## Abstract

The mode-selection method based on a single-mode microstructured optical fiber (MOF) in the multicore fiber (MCF) lasers is presented. With an appropriate choice of the designed parameters of the MOF, the power coupling coefficient between the fundamental mode (FM) of the MOF and the in-phase mode can be much higher than those between the FM and the other supermodes. As a result, the in-phase mode has the highest power reflection on the right-hand side of the MCF laser cavity, and dominates the output laser power. Compared to the MCF lasers based on the free-space Talbot cavity method, the MCF lasers with the MOF as a mode-selection component have higher effectiveness of the in-phase mode selection.

© 2008 Optical Society of America

## 1. Introduction

Over the past few years, multicore fibers have been the subject of extensive research in an effort to achieve high powers and good beam quality in fiber lasers. The experimental studies on the MCF laser with a Yb-doped 7-core hexagonal structure demonstrated the far-field pattern that is typical for a single-mode output at laser power that is higher than 100 W [1]. In-phase mode operation of a Yb-doped 19-core fiber amplifier was demonstrated with a Gaussain beam used as a seed [2]. In a multicore fiber laser, each core is designed to support a single mode only, and optical fields propagating in these cores are coupled, resulting in what are called supermodes. It has been demonstrated that diminishing the core-cladding index step can lead to a reduction of the amount of the guided modes due to extension of the field from the cores [3]. Among all supermodes, only the in-phase mode, where all cores have the same phase, has the preferable Gaussian-like far-field intensity distribution. Since all supermodes in MCF laser compete with each other by sharing the population inversion, it is very important to build a fiber laser cavity that establishes solely the in-phase mode and suppresses the other modes [4]. One particularly useful method for in-phase mode selection is to utilize the free-space Talbot effect, which states that a coherent one-dimensional periodic wave reproduces its initial field distribution after it propagates a certain distance [5, 6]. Figure 1(a) shows the experimental setup of the MCF laser with a free-space Talbot cavity. The MCF supermodes exit the right-hand end of the fiber and re-enter the MCF after completing the propagation distance Z_{d}. However, a pure in-phase mode is not obtained because of mode mixing, and the inclusion of free-space optics in the resonator decreases the laser efficiency owing to the additional cavity losses [4]. Recently, a mode-selection approach based on the all-fiber Talbot resonator has been obtained: end-coated passive coreless fibers of controlled lengths were spliced to the ends of the active multicore fibers. This approach has been successfully employed for phase-locking 19-core and 37-core fiber lasers [7, 8].

In this paper, an approach that effectively selects the in-phase mode of MCF lasers is presented. A piece of passive MOF with an appropriate geometry is spliced to one end of the MCF as shown in Fig. 1(b). The contents of this paper are arranged as follows. In Section 2, the supermode patterns of the 19-core and 37-core fibers are calculated by a fully vectorial finite element method (FEM). In Section 3, the physical mechanism of the in-phase mode selection using the MOF is explained. The MOF formed by a single missing air-hole and three air-hole rings with two different kinds of air-hole diameters, is presented as a mode-selection component. Furthermore, the dependencies of the confinement losses of the FM (*C*
_{FM}) and cladding modes or leaky modes (*C*
_{LM}) on the air-hole diameter of the outmost air-hole ring are studied. The properties of the MCF lasers are analyzed based on the rate equations in Section 4. The numerical results show that the mode-selection mechanism based on the MOF can offer a high effectiveness of the in-phase mode selection. The MOF length can be shortened by decreasing the air-hole diameter of the outmost air-hole ring due to an increase of *C*
_{LM}.

## 2. The supermodes of the MCF

For the MCF (Fig. 2), each single-mode core has a radius *a _{MCF}* of 3.5

*µ*m, core separation

*d*of 10.5

_{MCF}*µ*m, and numerical aperture (NA) of 0.065. The inner cladding radius is assumed to be 150

*µ*m. The supermodes of the MCF are computed by the commercial software based on the finite element method (COMSOL Multiphysics™ [9]), using anisotropic perfectly matched absorbing boundary layers. Figures 3 and 4 show some important modes (intensity patterns) for the 19-core and 37-core fiber, respectively. The modes are numbered in increasing order of their propagation constants. The in-phase mode is the last mode (the 19th mode for the 19-core and the 37th mode for the 37-core fiber).

## 3. In-phase mode selection method based on the MOF

MOFs guide light with two kinds of effects: one is based on total internal reflection (TIR); the other is based on photonic bandgap (PBG). In this paper, the first kind of MOF or index-guiding MOF is considered. It has been already demonstrated that triangular MOF with a silica core formed by removing the central air-hole can be designed to be Endlessly Single-Mode (ESM) [10, 11]. This property can be useful in designing Large-Mode Area (LMA) single-mode PCF [12, 13].

It was demonstrated theoretically and experimentally that a high energy 19-core Yb-doped fiber amplifier can operate in its fundamental in-phase mode with a Gaussian beam as seed [2]. The reason is that the in-phase mode of the MCF can be excited strongly by the Gaussian beam with an optimum beam waist, and thus dominates the output laser power in the MCF amplifier. In Ref. [2], the output laser beam of the single-core fiber laser with single transverse mode was used as seed source because that, the FM of the fiber laser is close to a Gaussian beam. Similarly, the FM of the MOF can also be approximated by the Gaussian beam, and the mode field area of the FM depends on the air-hole diameter and hole-to-hole spacing for a given operating wavelength [13, 14]. In this paper, the MOF is used as a mode-selection component, and the physical mechanism can be understood as follows. When the supermodes of the MCF are coupled into the left-hand end of the MOF, the FM and cladding modes are excited and propagate along the MOF. After reflection at the right-hand end of the MOF, the FM and cladding modes are coupled into the right-hand end of the MCF, where the supermodes of the MCF are excited. Assuming that the confinement losses [15] of the cladding modes excited by the supermodes are much higher than that of the FM, the FM will dominate after a certain propagation distance in the MOF. Additionally, the parameters of the MOF should be optimized so that a maximum power coupling coefficient between the FM of the MOF and the in-phase mode can be obtained, meanwhile, the power coupling coefficients between the FM and the other supermodes decrease to the minima. As a result, the in-phase mode has the highest power reflection at the right-hand side of the laser cavity, and thus dominates the output power in the MCF laser. Based on the above analysis, the power coupling coefficients and the confinement losses of the FM and cladding modes must be considered in the process of determining the parameters of the MOF.

Figure 5 shows the cross section of the MOF, which is formed by a single missing air-hole and three air-hole rings, with two different kinds of air-hole diameters, *d*
_{1} and *d*
_{2}. *Λ* is the hole-to-hole spacing. The mode fields and associated confinement losses are calculated by the FEM.

#### 3.1 Coupling between the supermodes and the FM of the MOF

The power coupling between the in-phase mode and the FM of the MOF plays an important role in the mode-discrimination mechanism. The power coupling coefficient is given by [4]

where *A _{i}* and

*A*are the amplitudes of the

_{FM}*i*th supermode and the FM of the MOF (normalized with respect to power), and

*η*is the power coupling coefficient of the

^{FM}_{i}*i*th supermode to the FM.

*η*and

_{19}^{FM}*η*are the power coupling coefficients of the in-phase mode to the FM for the 19-core and 37-core fibers, respectively.

_{37}^{FM}It has been demonstrated that, for a given MCF, the power coupling coefficient between the Gaussian beam and the in-phase mode can reach the maximum value by optimizing the Gaussian beam waist [2]. Similarly, for a given in-phase mode, *η _{19}^{FM}* (

*η*) can also reach the maximum value by optimizing the mode field diameter (MFD) of the FM of MOF, which depends strongly on the hole-to-hole spacing and slightly on the air hole size [14]. Figures 6(a) and (b) show the dependencies of

_{37}^{FM}*η*(19-core) and

_{19}^{FM}*η*(37-core) on

_{37}^{FM}*d*

_{1}/

*Λ*respectively. It can be seen that, for a given air-hole diameter, the optimum value of

*d*

_{1}/

*Λ*can be obtained by calculating the power coupling coefficient of the in-phase mode to the FM of the MOF, where

*d*

_{1}/

*Λ*<0.43. For the 19-core fiber, when

*Λ*=31

*µ*m,

*η*reaches the maximum value of 0.9675,

_{19}^{FM}*η*=0.0034,

_{7}^{FM}*η*=0.0042. The other power coupling coefficients are much less than 0.001 and are hence negligible. For the 37-core fiber, when

_{1}^{FM}*Λ*=38

*µ*m,

*η*reaches the maximum value of 0.9588,

_{37}^{FM}*η*

_{23}

^{FM}=0.001,

*η*=0.0067, and the others are negligible.

_{19}^{FM}#### 3.2 Confinement losses of the FM and cladding modes of the MOF

In a MOF with a finite number of air-hole rings, no true bound modes exist and every mode guided by the fiber (including the FM) has a confinement loss [15]. As a result, the confinement losses of the FM and cladding modes must be taken into account when determining the length of the MOF.

Figure 7 shows the intensity profiles of the FM and seven lowest-order cladding modes of the MOFs. The cladding modes are classified into four mode groups, which are group A including mode A_{1} and A_{2}, group B, group C including mode C_{1} and C_{2}, and group D including mode D_{1} and D_{2}. The modes of each mode group are degenerate, and the confinement losses of the modes are nearly equal.

So far, only the case of *d*
_{1}=*d*
_{2} has been considered. We next study the dependencies of the confinement losses of the modes on the air-hole diameter of the outmost air-hole ring *d*
_{2}. The confinement losses of the FM and cladding modes are depicted as a function of *d*
_{2} in Fig. 8. It can be seen that the confinement losses of the FM and cladding modes increase with decreased *d*
_{2}. For example, for the MOF with *d*
_{1} of 12.4 *µ*m and *Λ* of 38 *µ*m, the confinement losses corresponding to the mode group A, B, C, and D increase from 37, 54, 93 and 104 dB/m to 176, 199, 245 and 140 dB/m with a decrease of *d*
_{2} from 12.4 to 9 *µ*m, respectively, while the corresponding confinement loss of the FM increases from 0.007 to 0.03 dB/m. Since the air-holes provide the confinement mechanism by the total internal reflection [13], decreasing *d*
_{2} allows the field to penetrate farther into the cladding. Consequently, the confinement losses of the modes increase with the decreased *d*
_{2}. Based on the above analysis, there is a trade off between *C*
_{FM} and *C*
_{LM} in the process of determining *d*
_{2}. In both cases of *d*
_{1}=12 *µ*m and *Λ*=31 *µ*m, and *d*
_{1}=12.4 *µ*m and *Λ*=38 *µ*m, we take *d*
_{2}=10 *µ*m. Figure 9 shows the corresponding intensity profiles of modes. Compared to the results of Fig. 7, due to a decrease of *d*
_{2}, the cladding modes are less confined as shown in Fig. 9. As a result, the intensity distributions of the cladding modes shift into the outer ring region, especially for the mode group A, B, and C.

#### 3.3 Theory model of the MCF laser

The rate equations, which describe the interplay between the pump, signal, and ion populations in the MCF laser based on the free-space Talbot cavity, are presented in Ref. [4]. The rate equations are also applicable to the MCF lasers with the MOF as a mode-selection component by applying the appropriate boundary condition on the right-hand side of the MCF. On the right-hand side of the MCF, the boundary condition is given by

where *P*
^{+}
* _{si}* and

*P*

^{-}

*represent the forward and backward signal powers of the*

_{si}*i*th supermode,

*L*

_{MCF}is the MCF length, and {

*P*

^{+}

*(*

_{si}*L*

_{MCF})} ({

*P*

^{-}

*(*

_{si}*L*

_{MCF})}) represents a column vector containing all the forward (backward) signal powers at

*z*=

*L*

_{MCF}. {

*η*

^{Ω}

*} is also a column vector, where*

_{i}*η*

^{Ω}

*represents the power coupling coefficient between the ith supermode and the mode Ω of the MOF. The index Ω is a symbolic variable, and is used to denote the modes of the MOF, including the FM and the seven lowest-order cladding modes (A*

_{i}_{1}~D

_{2}).

*α*

^{Ω}is the power attenuation factor corresponding to the mode Ω due to the confinement loss, and is given by

where Im(*β*
^{Ω}) represents the imaginary part of the propagation constant corresponding to the mode Ω, and *L*
_{MOF} is the MOF length.

## 4. Numerical results

#### 4.1 The length of the MOF

For the MCF lasers the effectiveness of the in-phase mode selection can be defined as the degree of suppressing all modes but the in-phase mode. For the purpose of investigating the effectiveness, the factor *Q* is defined as the ratio of the output laser power of the in-phase mode to the total output power, and is given by

where *P _{si}*

^{out}represents the output laser power of the

*i*th supermode, and

*P*

_{sm}^{out}is the output laser power of the in-phase mode (

*P*

_{s19}^{out}for the 19-core fiber and

*P*

_{s37}^{out}for the 37-core fiber). In this definition, a high value of

*Q*corresponds to a high effectiveness of the in-phase mode selection.

From the viewpoint of the compactness of the fiber lasers, the MOF with shorter fiber length is desired. Figure 10 shows the output laser power of the in-phase mode and the corresponding factor *Q* as a function of *L*
_{MOF} for the 19-core lasers, where two MOFs with different *d*
_{2} are compared. Similar results for the 37-core fiber lasers are shown in Fig. 11. The detail parameters used in simulations are given in table 1. One can see that, for the MOF with *d*
_{1}=*d*
_{2}=12 *µ*m as shown in Fig. 10(a), when *L*
_{MOF}<0.03 m, *P _{s19}*

^{out}and

*Q*are lower than 281.27 W and 86.07% respectively. Compared to the results of the Fig. 10(a), for the MOF with

*d*

_{1}=12

*µ*m and

*d*

_{2}=10

*µ*m,

*P*

_{s19}^{out}and

*Q*are higher than 334.2 W and 98.55% when

*L*

_{MOF}>0.02 m. One can conclude that the length of the MOF required to obtain a high output laser power of the in-phase mode and

*Q*can be shortened by decreasing the air-hole diameter

*d*

_{2}. Consequently, an appropriate decrease of the air-hole diameter

*d*

_{2}is a good solution to shorten the fiber length of the MOF when the output laser power of the in-phase mode does not suffer from the increase of

*C*

_{FM}.

#### 4.2 Comparison between two mode-selection methods based on the free-space Talbot cavity and MOF

As can been seen in Fig. 12, the output laser powers of the supermodes are plotted as a function of Z_{d} for the 19-core and 37-core fiber lasers respectively. The parameters are identical to those of Fig. 10 and Fig. 11. One can see that, the output laser powers of the in-phase mode reach the maximum values of 186.6 and 333.65 W when Z_{d}=2.2 and 2.8 mm respectively, and the corresponding *Q* are 79.06% and 72.49%. The effect of mode mixing can explain why a pure in-phase mode is not obtained [4]. Compared to the results, the mode-selection method based on the MOF has a better effectiveness of the in-phase mode selection.

## 5. Conclusion

In this paper, the mode-selection method based on a single-mode MOF in the MCF lasers is presented. The designed MOF has a central core region formed by a missing air-hole, and three air-hole rings. With an appropriate choice of the design parameters of the MOF, the power coupling between the FM of the MOF and the in-phase mode can be much higher than those between the FM and the other supermods. Additionally, because that the confinement loss of the FM is much less than those of the cladding modes of the MOF, all modes but the FM are radiated off, thus leading to a single-mode operation after a certain propagation distance. As a result, the in-phase mode has the highest power reflection at the right-hand side of the laser cavity, and dominates in the MCF laser. From the viewpoint of the compactness of the MCF laser, the MOF with shorter length is desired. In this respect, increasing the confinement losses of the cladding modes is a good solution. Numerical simulations show that, with an appropriate decrease of the air-hole diameter of the outmost air-hole ring, the confinement losses of the cladding modes have a more considerable increase compared with that of the FM. This is helpful to a further shortening of the MOF length. The effectiveness of the in-phase mode selection is compared between the mode-selection method based on the free-space Talbot cavity and that based on the MOF for the 19-core and 37-core fiber lasers. The numerical results show that the MCF laser with the MOF as a mode-selection component has higher output laser power of the in-phase mode and corresponding ratio *Q*.

## Acknowledgments

This work is supported by National 863 High Technology Project (Grant No. 2007AA01Z 258).

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