05 [21

G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for experimenters: An Introduction to Design, Data Analysis and Model Building (John Wiley & Sons, New York, 1978).

]. Any effect that extends past this reference line is potentially important with 95%(=1-α) confidence. From these charts, we can see that the important factors in our design are the inner cladding radius r, the hole width w and the bridge thickness t at 1310nm and that, at 1550nm, the core radius c also turns out to be a significant factor. Therefore we select all 4 factors for the subsequent Box-Behnken design to fit the total link loss response surface.

3.1.2. Refined Box-Behnken analysis

The three-level Box-Behnken design with the four significant factors is shown in table 2. We use the same high and low levels for the factors as the ones used in the Plackett-Burman design,

Table 1. Plackett-Burman design table for hole-assisted fiber Type A. The factors are the core radius c, the inner cladding radius r, the bridge thickness t and the hole width w. Two responses, the loss at 1310nm and at 1550nm, are calculated using MODE Solutions.

Run	c (µm)	t (µm)	r (µm)	w (µm)	Loss 1310nm (dB)	Loss 1550nm (dB)
1	4.3	1.0	12.0	5.0	5.345×10^-3	2.327×10^-2
2	4.3	3.0	18.0	2.5	1.425×10^-3	3.997×10^-3
3	4.0	1.0	18.0	5.0	2.055×10^-3	1.484×10^-3
4	4.3	3.0	12.0	5.0	5.166×10^-3	2.228×10^-2
5	4.3	3.0	18.0	2.5	1.425×10^-3	3.997×10^-3
6	4.3	1.0	18.0	5.0	2.167×10^-3	1.738×10^-3
7	4.0	3.0	18.0	5.0	1.953×10^-3	6.962×10^-4
8	4.0	3.0	12.0	5.0	5.237×10^-3	2.148×10^-2
9	4.0	3.0	12.0	2.5	5.129×10^-3	2.114×10^-2
10	4.3	1.0	12.0	2.5	5.345×10^-3	2.327×10^-2
11	4.0	1.0	18.0	2.5	2.055×10^-3	1.480×10^-3
12	4.0	1.0	12.0	2.5	5.435×10^-3	2.251×10^-2

but a middle level is now added. The bridge thickness t is varied between 1.0µm (the expected lower limit for practical fabrication of the fiber) and 3.0µm. The boundary values for the inner cladding radius r are 12.0µm and 18.0µm. The core radius c ranges from 4.0µm and 4.3µm and finally, the hole width w is varied between 2.5µm and 5.0µm. Notice that the last two runs in table 2 are replications of the center point, for which we changed the simulation grid spacing by ±8%. Including these extra runs allows the Box-Behnken analysis to take an estimate of the simulation error into account [7

D. C. Montgomery, Design and Analysis of Experiments, 5th ed., (John Wiley & Sons, New York, 2001).

The resulting quadratic response surface is generated by Minitab from the Box-Behnken design in Table 2 according to Eq. (4). The coefficient of determination R ² is 98.7% for 1310nm and 99.7% for 1550nm, indicating that the model fits the data very well. The minimum of this response surface gives us the optimal system parameters for both wavelengths: c=4.15µm, r=16.9µm, w=5.0µm and t=2.0µm. However, as the response surface is only quadratically fitted, we end up with total link loss values at this optimum that are negative, which is not physically possible. Because we want to use the response surface for a Monte-Carlo tolerance analysis, we decide to perform a second Box-Behnken analysis, in which the ranges for the factor levels are tightened around the found optimum, such that the resulting response surface would better fit the physical system (and thus avoid negative total link loss values). The new factor ranges for this second Box-Benhken analysis are: 4.05µm<c<4.25µm, 16.5µm<r<17.3µm, 4.6µm<w<5.4µm and 1.8µm<t<2.2µm. The quadratic fit parameters resulting from the Box-Behnken analysis by Minitab are given in Table 3. The coefficient of determination R ² has now increased to 100.0% for both the 1310nm and the 1550nm total link loss response.

3.1.3. Monte-Carlo tolerance for manufacturability analysis

An important advantage of having the response surface generated by the Box-Behnken analysis is that it can subsequently be used for Monte-Carlo simulations to acquire information about the tolerance for manufacturability of the hole-assisted fiber design. Based on the quadratic fit parameters in Table 3, we calculate the effect of parameter tolerance by performing Monte-Carlo

Table 2. Refined Box-Behnken design table for hole-assisted fiber Type A. The factors are the core radius c, the inner cladding radius r, the bridge thickness t and the hole width w. Two responses, the loss at 1310nm and at 1550nm, are calculated using MODE Solutions.

Run	c (µm)	t (µm)	r (µm)	w (µm)	Loss 1310nm (dB)	Loss 1550nm (dB)
1	4.0	2.0	12.0	3.75	5.354×10^-3	2.209×10^-2
2	4.3	2.0	12.0	3.75	5.272×10^-3	2.287×10^-2
3	4.0	2.0	18.0	3.75	2.048×10^-3	1.412×10^-3
4	4.3	2.0	18.0	3.75	2.166×10^-3	1.721×10^-3
5	4.15	1.0	15.0	2.5	2.627×10^-4	3.156×10^-3
6	4.15	1.0	15.0	5.0	2.627×10^-4	3.156×10^-3
7	4.15	3.0	15.0	2.5	1.017×10^-4	2.361×10^-3
8	4.15	3.0	15.0	5.0	2.481×10^-4	3.027×10^-3
9	4.0	2.0	15.0	2.5	2.262×10^-3	3.790×10^-3
10	4.3	2.0	15.0	2.5	2.408×10^-3	4.541×10^-3
11	4.0	2.0	15.0	5.0	2.269×10^-3	3.833×10^-3
12	4.3	2.0	15.0	5.0	2.412×10^-3	4.563×10^-3
13	4.15	1.0	12.0	3.75	3.338×10^-3	2.196×10^-2
14	4.15	1.0	18.0	3.75	2.052×10^-5	4.451×10^-4
15	4.15	3.0	12.0	3.75	3.137×10^-3	2.091×10^-2
16	4.15	3.0	18.0	3.75	2.098×10^-4	1.476×10^-3
17	4.0	1.0	15.0	3.75	2.274×10^-3	3.877×10^-3
18	4.3	1.0	15.0	3.75	2.417×10^-3	4.610×10^-3
19	4.0	3.0	15.0	3.75	2.235×10^-3	3.630×10^-3
20	4.3	3.0	15.0	3.75	2.389×10^-3	4.426×10^-3
21	4.15	2.0	12.0	2.5	3.256×10^-3	2.153×10^-2
22	4.15	2.0	18.0	2.5	2.502×10^-5	3.786×10^-5
23	4.15	2.0	12.0	5.0	3.261×10^-3	2.155×10^-2
24	4.15	2.0	18.0	5.0	2.008×10^-5	4.382×10^-4
25	4.15	2.0	15.0	3.75	2.577×10^-4	3.108×10^-3
26	4.15	2.0	15.0	3.75	2.577×10^-4	3.106×10^-3
27	4.15	2.0	15.0	3.75	2.572×10^-4	3.105×10^-3

simulations [22

I. M. Sobol, A Primer for the Monte Carlo Method (CRC Press, 1994).

, 23

Crystal ball predictive modeling software, http://www.crystalball.com/cbpro/index.html.

] with the following parameter distributions: inner cladding radius target of 16.9µm with σ=0.1µm, hole width w target of 5µm with σ=0.1µm, bridge thickness t target of 2µm with σ=0.15µm and finally the core radius c target of 4.15µm with σ=0.05µm. These parameters are independent and were chosen in accordance with the expected fabrication errors. The total link loss probability charts resulting from these Monte-Carlo simulations for 500000 runs are shown in Fig. 6. The mean loss value is 2.8×10^-4 dB, with a standard deviation of 3.2×10^-4 dB for 1310nm and 1.0×10^-3 dB±2.0×10^-4 dB for 1550nm. From these charts, it is clear that the designed hole-assisted fiber design Type A conforms with the fabrication method. However, the sensitivity values (σ) assigned to the parameters might have to be adjusted once the fabrication process for the hole-assisted fiber is established.

Table 3. Quadratic surface response model coefficients resulting from the refined Box-Behnken analysis of hole-assisted fiber design Type A for the following parameters: core radius c, inner cladding radius r, bridge thickness t and hole width w. R ²=100%.

Coefficient	Term	for 1310nm	for 1550nm
β ₀	constant	1.57593	8.91364×10^-1
β ₁	c	-7.57269×10^-1	-4.06006×10^-1
β ₂	r	-5.62109×10^-4	-5.37657×10^-3
β ₃	w	-1.02990×10^-6	-1.37302×10^-6
β ₄	t	-2.60304×10^-5	-1.87920×10^-4
β ₁₁	c ²	9.13375×10^-2	5.00255×10^-2
β ₂₂	r ²	1.90922×10^-5	1.98202×10^-4
β ₃₃	w ²	1.06484×10^-6	4.37578×10^-6
β ₄₄	t ²	1.18719×10^-5	3.30625×10^-5
β ₁₂	c×r	-3.16313×10^-5	-4.67938×10^-4
β ₁₃	c×w	1.86250×10^-6	6.50000×10^-6
β ₁₄	c×t	-8.15000×10^-6	-3.02500×10^-5
β ₂₃	r×w	-3.12500×10^-7	-1.49375×10^-6
β ₂₄	r×t	2.81250×10^-6	1.76719×10^-5
β ₃₄	w×t	-6.56205×10^-6	-2.40250×10^-5

Fig. 6. Results of the Monte-Carlo simulations for design Type A: 2.8×10^-4 dB±3.2×10^-4 dB for 1310nm and 1.0×10^-3 dB±2.0×10^-4 dB for 1550nm.

3.2. Hole-assisted fiber Type B

From a manufacturing point of view, it might be advantageous to use capillaries as connection structures between the inner and the outer cladding since it is easier to stack capillaries and rods to build up the preform of the fiber to be drawn, rather than having to create the bridges used in fiber Type A. To this aim, we introduce a hole-assisted fiber design Type B. In this fiber design, schematically shown in Fig. 4, the core radius c, the inner cladding radius r and the hole width w remain parameters of the system. However, two extra parameters are introduced here: the capillary width d and the capillary overlap o. The latter can be considered as being more or less the equivalent of the bridge thickness t used in fiber design Type A for the mode calculation simulations.

In a similar way as we handled the hole-assisted fiber Type A, we repeat our analysis for the Type B fiber. The range of the parameters c, r and w are chosen the same as for the Plackett-Burman design of hole-assisted fiber Type A. The range for d was set from 1µm to 2µm and for o from 1µm to 3µm. From the Plackett-Burman screening analysis of this system, the only significant parameter turns out to be the inner cladding radius r at a significance level α=0.05, as can be seen from the Pareto charts shown in Fig. 7. However, we chose to include the capillary overlap o and the hole width w in our Box-Behnken analysis, in view of their relatively large importance over the factors c and d. The latter factors are kept at their baseline level (c=4.15µm and d=1.5µm) for the Box-Behnken analysis. The response surface defined in Eq. (4) resulting from this three-factor three-level Box-Behnken analysis allows us again to determine the optimal system parameters. The coefficient of determination R ² is 96.4% for 1310nm and 99.2% for 1550nm. The minimum of the response surface is achieved for r=16.9µm, o=1.0µm and w=5.0µm. The response surface gives again negative total link loss values around this optimum, hence we perform a second analysis with refined ranges for the factors around the found optimum, in analogy to the approach followed for hole-assisted fiber Type A. The ranges for the second Box-Behnken analysis are: 16.5µm<r<17.3µm, 0.8µm<o<1.2µm and 4.6µm<w<5.4µm. The coefficient of determination R ² has increased to 100.0% for both the 1310nm and the 1550nm total link loss response.

Fig. 7. Pareto charts resulting from the Plackett-Burman design analysis for Type B, showing that the inner cladding radius r is the only significant factor at significance level α=0.05.

Fig. 8. Results of the Monte-Carlo simulations for design Type B: 5.2×10^-5 dB±4.5×10^-6 dB for 1310nm and 9.0×10^-4 dB±6.0×10^-5 dB for 1550nm.

Based on the quadratic fit parameters resulting from this second Box-Behnken analysis, we perform Monte-Carlo simulations to acquire the tolerance for manufacturability of the optimal hole-assisted fiber design. The effect of parameter tolerance was determined by performing Monte-Carlo simulations with the following parameter distributions: inner cladding radius r target of 16.9µm with σ=0.1µm, capillary overlap o target of 1.0µm with σ=0.05µm, hole width w target of 5µm with σ=0.1µm, core radius c target of 4.15µm with σ=0.1µm and finally capillary width d target of 1.5µm with σ=0.1µm. The total link loss probability charts resulting from these Monte-Carlo simulations for 500000 runs are shown in Fig. 8. The tolerance window for the manufacturability of the hole-assited fiber design Type B is again excellent, with a mean loss value of 5.2×10^-5 dB, and a standard deviation of 4.5×10^-6 dB for 1310nm and 9.0×10^-4 dB±6.0×10^-5 dB for 1550nm.

4. Detailed optical simulation of the resulting hole-assisted fiber designs

The detailed MODE simulation results for the optimal designs of hole-assisted fiber Type A (c=4.15µm, r=16.9µm, w=5.0µm and t=2.0µm) and Type B (c=4.15µm, r=16.9µm, w=5.0µm, o=1.0µm and d=1.5µm) are shown in respectively Table 4 and Table 5. These tables show the coupling loss to SMF-28 and the bending loss (at 5mm radius) at 1310nm and 1550nm respectively. The bending loss of the fundamental mode is mentioned in the third column, whereas the bending loss BL in the fourth column is calculated using Eq. (3). Notice that the loss for a fiber exposed to a macro-bend of 5mm is calculated for the worst case, i.e. when the fiber is bent in the direction of a bridge. It is clear that the bending loss at 5mm radius is negligible in comparison to the coupling loss to SMF-28. As mentioned in section 3.1, the bending loss calculated using Eq. (3) is higher than when taking bending loss of only the fundamental mode into account. The difference is smaller than one order of magnitude. The total link loss, comprising twice the coupling loss to SMF-28 and the loss for one and a half turn of 5mm radius, results in 5.2×10^-5 dB at 1310nm and 9.0×10^-4 dB at 1550^nm for design Type A and 5.1×10^-5 dB at 1310nm and 9.0×10^-4 dB at 1550nm for design Type B. The total link loss is thus virtually the same for both hole-assisted fiber designs. Notice that the simulation for a single system takes about 20 minutes to complete on a dual-core Pentium D system running at 3.0 Ghz with 2GB of memory. This clearly illustrates the advantage of a statistical approach over a traditional sensitivity analysis by sweeping all parameters of the hole-assisted fiber one by one to come to an optimal design.

Table 4. Simulation results of the hole-assisted fiber design Type A: coupling loss, bending loss of the fundamental mode and bending loss BL calculated using Eq. (3).

Wavelength (nm)	Coupling Loss (dB)	Bend Loss (dB/m)	BL (dB/m)
1310	2.57×10^-5	5×10^-6	9×10^-5
1550	4.50×10^-4	2×10^-5	2×10^-4

Table 5. Simulation results of the hole-assisted fiber design Type B: coupling loss, bending loss of the fundamental mode and bending loss BL calculated using Eq. (3).

Wavelength (nm)	Coupling Loss (dB)	Bend Loss (dB/m)	BL (dB/m)
1310	2.57×10^-5	1×10^-7	3×10^-6
1550	4.50×10^-4	1×10^-6	4×10^-5

4.1. Increasing the number of bridges in design Type A

Because the fabrication of a hole-assisted fiber with three bridges turned out to be extremely tedious, we investigate the influence of increasing the number of bridges connecting the inner cladding to the outer cladding. This might moreover be interesting from a practical point of view, e.g. when cleaving the fiber. When we increase the number of bridges in design Type A from 3 to 24, and keep the other parameters fixed at the same value as used for the Type A (core radius c=4.15µm, inner cladding radius r=16.9µm, hole width w=5µm and bridge thickness t=2µm), the simulation results are shown in Table 6. The total link loss becomes 6.4×10^-5 dB at 1310nm and 8.8×10dB at 1550nm.

Fig. 9. Bend loss performance of hole-assisted fiber Type A (3 bridges), Type A (24 bridges) and Type B versus conventional single mode fiber SMF-28 at 1550nm.

Table 6. Simulation results of the design Type A with 24 bridges instead of 3: coupling loss, bending loss of the fundamental mode and bending loss BL calculated using Eq. (3).

Wavelength (nm)	Coupling Loss (dB)	Bend Loss (dB/m)	BL (dB/m)
1310	2.13×10^-5	2×10^-5	3×10^-4
1550	3.88×10^-4	1×10^-4	6×10^-4

4.2. Bending performance compared to conventional single-mode fiber SMF-28

Figure 9 shows the bending loss performance for various bending radii of the designed hole-assisted fibers in comparison to standard telecom single-mode fiber SMF-28. In these calculations, the waveguide loss of the fundamental mode is being tracked as a function of the radius of the macro-bend to which the fiber is exposed. This graph clearly shows that the bending loss of all the hole-assisted fiber designs presented in this paper is several orders of magnitude smaller than the bending loss of conventional SMF-28. Although all HAF designs show a low bending loss, the hole-assisted fiber of design Type B has the best performance in relative terms. We believe that the small dips in the curves for the hole-assisted fibers are due to a resonance effect that is caused by interference of reflections at the index interfaces [12

L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15, 671–679 (1997). [CrossRef]

], e.g. at the air hole boundaries. Due to a much lower number of air holes, the peaks are in our case much less outspoken than the bending loss oscillations occurring in photonic crystal fibers [24

T. Martynkien, J. Olszewski, M. Szpulak, G. Golojuch, W. Urbanczyk, T. Nasilowski, F. Berghmans, and H. Thienpont, “Experimental investigations of bending loss oscillations in large mode area photonic crystal fibers,” Opt. Express 15, 13547–13556 (2007). [CrossRef] [PubMed]

4.3. Cut-off wavelength calculation

Finally, we check whether the hole-assisted fibers are still operating in the single-mode regime at 1250nm (for a straight fiber) to ensure that the cut-off frequency is far enough below 1310nm as required. We define the fibers as being single mode when the waveguide loss of the fundamental mode is at least three orders of magnitude smaller than the waveguide loss of the higher-order modes. Notice that we use this approach because it is difficult to determine the exact cut-off wavelength of hole-assisted fibers since higher-order modes may exist as leaky modes. In a strict sense, these low bending-loss fibers are multimode fibers as there will exist many modes with a reasonably low loss in the inner cladding. These modes have as a consequence only a small leakage to radiative modes through the bridges (or capillaries) connecting the inner cladding to the outer cladding. However, there is only one mode with significant energy in the core region of the fibers. This mode has furthermore an extremely high overlap (more than 99%) with the fundamental mode of standard single mode fiber (SMF-28). In practice, experimental measurement of the cut-off wavelength of the fabricated hole-assisted fibers will have to prove the validity of our approach to ensure single-mode operation. However, the experimental measurement of the cut-off wavelength of hole-assisted fibers is very tedious because the conventional bending method for cut-off wavelength measurement can not be used in a fiber with superior bending loss performance [25

K. Nakajima, et al., “Cutoff wavelength measurement in a fiber with improved bending loss,” IEEE Photon. Technol. Lett. 16, 1918–1920 (2004). [CrossRef]

5. Conclusion

For the first time to our knowledge, statistical methodology and tools were used for the design of low bending loss hole-assisted fibers. Unlike a sensitivity analysis for one parameter at a time, this method allows to consider interactions between different parameters. Moreover, the statistical approach has a clear advantage over a traditional sensitivity analysis for design optimization in terms of required simulation time. A Plackett-Burman fractional factorial design was used for a first parameter screening for the identification of significant factors, followed by a refined Box-Behnken analysis to generate a response surface for the selected significant factors. From this response surface, the optimal system parameters can be found. The optical simulations were performed by a commercially available fully vectorial mode solver based on a finite difference engine. The knowledge of the response surface allowed us also to easily perform a Monte-Carlo tolerancing analysis for manufacturability of the resulting fiber designs. Finally, the optimal designs resulting from the the statistical approach were thoroughly investigated and their superior bending performance compared to conventional single mode fiber (SMF-28) was shown. The hole-assisted fiber design Type B appeared to have the best performance in terms of bending loss. In [9

J. Van Erps, et al., “Mass manufacturable 180°-bend single mode fiber socket using hole-assisted low bending loss fiber,” IEEE Photon. Technol. Lett. 20, 187–189 (2008). [CrossRef]

], we show experimentally that the hole-assisted fiber Type A with 24 bridges is manufacturable and that no bending loss could be measured for bending radii down to 2mm.

Acknowledgments

This work was supported in part by DWTC-IAP, FWO, GOA, IWT-SBO, the European Network of Excellence on Micro-Optics NEMO, and by the OZR of the Vrije Universiteit Brussel. The work of J. Van Erps and C. Debaes was supported by the Fund for Scientific Research-Flanders (FWO) under a research fellowship.

References and links

1.	D. B. Payne and R. P. Davey, “The future of fibre access systems?,” B T Technol. J. 20, 104–114 (2002). [CrossRef]
2.	K. Himeno, S. Matsuo, N. Guan, and A. Wada, “Low-bending-loss single-mode fibers for fiber-to-the-home,” J. Lightwave Technol. 23, 3494–3499 (2005). [CrossRef]
3.	K. Nakajima, K. Hogari, J. Zhou, K. Tajima, and I. Sankawa, “Hole-assisted fiber for small bending and splice losses,” IEEE Photon. Technol. Lett. 15, 1737–1739 (2003). [CrossRef]
4.	Y. Tsuchida, K. Saitoh, and M. Koshiba, “Design and characterization of single-mode holey fibers with low bending losses,” Opt. Express 13, 4770–4779 (2005). [CrossRef] [PubMed]
5.	N. Guan, et al., “Holey fibers for low bending loss,” IEICE Trans. Electron. E89, 191–196 (2006). [CrossRef]
6.	Y. Bing, K. Oshono, Y. Kurosawa, T. Kumagai, and M. Tachikura, “Low-loss holey fiber,” Hitachi Cable Review 24, 1–4 (2005).
7.	D. C. Montgomery, Design and Analysis of Experiments, 5th ed., (John Wiley & Sons, New York, 2001).
8.	T. J. Santner, B. J Williams, and W. I. Notz, The Design and Analysis of Computer Experiment (Springer-Verlag, 2003).
9.	J. Van Erps, et al., “Mass manufacturable 180°-bend single mode fiber socket using hole-assisted low bending loss fiber,” IEEE Photon. Technol. Lett. 20, 187–189 (2008). [CrossRef]
10.	Lumerical MODE Solutions™, http://www.lumerical.com/mode.php.
11.	H. R. D. Sunak and S. P. Bastien, “Refractive index and material dispersion of doped silica in the 0.6–1.8µm wavelength region,” IEEE Photon. Technol. Lett. 1, 142–145 (1989). [CrossRef]
12.	L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15, 671–679 (1997). [CrossRef]
13.	Corning HPFS® Standard Grade, http://www.corning.com/docs/specialtymaterials/pisheets/H0607_hpfs_Standard_ProductSheet.pdf.
14.	J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computational Phys. 114, 185–200 (1994). [CrossRef]
15.	A. S. Webb, F. Poletti, D. J. Richardson, and J. K. Sahu, “Suspended-core holey fiber for evanescent-field sensing,” Opt. Eng. 46, 010503 (2007). [CrossRef]
16.	A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
17.	J. D. Love and C. Durniak, “Bend loss, tapering, and cladding-mode coupling in single-mode fibers,” IEEE Photon. Technol. Lett. . 19 1257–1259 (2007). [CrossRef]
18.	R. L. Plackett and J. P. Burman, “The design of multifactorial experiments,” Biometrika 33, 305–325 (1946). [CrossRef]
19.	G. E. P. Box and D.W. Behnken, “Some new three level designs for the study of quantitative variables,” Technometrics 2, 455–476 (1960). [CrossRef]
20.	Minitab Statistical Software, http://www.minitab.com/products/minitab/.
21.	G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for experimenters: An Introduction to Design, Data Analysis and Model Building (John Wiley & Sons, New York, 1978).
22.	I. M. Sobol, A Primer for the Monte Carlo Method (CRC Press, 1994).
23.	Crystal ball predictive modeling software, http://www.crystalball.com/cbpro/index.html.
24.	T. Martynkien, J. Olszewski, M. Szpulak, G. Golojuch, W. Urbanczyk, T. Nasilowski, F. Berghmans, and H. Thienpont, “Experimental investigations of bending loss oscillations in large mode area photonic crystal fibers,” Opt. Express 15, 13547–13556 (2007). [CrossRef] [PubMed]
25.	K. Nakajima, et al., “Cutoff wavelength measurement in a fiber with improved bending loss,” IEEE Photon. Technol. Lett. 16, 1918–1920 (2004). [CrossRef]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.4005) Fiber optics and optical communications : Microstructured fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: January 18, 2008
Revised Manuscript: March 22, 2008
Manuscript Accepted: March 24, 2008
Published: March 28, 2008

Citation
Jurgen Van Erps, Christof Debaes, Tomasz Nasilowski, Jan Watté, Jan Wojcik, and Hugo Thienpont, "Design and tolerance analysis of a low bending loss hole-assisted fiber using statistical design methodology," Opt. Express 16, 5061-5074 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-5061

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References

D. B. Payne and R. P. Davey, "The future of fibre access systems?," B T Technol. J. 20, 104-114 (2002). [CrossRef]
K. Himeno, S. Matsuo, N. Guan, and A. Wada, "Low-bending-loss single-mode fibers for fiber-to-the-home," J. Lightwave Technol. 23, 3494-3499 (2005). [CrossRef]
K. Nakajima, K. Hogari, J. Zhou, K. Tajima, and I. Sankawa, "Hole-assisted fiber for small bending and splice losses," IEEE Photon. Technol. Lett. 15, 1737-1739 (2003). [CrossRef]
Y. Tsuchida, K. Saitoh, and M. Koshiba, "Design and characterization of single-mode holey fibers with low bending losses," Opt. Express 13, 4770-4779 (2005). [CrossRef] [PubMed]
N. Guan, et al., "Holey fibers for low bending loss," IEICE Trans. Electron. E89, 191-196 (2006). [CrossRef]
Y. Bing, K. Oshono, Y. Kurosawa, T. Kumagai, and M. Tachikura, "Low-loss holey fiber," Hitachi Cable Review 24, 1-4 (2005).
D. C. Montgomery, Design and Analysis of Experiments, 5th ed. (John Wiley & Sons, New York, 2001).
T. J. Santner, B. J. Williams, and W. I. Notz, The Design and Analysis of Computer Experiment (Springer-Verlag, 2003).
J. Van Erps, et al., "Mass manufacturable 180?-bend single mode fiber socket using hole-assisted low bending loss fiber," IEEE Photon. Technol. Lett. 20, 187-189 (2008). [CrossRef]
LumericalMODE Solutions™, http://www.lumerical.com/mode.php.
H. R. D. Sunak and S. P. Bastien, "Refractive index and material dispersion of doped silica in the 0.6-1.8um wavelength region," IEEE Photon. Technol. Lett. 1, 142-145 (1989). [CrossRef]
L. Faustini and G. Martini, "Bend loss in single-mode fibers," J. Lightwave Technol. 15, 671-679 (1997). [CrossRef]
Corning HPFS® Standard Grade, http://www.corning.com/docs/specialtymaterials/pisheets/H0607 hpfs Standard ProductSheet.pdf.
J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Computational Phys. 114, 185-200 (1994). [CrossRef]
A. S. Webb, F. Poletti, D. J. Richardson, and J. K. Sahu, "Suspended-core holey fiber for evanescent-field sensing," Opt. Eng. 46, 010503 (2007). [CrossRef]
A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
J. D. Love and C. Durniak, "Bend loss, tapering, and cladding-mode coupling in single-mode fibers," IEEE Photon. Technol. Lett. 191257-1259 (2007). [CrossRef]
R. L. Plackett and J. P. Burman, "The design of multifactorial experiments," Biometrika 33, 305-325 (1946). [CrossRef]
G. E. P. Box and D.W. Behnken, "Some new three level designs for the study of quantitative variables," Technometrics 2, 455-476 (1960). [CrossRef]
Minitab Statistical Software, http://www.minitab.com/products/minitab/.
G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for experimenters: An Introduction to Design, Data Analysis and Model Building (John Wiley & Sons, New York, 1978).
I. M. Sobol, A Primer for the Monte Carlo Method (CRC Press, 1994).
Crystal ball predictive modeling software, http://www.crystalball.com/cbpro/index.html.
T. Martynkien, J. Olszewski, M. Szpulak, G. Golojuch, W. Urbanczyk, T. Nasilowski, F. Berghmans, and H. Thienpont, "Experimental investigations of bending loss oscillations in large mode area photonic crystal fibers," Opt. Express 15, 13547-13556 (2007). [CrossRef] [PubMed]
K. Nakajima, et al., "Cutoff wavelength measurement in a fiber with improved bending loss," IEEE Photon. Technol. Lett. 16, 1918-1920 (2004). [CrossRef]

B T Technol. J.

D. B. Payne and R. P. Davey, "The future of fibre access systems?," B T Technol. J. 20, 104-114 (2002). [CrossRef]

Biometrika

R. L. Plackett and J. P. Burman, "The design of multifactorial experiments," Biometrika 33, 305-325 (1946). [CrossRef]

Hitachi Cable Review

Y. Bing, K. Oshono, Y. Kurosawa, T. Kumagai, and M. Tachikura, "Low-loss holey fiber," Hitachi Cable Review 24, 1-4 (2005).

IEEE Photon. Technol. Lett.

J. Van Erps, et al., "Mass manufacturable 180?-bend single mode fiber socket using hole-assisted low bending loss fiber," IEEE Photon. Technol. Lett. 20, 187-189 (2008). [CrossRef]
K. Nakajima, K. Hogari, J. Zhou, K. Tajima, and I. Sankawa, "Hole-assisted fiber for small bending and splice losses," IEEE Photon. Technol. Lett. 15, 1737-1739 (2003). [CrossRef]
H. R. D. Sunak and S. P. Bastien, "Refractive index and material dispersion of doped silica in the 0.6-1.8um wavelength region," IEEE Photon. Technol. Lett. 1, 142-145 (1989). [CrossRef]
J. D. Love and C. Durniak, "Bend loss, tapering, and cladding-mode coupling in single-mode fibers," IEEE Photon. Technol. Lett. 191257-1259 (2007). [CrossRef]
K. Nakajima, et al., "Cutoff wavelength measurement in a fiber with improved bending loss," IEEE Photon. Technol. Lett. 16, 1918-1920 (2004). [CrossRef]

IEICE Trans. Electron.

N. Guan, et al., "Holey fibers for low bending loss," IEICE Trans. Electron. E89, 191-196 (2006). [CrossRef]

J. Computational Phys.

J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Computational Phys. 114, 185-200 (1994). [CrossRef]

J. Lightwave Technol.

L. Faustini and G. Martini, "Bend loss in single-mode fibers," J. Lightwave Technol. 15, 671-679 (1997). [CrossRef]
K. Himeno, S. Matsuo, N. Guan, and A. Wada, "Low-bending-loss single-mode fibers for fiber-to-the-home," J. Lightwave Technol. 23, 3494-3499 (2005). [CrossRef]

Opt. Eng.

A. S. Webb, F. Poletti, D. J. Richardson, and J. K. Sahu, "Suspended-core holey fiber for evanescent-field sensing," Opt. Eng. 46, 010503 (2007). [CrossRef]

Opt. Express

Technometrics

G. E. P. Box and D.W. Behnken, "Some new three level designs for the study of quantitative variables," Technometrics 2, 455-476 (1960). [CrossRef]

Other

Minitab Statistical Software, http://www.minitab.com/products/minitab/.
G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for experimenters: An Introduction to Design, Data Analysis and Model Building (John Wiley & Sons, New York, 1978).
I. M. Sobol, A Primer for the Monte Carlo Method (CRC Press, 1994).
Crystal ball predictive modeling software, http://www.crystalball.com/cbpro/index.html.
A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
Corning HPFS® Standard Grade, http://www.corning.com/docs/specialtymaterials/pisheets/H0607 hpfs Standard ProductSheet.pdf.
LumericalMODE Solutions™, http://www.lumerical.com/mode.php.
D. C. Montgomery, Design and Analysis of Experiments, 5th ed. (John Wiley & Sons, New York, 2001).
T. J. Santner, B. J. Williams, and W. I. Notz, The Design and Analysis of Computer Experiment (Springer-Verlag, 2003).

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