Analysis of the multifilament core fiber using the effective index theory

Guillaume Canat; Ron Spittel; Sylvia Jetschke; Laurent Lombard; Pierre Bourdon

doi:10.1364/OE.18.004644

1. Introduction

Power scaling of the fiber laser technology has been quite impressive in the recent years. In pulsed regime, multi-megawatt peak power has been obtained for nanosecond pulses in large core fibers operating beyond the single mode regime. Peak power is limited in singlemode fibers by either power handling of the core or nonlinear effects. Special structures have been proposed to decrease the effective non-linear gain in order to scale to higher power. For instance, W like waveguide structures can be designed to reduce the overlap between signal and Raman Stokes wavelength [1]. This solution increases the Raman non-linear threshold but does not address other non-linearities. Moreover the damage threshold is unaffected by this technique. Another approach is then to increase the mode field diameter (MFD) while decreasing the numerical aperture (NA). Standard large mode area (LMA) single mode fibers have a diameter typically limited to 30 µm with 0.06 NA for single mode operation. Photonic crystal fibers (PCF) can be designed to operate in the endlessly single mode regime but are limited by bending losses in the same range of core diameter [2]. Recently, leakage channel fibers have been introduced to address these limitations [3]. These fibers are formed by a core surrounded by a single ring of large holes. These fibers require the core refractive index to be equal to the silica index. For fibers operating at eyesafe wavelengths (1.5 and 2µm) this is typically more difficult than for 1 µm amplification because of high dopant index. In 1.5µm Erbium-Ytterbium fibers, a large amount of phosphorous is used which increases the refractive index. In the same manner, in 2 µm Thulium fibers high efficiency can be obtained at high doping level. But the core refractive index is increased accordingly. Pedestal fibers allow to compensate for this larger NA but their NA is still higher than 0.1 [4]. There are thus still many efforts to propose special fiber designs with large mode effective area operating in the single mode regime or close to this regime and with low bending losses.

Recently, such a new type of large mode fiber has been proposed. Its core is composed of high index filaments surrounded by undoped silica (Fig. 1 ). An Erbium-Ytterbium doped 37-filaments fiber has been prepared and tested in MOPA configuration [5]. The fiber core size was 24 x 32 µm. The amplifier operated in a quasi-single mode regime with M² = 1.3. More recently, a passive 19-filaments fiber has been prepared and tested in the single mode regime [6]. Both experiments show that the MFC fibers exhibit low bending losses at bending radius down to 15 cm. This fiber structure is thus promising. However, the source of these good performances remained unclear. These multifilament fibers can be seen as a version of the multicore fibers with core radius a≤λ. In these fibers, there is a strong coupling between the filaments which enables only a few supermodes to propagate in the core. This situation is different of usual multicore fibers in which the coupling between the cores is low: usual multicore fibers with N cores guide N supermodes which can be studied using the coupled mode theory.

Fig. 1 Cleave of a 28 x 31 µm multifilament core fiber with N = 37 filaments.

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In this paper, we propose a theoretical analysis of the multifilament core fibers. Our theory explains the most significant properties of the MFC. We first developed a finite element modelling (FEM) of the MFC fiber that supports the good experimental results. This model is presented in part 2. This method provides little insight into the fiber properties and is computationally demanding. In part 3, we show that the equivalent step index fiber theory (ESI) fails to explain the fiber performances. In part 4, we model the MFC fiber by an improved equivalent step index fiber with a core refractive index dependent on the wavelength. The equivalent core index is equal to the so-called fundamental space filling mode (FSM), similar to the one used in photonic crystal fibers (PCF) [7]. This semi-analytical model will be referred as the FSM based step index method (ESI-FSM). It is shown to agree with the FEM computations. Part 5 uses this ESI-FSM model to predict the fiber performances.

2. The multifilament core (MFC) fiber

The MFC structure can be described as a microstructured core surrounded by a silica cladding with refractive index n_clad = 1.444 (Fig. 2 ). The core has a periodic structure composed of Erbium-Ytterbium doped filaments with radius a and refractive index n₁ surrounded by a fluorine doped glass with refractive index n₂. The fiber that has been tested has a hexagonal structure with pitch Λ = 5.1 µm. Through this paper we will consider the structure of Fig. 2 with a = 0.9 µm and Λ = 5.1 µm as well as n₁ = 1.455 and n₂ = 1.443 corresponding to Erbium-Ytterbium filaments and fluoride doped glass. The fiber on Fig. 2 has 37 filaments. On an hexagonal lattice, N filaments can be packed into P rings in a regular MFC if N = 1 + 3P(P + 1).

Fig. 2 The multifilament fiber core with hexagonal lattice. The high index filament (red disks) radius is a and the lattice pitch is Λ.

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Modelling can be performed using vectorial finite elements method. Using the commercial software COMSOL Multiphysics, we were able to compute the variations of the fundamental mode effective index $n_{e f f}^{F E M}$ and effective area with wavelength (Fig. 3 ). This method will be used as reference through this paper. At 1.5 µm, the effective area reaches more than 800 µm².

Fig. 3 Evolution of the effective index and effective area of the fundamental mode as a function of wavelength computed with the finite element method.

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The bending losses α for the power propagating in the fundamental mode in a fiber with bending radius R can be computed by surrounding the fiber by a perfectly matched cladding layer (PML). The bending of the fiber was taken into account by transforming the index profile n(r) to n_bend(r) with [8], [9]:

n_{b e n d} (r) = n (r) (1 + 2 \frac{x}{ξ R_{b e n d}}) = n (r) (1 + 2 \frac{x}{R_{e f f}})

where ξ = 1.27 accounts for the bend induced stress in the fiber.

The propagation constant β of the fundamental mode is complex: its imaginary part accounts for attenuation. The propagation losses are computed using the following equation:

α (d B / m) = \frac{20}{\ln (10)} Im β

As it was pointed out the multifilament core fibers exhibit low sensitivity to bending considering the very large area of the fundamental mode (Fig. 4 ). At 1545 nm for instance, the effective area is A_eff = 855 µm² and the critical bending radius where the bending loss reaches 1 dB/m is around 20 cm.

Fig. 4 Modelling of the bending loss as a function of the bending radius for the MFC fiber using the finite element method (circles) and the equivalent step index fiber from moments (squares).

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3. The equivalent step index fiber (ESI) using the theory of moments

To draw a comparison, we can make calculations on an equivalent step index (ESI) fiber. The constant refractive index in the equivalent core will be an averaged version of the MFC fiber refractive index profile. At first sight, the best equivalent step index fiber can be designed using the two first moments of the refractive index distribution [10]. We define the moments G₀ and G_2p of the refractive index profile n(r,θ) by

G_{0} = \frac{1}{2 π} \iint [n {(r, θ)}^{2} - n_{c l a d}^{2}] r d r d θ

G_{2 p} = \frac{1}{2 π} \iint [n {(r, θ)}^{2} - n_{c l a d}^{2}] r^{2 p + 1} d r d θ

It can be shown that in the limit λ→∞, two fibers have the same effective index of the fundamental mode if they have the same moments G_p of all orders [10]. In the long wavelength limit, this equivalence is exact. For a step index fiber with core radius a and core index n_core it implies:

G_{0} = (n_{c o r e}^{2} - n_{c l a d}^{2}) \frac{a^{2}}{2}

G_{2} = (n_{c o r e}^{2} - n_{c l a d}^{2}) \frac{a^{4}}{4}

We can thus compare the MFC fiber with a step index fiber with core radius a_step and numerical aperture NA_step given by

a_{e q}^{E S I} = \sqrt{\frac{2 G_{2}}{G_{0}}}

and

N A_{e q}^{E S I} = \frac{G_{0}}{\sqrt{G_{2}}}

For the MFC of Fig. 2, G₀ = 0.138 µm² and G₂ = 17.1 µm⁴. We thus get

a_{e q}^{E S I}

= 15.74 µm and

N A_{e q}^{E S I}

= 0.034. The corresponding equivalent core index is

n_{e q}^{E S I}

= 1.444386. In this ESI fiber, we can compute the effective index of the fundamental mode using the characteristic equation for step index fibers (Fig. 5 ) [11]. We observe that the effective index predicted by the equivalent step index fiber is always far below the FEM effective index. At 1545 nm for instance, the effective index using the ESI fiber is

n_{e f f}^{E S I}

= 1.444177 whereas

n_{e f f}^{F E M}

= 1.444361 computed using the FEM solver. At large wavelengths, the field only sees the averaged index profile and both values converge.

Fig. 5 Variation of the effective index of the fundamental mode in the MFC and in the equivalent step index fiber (ESI). The dashed line corresponds to λ = 1545nm where the bending loss modeling has been performed.

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The bending loss of the fundamental mode in a bent fiber has an exponential dependence on the bending radius normalized by a. critical bending radius R_crit defined by [12]:

R_{c r i t} = \frac{8 π^{2} n_{s t e p}^{2} a_{s t e p}^{3}}{λ^{2} W^{3}} \approx \frac{8 π^{2} n_{c l a d}^{2}}{λ^{2} w^{3}}

with

W = k a_{s t e p} \sqrt{n_{e f f}^{2} - n_{c l a d}^{2}} = w a_{s t e p}

.

This definition is quite general and can be used for arbitrary refractive index profiles with cylindrical symmetry. The much lower value of $n_{e f f}^{E S I}$ predicted by the ESI theory leads to a value of w lower than in the real MFC using $n_{e f f} = n_{e f f}^{F S M}$ . This explains why we cannot compare the bending loss of the MFC and the bending loss of a step index with the same averaged core refractive index. Figure 4 shows that the bending losses computed with this equivalent fiber are overestimated. The equivalent fiber built using the moments is an insufficient first order approximation of the real MFC profile. The real fiber structure has large index fluctuations due to the high index filament. For wavelengths with realistic values, the effective index of the core is thus wavelength dependent: at short wavelength, the light is confined into the high index filament whereas at long wavelength, the light penetration into both glasses is similar. This phenomenon is similar to the dependence of the effective cladding index in PCF on wavelength. We will thus adapt the effective index model developed for the PCF analysis with a wavelength-dependent cladding index to the MFC with a wavelength-dependent core index.

4. The equivalent step index fiber based on the fundamental space filling mode (ESI-FSM) calculation

Due to the total internal reflection mechanism, a mode will propagate in the MFC if it can travel with a propagation constant β such that the effective index n_eff = β/k is larger than n_clad. If the radius of the MFC core is increased to infinity, the light will only see a periodic composite medium with unit cell identical to the unit cell of the MFC. It will travel with an effective propagation constant β_FSM which is the fundamental space filling mode (FSM) constant [13]. In this ESI-FSM model, the equivalent core index is defined by $n_{e q}^{F S M}$ = β_FSM/k. For a core of finite size, we have $n_{e q}^{F S M} > n_{e f f}^{F S M} > n_{s i l i c a}$ . The value of β_FSM can be determined by analyzing only the elementary cell of the lattice which is a hexagonal cell with a filament at the center and edge size $c = \sqrt{3} / 3 Λ$ (Fig. 6 ). As noticed by Midrio [13], the periodicity of the lattice and the hexagonal geometry impose symmetry properties to the electric and magnetic field at the point P on the symmetry axes which can only be satisfied if the longitudinal components of the fields satisfy E_z(P) = H_z(P) = 0.

Fig. 6 The unit cell of the infinite hexagonal lattice in the fundamental space filling mode theory.

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The geometry of the problem can be simplified by considering that the cell is circular with radius R. Equating the filling ratio in both geometries yields

R = Λ {(\sqrt{3} / 2 π)}^{1 / 2}

In this cylindrical geometry, the FSM can be computed using a vectorial semi-analytical technique developed for PCF (same as in reference [13]). However, the unit cell is different: in PCF fibers a low index central air hole is surrounded by silica, whereas in the MFC a high index filament is surrounded by a low index glass. In both cases, the circular approximation imposes the boundary condition E_z(R,θ) = H_z(R, θ) = 0.

Following conventional fiber optics, we can write the electric and magnetic longitudinal components using Eq. (11) and (12).

In the filament (r<a):

\begin{array}{l} E_{z}^{} (r, θ) = A J_{l} (u r) \exp i l θ \\ H_{z}^{} (r, θ) = B J_{l} (u r) \exp i l θ \end{array}

In the low index glass (r>a):

\begin{array}{l} E_{z}^{} (r, θ) = C P_{l} (w r) \exp i l θ \\ H_{z}^{} (r, θ) = D P_{l} (w r) \exp i l θ \end{array}

where

(r, θ)

are the polar coordinates,

u^{2} = k^{2} (n_{1}^{2} - n_{}^{F S M}^{2})

,

w^{2} = k^{2} (n_{}^{F S M}^{2} - n_{2}^{2})

, l is an integer and A, B, C, D are constants.

Function P_l is determined from the boundary condition at r = R:

P_{l} (W) = I_{l}^{} (W) K_{l}^{} (W \frac{R}{a}) - K_{l}^{} (W) I_{l}^{} (W \frac{R}{a})

U = u a

We obtain the following characteristic equation for the hybrid modes guided by the filaments.

[{\hat{j}}_{l}^{} (U) + {\hat{p}}_{l}^{} (W)] [n_{1}^{2} {\hat{j}}_{l}^{} (U) + n_{2}^{2} {\hat{p}}_{l}^{} (W)] = l^{2} (\frac{1}{U^{2}} + \frac{1}{W^{2}}) (\frac{n_{1}^{2}}{U^{2}} + \frac{n_{2}^{2}}{W^{2}})

with

\begin{array}{l} {\hat{j}}_{l}^{\begin{array}{l}  \end{array}} (U) = \frac{J_{l}^{'} (U)}{U J_{l} (U)} \\ {\hat{p}}_{l}^{\begin{array}{l}  \end{array}} (W) = \frac{P_{l}^{'} (W)}{W P_{l} (W)} \end{array}

where the primes denote derivatives with respect to the argument and I, J and K denote the Bessel functions. The characteristic Eq. (14) can be solved for

n_{}^{F S M}

. Its highest value is obtained for l = 1 corresponding to the HE modes.

n_{e q}^{F S M}

is defined to be this highest value. It decreases with wavelength (Fig. 7 ). In the limit of large λ,

n_{e q}^{F S M}

converges to the geometric average index value:

\lim_{λ \to \infty} n_{e q}^{F S M} = f n_{1} + (1 - f) n_{2} = n_{a v}

with f the filling ratio

f = 2 π / \sqrt{3} {(a / Λ)}^{2}

. In the short wavelength region, we get the filament refractive index value n₁. The large variation of the equivalent refractive index introduces a large core index dispersion which can be interpreted as waveguide dispersion in the core elementary cell.

Fig. 7 Variation of the core equivalent index from FSM as function of the normalized wavelength. The dashed line corresponds to the equivalent step index computed by the ESI-FSM theory.

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Having computed the equivalent step index $n_{e q}^{F S M}$ , we need to define an equivalent core radius $a_{e q}^{F S M}$ _. We choose $a_{e q}^{F S M}$ in order to have the same core area in the cylindrical equivalent core and in the hexagonal MFC. For a N filaments MFC fiber, we have:

a_{e q}^{F S M} = Λ {(\frac{N \sqrt{3}}{2 π})}^{1 / 2}

For the fiber of Fig. 2, we get

a_{e q}^{F S M}

= 16.28 µm and at 1545 nm

n_{e q}^{F S M}

= 1.444582. We can define an effective normalized frequency by the equation

V_{e f f} = \frac{2 π}{λ} Λ {(\frac{N \sqrt{3}}{2 π})}^{1 / 2} {(n_{F S M}^{2} (λ) - n_{c l a d}^{2})}^{1 / 2}

Contrary to PCF where the variation of n_clad can compensate for the λ⁻¹ behaviour, the variation of

n_{e q}^{F S M}

in MFC enhances the V variation with λ. We check that the

a_{e q}^{F S M}

definition using (17) is correct by comparing the cut-off wavelengths of the LP₁₁ mode for the MFC and for the equivalent step index fiber for V_eff = 2.405.

5. MFC fiber properties from the effective index model

We can thus model the MFC as a step index fiber with core index n_core = $n_{e q}^{F S M}$ and core radius $a_{e q}^{F S M}$ surrounded by silica with refractive index n_clad. We can then compute the value of $n_{e f f}^{F S M}$ using the characteristic equation of ESI-FSM fiber and compare it to the value $n_{e f f}^{F E M}$ computed using the Finite Element Method on the MFC structure. Figure 8 shows that an excellent agreement between both values of the effective index is obtained.

Fig. 8 Comparison of the effective index of the fundamental mode computed using the finite elements method (FEM) (squares) and the Effective Step Index Approximation (ESI-FSM) (disks) for a = 0.9 µm and Λ = 4.5 µm, 5.1 µm and 6.5 µm.

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This ESI-FSM theory can predict any property of the MFC which is only related to $n_{e f f}^{F S M}$ or $n_{e q}^{F S M}$ . Equation (18) can be used to determine the single mode regime. Figure 9 depicts the V_eff value as a function of the fiber design parameters a/Λ and λ/Λ. In step index fibers, the useful operation region is limited at high V values by the single mode limit at V = 2.405 and at the low V values by the bending losses at around V = 1.5. The equivalence between both fibers implies that the useful operating region can be extracted from our theory.

Fig. 9 Variation of the effective normalized frequency V_eff with the normalized wavelength and the a/Λ parameter. The single mode region is in the direction of the arrows. The black circle shows the fiber described in Fig. 2.

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The fundamental mode radius of this equivalent step index fiber is given by the Marcuse relation:

ω_{e q} = \sqrt{\frac{N \sqrt{3}}{2 π}} Λ (0.65 + \frac{1.619}{V_{e f f}^{3 / 2}} + \frac{2.879}{V_{e f f}^{6}})

where V_eff is the (effective) normalized frequency given by Eq. (18).

If we use this relation to estimate the effective area of the fundamental mode A_eff = πω² _eqwe find results very closed to the finite elements method (Fig. 10(a) ). The difference tends to increase at short and long wavelengths corresponding to very large and very small V_eq but it is smaller than 10%. We can expect deviation to the Marcuse relation at large V_eff as the mode shape will differ more and more from a gaussian shape.

Fig. 10 Comparison of the fundamental mode effective area (a) and bending loss (b) computed at λ = 1545 nm using the Marcuse equations and the effective index computed with the ESI-FSM compared (line) to numerical calculations using COMSOL (squares).

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Bending loss for the MFC can be computed using the Marcuse equation and the value of $n_{e f f}^{F S M}$ [9]:

2 α = \frac{π^{1 / 2} u^{2} \exp (- \frac{2}{3} \frac{w^{3} R_{e f f}}{k^{2} n_{e f f}^{2}})}{2 R_{e f f}^{1 / 2} w^{3 / 2} V_{e f f}^{2} K {(w a_{e q})}^{2}}

where

u^{2} = k^{2} (n_{e q}^{F S M}^{2} - n_{e f f}^{F S M}^{2})

and

w^{2} = k^{2} (n_{e f f}^{F S M}^{2} - n_{c l a d}^{2})

. Figure 10(b) shows once again an excellent agreement between numerical and semi-analytical calculations.

6. Conclusion

We have proposed a semi-analytical analysis of the multifilament core fiber using the equivalent step index technique based on fundamental space mode (ESI-FSM) technique. The fundamental space filling mode was computed on a circular cell providing an equivalent step index model for the MFC much more accurate than modeling based on the moment theory. It allows to predict the fiber properties using semi-analytical expressions. Using this technique we were able to determine the useful design domains of the MFC from the effective normalized frequency value. The low bending loss and large effective area of the fundamental mode were also recovered.

The main interest of the MFC fiber structure is its emulation of a low NA step index fiber using large NA doped filaments for eyesafe wavelength amplification. This theory will be useful to optimize the MFC fiber design.

References and links

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4. K. TankalaB. Samson, et al.., “New developments in High Power Eye-Safe LMA fibers,” Proc. SPIE 6102, 610206 (2006). [CrossRef]

5. G. Canat, S. Jetschke, L. Lombard, S. Unger, P. Bourdon, J. Kirchhof, A. Dolfi, V. Jolivet, and O. Vasseur, “Multifilament-core fibers for high energy pulse amplification at 1.5 µm with excellent beam quality,” Opt. Lett. 33(22), 2701–2073 (2008). [CrossRef] [PubMed]

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Analysis of the multifilament core fiber using the effective index theory

Abstract

1. Introduction

2. The multifilament core (MFC) fiber

3. The equivalent step index fiber (ESI) using the theory of moments

4. The equivalent step index fiber based on the fundamental space filling mode (ESI-FSM) calculation

5. MFC fiber properties from the effective index model

6. Conclusion

References and links

Cited By

Figures (10)

Equations (22)

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