## Abstract

One of the major trends in optical fiber science is to be able to obtain fibers with large-mode-area (LMA), optimized for various applications such as high power delivery, fiber amplifiers, and fiber lasers. In order to ensure the high beam quality and the ultimate controllability of the damage threshold in the fiber’s material, it is required to have a LMA property and of course to operate in a single mode fashion. While the conventional fibers have some difficulties in providing simultaneously LMA, single mode operation, as well as low macro-bending loss characteristics, all-silica holey fibers are highly attractive candidates for realizing LMA single-mode fibers with low bending losses. In this paper, we present a novel type of effectively single-mode holey fibers with effective mode area of about 1400 μm^{2}, small allowable bending radius as small as 5 cm, good beam quality factor of 1.15, and high confinement losses exceeding 1 dB/m for the higher-order mode at 1.064-μm wavelength.

© 2007 Optical Society of America

## 1. Introduction

Large-mode-area (LMA) single mode optical fibers have attracted much attention regarding applications such as high power delivery [1], fiber amplifiers [2, 3], and fiber lasers [4]. The main drawback in the development of high power applications is the nonlinear effect, which can be partly overcome by using optical fibers with LMA, which can maintain the single-mode operation. In principle, LMA single-mode optical fibers can be achieved by using the conventional LMA step-index fiber [5, 6]. However, the conventional LMA step-index fiber has certain disadvantages, for example the minimum allowable radius for the fiber to be bent is quite large (around 30 cm), and in addition few higher-order modes (HOMs) are supported. On the other hand, holey fibers have many interesting properties compared to the conventional fibers. Some of the most interesting properties are small bending losses [7] because of the large index difference between the silica and air, and easy realization of LMA [1] by increasing the air hole spacing. Recently, a holey fiber with one air-hole ring, which has an effective mode area of 1417 μm^{2} at 1.064-μm wavelength and small bending losses, has been reported [3]. However, from the theoretical point of view, the holey fiber reported in Ref. [3] has small confinement losses (less than 0.01 dB/m at 1.064-μm wavelength based on our calculation taking into account the fiber characteristics given in Ref. [3]) for the HOM, which means that the proposed holey fiber can not operate in an effectively single-mode fashion.

In this paper, we propose a novel design method for realizing LMA single-mode holey fibers with effective mode area of 1400 μm^{2}, small allowable bending radius as small as 5 cm, good beam quality factor of 1.15, and high confinement losses exceeding the 1 dB/m for the HOM at 1.064-μm wavelength. The designed holey fiber has a central core region formed by seven missing neighboring air-holes, and a ring core region between the innermost and the outermost air-hole rings. With an appropriate choice of the design parameters, the HOMs in the central core can be resonantly coupled to the much leaky ring-core mode around a certain wavelength range, while on the other hand the fundamental mode is strongly confined into the central core, leading to low bending losses and effectively single-mode operation around the desired operating wavelength.

## 2. Holey fiber design

We consider four types of holey fibers as shown in Fig. 1. In particular Fig. 1(a) shows the cross section of a conventional holey fiber formed by a single missing air hole with only one air-hole ring (denoted from now on as HF1), where the hole pitch is Λ and the air-hole diameter is *d*. Figure 1(b) shows the cross section of a holey fiber formed by seven missing neighboring air-holes with one air-hole ring (denoted through the paper as HF7-1), where the distance *l* is a gap between the air-holes. Figure 1(c) shows the cross section of a holey fiber formed by seven missing neighboring air-holes and only one air-hole ring, with two different kinds of air-hole diameters, *d*
_{1} and *d*
_{2} (HF7-2). We also consider a novel type of holey fiber formed by seven missing neighboring air holes, and a ring core region around the inner air-hole ring as shown in Fig. 1(d) (HF7-3). The dashed circles in Fig. 1(d) denote the silica-based ring-core region.

For applications such as high power beam delivery, fiber amplifiers, and fiber lasers, the optical fiber should exhibit a LMA, single-mode propagation characteristics, with low bending losses. Although it is difficult for the conventional fiber to meet all these requirements at the same time, holey fibers have the possibility of realizing all these specifications, because of their great design flexibility. At first, we consider the holey fiber structure as shown in Fig. 1(a) and we determine the structural parameters, such as the hole pitch Λ and the air-hole diameter-*d*, exhibiting various values of effective mode area *A _{eff}*, while the confinement losses of the HOM-

*L*

^{2nd}, are 0.1 dB/m, 1dB/m, and 10 dB/m, by using a full vectorial modal solver based on the finite element method (FEM) incorporating anisotropic perfectly matched layers as absorbing boundary conditions to evaluate confinement losses described in detail in Ref. [8] and references therein. Here, we briefly list the most important definitions. For a given frequency ω =

*ck*=

*c*2π/λ (where

*c*is velocity of light, and

*k*and λ are the free-space wavenumber and wavelength, respectively) the numerical calculation provides us with a complex propagation constant γ(ω) = β(ω) +

*i*α(ω) where β is the usual propagation constant component along the fiber axis associated with the mode and α is the attenuation constant associated with the exponential decay along the fiber axis. We present the attenuation on a dB-scale by 20 × log

_{10}(

*e*) × α ≅ 8.686 × α. Regarding the effective node area of the fiber core, we use the following definition:

where ** E** is the electric field vector and

*S*denotes the whole fiber cross section.

Figure 2 shows the contour plot of the effective mode area (color map) of the fundamental mode at 1.064-μm wavelength mapped into the range of the design parameters *d*/Λ and Λ, while the solid black curves depicted inside the graph correspond to different values of the confinement losses of the HOM of HF1, shown Fig. 1(a). In this study, we target an effective mode area of 1400 μm^{2} as in Ref. [3] with sufficiently large confinement losses of 1 dB/m for the HOM at a wavelength of 1.064 μm. A holey fiber with an effective mode area of 1400 μm^{2}, can significantly reduce the nonlinear effects even for high-power delivery applications. In addition, we define that the holey fiber with *L*
^{2nd} of 1 dB/m can operate in an effectively single-mode fashion. In order to satisfy all the above mentioned characteristics, we can determine the structural parameters to be: Λ=41.9 μm and *d*/Λ=0.68 for HF1, and these parameters correspond to the black filled circle in Fig. 2. However, we have numerically confirmed that the HF1 with Λ=41.9 μm and *d*/Λ=0.68 has relatively large bending losses of 8 dB/m for 20 cm bending radius at 1.064 μm wavelength, so it is difficult to realize LMA single-mode holey fiber with low bending losses, using the conventional HF1 structure. Notice that the holey fiber structure reported in Ref. [3] has relative air-hole diameters about 0.90 and 0.77 and large hole pitch of 51 μm, which means that we can not plot this structure in Fig. 2 and the reported holey fiber has small confinement losses of the HOM less than 10^{-2} dB/m at 1.064-μm wavelength.

Next, we consider the holey fiber structure as shown in Fig. 1(b) (HF7-1). Due to the core shape of HF7-1, this type of fiber can easily achieve the LMA property when compared with the HF1 by using a small hole pitch Λ. Figure 3 shows the contour plot of the effective mode area (color map) of the fundamental mode at 1.064-μm wavelength mapped into the range of the design parameters *d*/Λ and Λ, while the solid black curves (depicted inside the graph) correspond to different values of the confinement losses of the HOM of HF7-1, while the solid white curves correspond to different values of the distance l between air-holes of HF7-1, shown Fig. 1(b). In order to achieve effective mode area of 1400 μm^{2} and large confinement losses of the HOM, *L*
^{2nd}, above 1 dB/m, we can determine the structural parameters to be: Λ=18.5 μm and *d*/Λ=0.53 for HF7-1, and these parameters correspond to the black filled circle in Fig. 3. However, we have numerically confirmed that the HF7-1 with Λ=18.5 μm and *d*/Λ=0.53 has relatively large bending losses of 2 dB/m for 20 cm bending radius at 1.064 μm wavelength. In order to achieve low bending losses at 1.064-μm wavelength, small value of *l* of about 5 μm is required. Therefore, for low bending losses, we should choose the following values of the design parameters: of Λ≈20 μm and *d*/Λ≈0.75, and these parameters correspond to the white filled circle in Fig. 3. However, the confinement losses of the HOM with these structural parameters are lower than 0.5 dB/km at 1.064-μm wavelength. As a result, the HF7-1 with Λ=20 μm and *d*/Λ=0.75 can not operate in an effectively single-mode fashion at 1.064-μm wavelength because of the low confinement losses of the HOM. So, in the aim of an enhancement of the confinement losses of the HOM we consider the holey fiber structure as shown in Fig. 1(c) (HF7-2). The 12 air-holes with two different diameters of *d*
_{1} and *d*
_{2} are arranged alternatively with hole-pitch Λ. The distance *l*, is a gap between the air-holes, defined as *l*=Λ-*d*
_{1}/2-*d*
_{2}/2.

Figure 4 shows the dependence of the confinement losses of the HOM normalized by the hole pitch Λ, on the value of *d*
_{2}/Λ, and for different values of *d*
_{1}/Λ, where the normalized operating wavelength is λ/Λ = 0.05 (which corresponds to Λ≈20 μm and λ≈-1.064 μm). The red and blue circles correspond to HF7-1 structures with *d*
_{1}/Λ=*d*
_{2}/Λ=0.7 and *d*
_{1}/Λ=*d*
_{2}/Λ=0.75, respectively. In addition, the red and blue dashed curves represent the variation of the normalized confinement losses of the HOM *L*
^{2nd}Λ by varying *d*
_{1}/Λ and *d*
_{2}/Λ while keeping the values of *l*/Λ=0.30 and *l*/Λ=0.25, respectively. From these results, we can clearly observe that the confinement losses of the HOM can be controlled only by a small fraction by changing the values of *d*
_{1}/Λ and *d*
_{2}/Λ in spite of keeping the same value of *l*/Λ. The value of *l*/Λ is an important parameter regarding the bending losses of the fundamental mode and it should be as small as possible. As a result, we can say that, to achieve the effective mode area of the fundamental mode *A _{eff}* = 1400 μm

^{2}as well as the low bending loss properties at λ= 1.064 μm, we should consider the following range for the value of

*l*/Λ<0.27 while Λ≈20 μm (

*l*≈5.4 μm). However, the confinement losses of the HOM within these structural parameters are still lower than 3 dB/km at 1.064-μm operating wavelength. Therefore, it is difficult to realize low bending loss single-mode LMA fibers using HF7-2 structure as in the case of HF1 topology, and we should enhance the confinement losses of the HOM further.

In order to maintain the LMA and to sufficiently enhance the confinement losses of the HOM, in a level of at least 1 dB/m, we propose a novel holey fiber structure as shown in Fig. 1(d) (HF7-3). The central core is formed by seven missing air-holes surrounded by 12 air-holes with diameters of *d*
_{1} and *d*
_{2} arranged alternatively with hole-pitch Λ, as in the case of HF7-2. To enhance the confinement losses of the HOM in the central core, a ring core of pure-silica region is introduced outside the central core. The enhancement of the confinement losses of the higher-order central-core mode can be achieved through the index-matching mechanism between the central-core and the ring-core modes [9–11]. The air-hole diameter-*d* in the outermost air-hole ring is determined in such a way as to match the effective index of the LP_{11}-like HOM in the central core with that in the ring core, at the desired wavelength, which in our case is λ= 1.064 μm. Around this resonant wavelength, the HOMs of the central core are strongly coupled to those of the ring-core, resulting in a sufficient enhancement of the confinement losses of the coupled modes (HOMs). To achieve *A _{eff}* =1400 μm

^{2}, index-matching around 1.064-μm wavelength, and low bending loss characteristics, we set the structural parameters as Λ=20 μm,

*d*

_{1}/Λ=0.95,

*d*

_{2}/Λ=0.51 (

*l*/Λ=0.27), and

*d*/Λ=0.451, and these structural parameters correspond to the black filled circle in Fig. 4.

In Figs. 5(a) and 5(b), the optical field distributions of the HOMs (LP_{11}-like modes) in the central core and the ring core are plotted, respectively. These two modes can be coupled through the resonant coupling effect and Fig. 5(c) shows the corresponding field distribution of the coupled mode at λ=1.064 μm, where Λ=20 μm, *d*
_{1}/Λ=0.95, *d*
_{2}/Λ=0.51, and *d*/Λ=0.451. There is an obvious coupling of the central-core HOM into the outer ring-core. On the other hand, in Fig. 5(d), we plot the optical field distribution of the fundamental mode (LP_{01}-like mode) at λ=1.064 μm. It is evident that the mode is confined strongly into the central core. Figure 6 shows the confinement losses of the HOM for HF7-3 as a function of the operating wavelength, where Λ=20 μm, *d*
_{1}/Λ=0.95, *d*
_{2}/Λ=0.51, and *d*/Λ=0.451. We can clearly see that large confinement losses of the HOM above 1 dB/m can be obtained over 100 nm wavelength range, leading to an effectively single-mode operation in this wavelength range. Due to the large core and the small value of *l*, the holey fiber proposed here supports several higher-order modes in the central core region, however, the bending losses of other modes higher than 2nd-order modes increase dramatically around 20 cm bending radius. As a result, an effectively single-mode operation could be achieved for practical application. Figure 7 shows the dependence of the numerically calculated bending losses of the fundamental mode on the bending radius at 1.064-μm wavelength through the vector FEM [12] in HF7-3. For comparison, we also plot the bending losses of the HF1 and HF7-1 with structural parameters of Λ=41.9 μm and *d*/Λ=0.68 and Λ=18.5 μm and *d*/Λ=0.53, respectively. Although the holey fibers of HF1, HF7-1, and HF7-3 have the same effective mode area of 1400 μm^{2} and the same level of confinement losses of the HOM at 1.064-μm wavelength, the bending losses of HF7-3 are much less than that of HF1, and HF7-3 can be bent down to a bending radius of 5 cm, without a significant increment of the bending losses. In Fig. 8, we plot the optical field distributions in the curved HF7-3, where the operating wavelength is 1.064 μm and the bending radius is (a) 30 cm, (b) 20 cm, and (c) 10 cm. As we can see, the fiber curvature can not affect the coupling phenomenon between the fundamental mode of the central core and the ring-core mode, so low bending losses can be maintained even at a small bending radius.

In Fig. 9, we show the impact of the singlemodeness of the proposed holey fiber (HF7-3) on the fiber curvature. Figure 9 shows the dependence of the wavelength where the phase matching condition between the HOM in the central core and that in the ring-core is satisfied on the bending radius. Black dashed line represents the desired wavelength of 1.064-μm to achieve the phase matching condition at a straight fiber. We can find that the phase matching wavelength shifts to shorter wavelength as the bending radius decreases and the shifting rate is about 2 nm/cm. However, as shown in Fig. 6, the enhancement of the confinement losses of the HOMs can be obtained over 100 nm wavelength range, therefore, we can achieve effectively single-mode operation even in the bent holey fiber.

## 3. Calculation of the beam quality factor

In this section we carry out an important calculation regarding the beam quality factor *M*
^{2} of the HF7-3 with the following design parameters: Λ= 20 μm, *d*
_{1}/Λ = 0.95, *d*
_{2}/Λ= 0.51, and *d*/Λ=0.451. *M*
^{2}
_{x} which is the *x*-component of *M*
^{2} can be expressed as follows [13]:

where the *z*
_{0} is the location of the beam waist and λ is the operation wavelength. The second-moment-based beam width *W _{x}* is defined as follows:

where in Eq. (3), *I*(*x*, *y*) stands for the beam intensity profile in the cross section. *M*
^{2}
_{y} which is *y*-dependency of *M*
^{2} can be also calculated by changing the subscript *x* to *y* in Eqs. (2) and (3). The *M*
^{2} value that we present in this paper is the average value of *M*
^{2}
_{x} and *M*
^{2}
_{y}. The value of *M*
^{2} can be evaluated by calculating the modal spot size variation when the mode propagates in free space, as suggested in Ref. [13]. We have used a vector beam propagation method [14] for the calculation of modal spot size variation, and we have obtained a value of *M*
^{2}=1.15 for the fundamental mode at 1.064-μm wavelength. The conclusion is that although the HF7-3 structure has a ring core region, we can achieve a relative high beam quality factor, a fact which can be advantageous utilized in applications such as high power beam delivery.

## 4. Sensitivity analysis of the proposed holey fiber

To ensure the feasibility of the proposed holey fiber topology, it is important to examine in a detailed way the sensitivity of the structure on the design parameters. Such analysis will inform the prospective reader on the impact of structural tolerances on the performance of the fabricated structure. Specifically we analyze the sensitivity of the *A _{eff}* and

*L*

^{2nd}to the tolerances of all the structural parameters that are Λ,

*d*

_{1}/Λ,

*d*

_{2}/Λ, and

*d*/Λ. Figure 10(a) shows the dependence of the

*A*(red curve, right axis) and

_{eff}*L*

^{2nd}(blue dashed curve, left axis) on the variation of the lattice constant-Λ, while keeping all the other design parameters fixed in their optimized values. In Fig. 10(b) we examine the variation of the same quantities as in Fig. 10(a) but this time as a function of the tolerance of the normalized design parameter

*d*

_{1}/Λ. The impact of the parameter variation

*d*

_{2}/Λ and

*d*/Λ is plotted in Figs. 10(c) and 10(d), respectively. From these results we can estimate the variation of the quantities

*A*and the

_{eff}*L*

^{2nd}at the operating wavelength of 1.064-μm for the case of possible tolerances from the ideal nominal values: Λ=20 μm,

*d*

_{1}/Λ=0.95,

*d*

_{2}/Λ=0.51, and d/Λ=0.451. We can clearly conclude that the

*A*of the proposed holey fiber is roughly constant as a function of the parameter variation Λ,

_{eff}*d*

_{1}/Λ,

*d*

_{2}/Λ, and

*d*/Λ. On the other hand, we see that the effectively single-mode operation can be achieved only by setting the structural parameters of

*d*

_{1}/Λ,

*d*

_{2}/Λ, and

*d*/Λ within a certain allowable range. This is because, by varying these parameters from their idealized values, the proposed holey fiber can not meet any more the condition necessary for the index-matching mechanism between the central-core and the ring-core modes at the desired 1.064-μm wavelength range. The crucial tolerance range within which the index-matching mechanism can result in an effectively single mode operation was estimated to be ± 0.5% to keep the level of

*L*

^{2nd}larger than 1 dB/m.

## 5. Conclusion

We have proposed a novel type of LMA single-mode holey fiber with low bending losses for applications such as, high power beam delivery. The core region of the proposed holey fiber is formed by removing seven neighboring air-holes. The air-hole diameters of *d*
_{1} and *d*
_{2} in the innermost air-hole ring can be tuned to enhance the confinement losses of the HOM and to reduce the bending losses of the fundamental mode. It has been shown through a detailed numerical analysis that it is possible to design a holey fiber with an effective mode area of 1400 μm^{2}, sufficiently enhanced confinement losses of the HOM above 1dB/m for effectively single-mode operation, and low bending losses below 1 dB/m at 5 cm bending radius at λ=1.064 μm. We have succeeded to enhance the confinement losses of the HOMs up to a level of 1400 dB/km at λ=1.064 μm, a result that is found to be superior when compared to the reported holey fiber in Ref. [3], which has confinement losses (not total attenuation) of the HOMs less than 10^{-2} dB/km.

## Acknowledgments

The authors would like to acknowledge stimulating discussions with Dr. N. J. Florous from Hokkaido University.

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