## Abstract

We demonstrate, theoretically and experimentally, a high power/energy 19-core Yb-doped fiber amplifier that operates in its fundamental in-phase mode. The calculated result using an improved coupled mode theory with gain shows that, with a Gaussian beam as seed, the in-phase supermode dominates. Experimentally, we use a Q-switched single-core fiber laser with single transverse mode as seed, and amplify it with a 5.8 m 19-core fiber. The measured near and far fields are close to the in-phase supermode. The measured M^{2} factor of the amplified beam is 1.5, which is close to the theoretical value. A pulse energy gain of 20 dB is obtained with the amplified pulse energy up to 0.65 mJ at a repetition frequency of 5 kHz. No appreciable stimulated Brillouin scattering is observed at this power level.

©2004 Optical Society of America

## 1. Introduction

Double-clad pumped Yb-doped fiber lasers and amplifiers operating at around 1 µm have many industrial, medical, and militarial applications due to their advantages of high power/energy, good beam quality, compactness, and excellent pump power conversion efficiency. Kilowatt single transverse-mode CW lasers [1] and multi-mJ Q-switched (few modes with M^{2} around 3) lasers [2] have been demonstrated. Some applications, such as coherent LIDAR, also require narrow linewidth besides high power/energy and good beam quality. In this case, the Master Oscillator Fiber Amplifier (MOFA) structure is more appropriate because narrow linewidth is not readily obtainable from a high power/energy fiber laser. A low power narrow linewidth laser can be used as the master oscillator, and a high power fiber can be used as the amplifier. A MOFA structure has been demonstrated using a multi-mode (MM) fiber amplifier [3], and Stimulated Brillouin Scattering (SBS) has been observed when average power is over 2 W.

In Ref. [4], multi-core fiber laser was proposed as a candidate for power scaling. A 7-core fiber laser with good beam quality was demonstrated [5], and later was used for second harmonic generation [6]. In a multi-core fiber, the cores are arranged isometrically and they are close enough to be coupled via evanescent wave so that light power can transfer from core to core. Every core supports only a single transverse mode (V value below 2.4). According to coupled mode theory (CMT), the array of cores will have N supermodes, where N is the number of cores. The in-phase supermode, where all cores have the same phase, has the best beam quality (smallest M^{2}) and is the desired operation mode. Although some analysis [7,8] suggests that the in-phase mode is favored in a multi-core fiber laser due to nonlinear refractive index in the array, the mode-competition mechanism is not completely known. Further theoretical and experimental investigation is needed.

In this paper, we show that the in-phase fundamental supermode dominates in a 19-core fiber amplifier when the seed is close to a Gaussian beam. In Section 2, an Improved Coupled Mode Theory (ICMT) [9] with gain is used to show that this is indeed the case by propagating and amplifying a Gaussian input and analyzing the output beam. In Section 3, experimental results are given using a custom-made 19-core fiber as the amplifier and a Q-switched standard single-core fiber laser as the oscillator. The measured near field, far field, and M^{2} factor are close to the in-phase mode. Pulse energy up to 0.65 mJ is obtained with M^{2} around 1.5. The multi-core fiber amplifier has larger effective mode area. This can effectively reduce the laser intensity and related nonlinear effects such as SBS. Appreciable SBS is not observed in our experiment.

## 2. Theory

Reference [10] shows that single transverse-mode can be excited in a passive MM fiber with ultrashort pulses. Here, using a scalar ICMT, we show the in-phase supermode of a 19-core fiber amplifier is favored when a Gassian beam seed is coupled to the input at its waist.

The configuration of the cores in a 19-core amplifier is shown in Fig. 1. The array consists of a central core and two surrounding hexagonal rings of cores. The radius of the cores is *a*, and the distance between adjacent cores is *d*. The cores and the inner cladding have refractive index *n _{1}* and

*n*respectively, so each core has $NA=\sqrt{{n}_{1}^{2}-{n}_{2}^{2}.}$. The array is located inside a D-shape inner cladding with area

_{2}*A*.

_{cl}#### 2.1 The array supermodes

First, we calculate the 19 supermodes of the array without gain. The coupled mode equations are given by

where *z* the propagation direction, **A** is the vector of amplitudes in the cores [**A**
_{1}
**A**
_{2} … **A**
_{19}]′, and **K** is the coupling matrix given by

where **B** is a diagonal matrix with element B_{mm} corresponding to the propagation constant of m^{th} core *β*
_{m}, and the elements of **C** and **K̄** are defined as

where *ε _{l}*(

*x, y*) [

*ε*(

_{m}*x, y*)] is the normalized electric field distribution when only

*l*

^{th}(

*m*

^{th}) core is present,

*k*

_{0}=2

*π/λ*is the wave vector in free space,

*λ*is the free space wavelength,

*n*(

*x, y*) is the refractive index distribution, and

*n*(

_{m}*x, y*) is the refractive index distribution when only

*m*

^{th}core is present. The ICMT differs from classical CMT in that the former includes of the off-diagonal elements of

**C**. By doing so, better accuracy is achieved especially when the V values of the cores are small, and the coupling between them is strong.

Assuming **A**=**A**(0)exp(*iγz*), Eq. (1) can be solved as an eigenvalue problem to obtain the propagation constants *γ* and amplitude distributions **A**(0) of all 19 supermodes. Using NA=0.067, *a*=3.5 µm, *d*=10.5 µm, and *λ*=1.04 µm, the supermodes are calculated and the near fields (intensity) are shown in the attached movie (supermodes), where the first one is the in-phase mode shown in Fig. 2. Once the near field is known, the far field can be calculated using diffraction equation and the M^{2} factor of each mode can be obtained accordingly [11]. The M^{2} factors are calculated to be: 1.29, 6.09, 3.53, 5.45 5.05, 4.95, 4.90, 2.44, 2.44, 3.67, 3.66, 4.75, 6.82, 6.82, 4.75, 4.70, 4.70, 6.14, and 6.14 corresponding to the supermodes in the movie. It is seen that the in-phase mode has much lower M^{2} (1.29) than all other modes (the lowest is 2.44). By observing the near/far field and the M^{2} of the output beam, one can tell whether it is dominated by the in-phase mode or a mix of different modes.

#### 2.2 ICMT with gain

In the case light propagation in the fiber amplifier, gain need to be included. Power gain *g* is equivalent to an imaginary effective index change Δ*n _{eff}*

where *λ _{s}* is laser (signal) wavelength and

*α*is power loss. Real index changes may exist due to two reasons. One is the Kerr nonlinearity [12] and the other is related to the gain through Kramers-Kronig relationship [7,8]. The latter is shown to saturate at high power levels in Ref. 8. Both are neglected here because, for the power level and fiber length under consideration, they have no considerable effects.

Following Ref. 7, the ICMT for light propagation in the multi-core array is given by

where $\left[\frac{\partial {\mathbf{K}}_{\mathit{lm}}}{\partial {n}_{q}}\right]$ is a 19×19×19 matrix describing the effect of the index change in the *q*
^{th} core on the coupling matrix element **K**
_{lm}, and *g _{q}* is the power gain of the

*q*

^{th}core.

The power gain in each core is related to the laser intensity in the core and the pump power. Assuming the amplifier is uniformly pumped with *P _{p}*, the pump intensity is given by

*I*=

_{p}*P*. The power gain in the

_{p}/A_{cl}*q*

^{th}core is given by

where *N _{2q}* (

*N*) is the population density of the upper (lower) energy band of the Yb ions, and

_{1q}*σ*(

_{es}*σ*) is the laser emission (absorption) cross section. The population densities are given by

_{as}where *N _{0}* is the total ion concentration,

*ν*(

_{p}*ν*) is the pump (laser) frequency,

_{s}*I*is the laser intensity in the

_{sq}*q*

^{th}core, and

*τ*is the upper level lifetime.

Because the cores are coupled through evanescent wave, the intensity in the *q*
^{th} core is not determined by **A**
_{q} alone, but should include contributions from other cores

where *A _{co}* is the core area, the “*” sign means complex conjugate, amplitude

**A**is scaled so that the square of its absolute value represents the power, and the 19×19×19 matrix

**E**is given by

where the integration is over the *q*
^{th} core. The total laser power at position *z* is given by

where the matrix **D** is given by

where the integration is over the whole xy plane.

Starting from initial values **A**(0), Eq. (6) is solved using Runge-Kutta method with the relations (7)–(13). The following values are used in the calculation: *N _{0}*=7×10

^{25}m

^{-3},

*λ*=1.04 µm,

_{s}*λ*=0.976 µm,

_{p}*A*=1.26×10

_{cl}^{-7}m

^{2},

*A*=3.85×10

_{co}^{-11}m

^{2},

*σ*=6.×10

_{ap}^{-25}m

^{2},

*σ*=4.×10

_{ep}^{-25}m

^{2},

*σ*=1.4×10

_{as}^{-27}m

^{2},

*σ*=2.5×10

_{es}^{-25}m

^{2},

*P*=25 W, and fiber length

_{p}*L*=5 m. A Gaussian beam is used with a distribution given by

*A*(

*r*)=

*A*

_{0}exp(-

*r*

^{2}/

*σ*

^{2}), where

*r*is the distance to the center and

*σ*is the 1/e point. It is assumed to be coupled to the amplifier. Figure 3(a) and (b) show the near field and far field intensity distributions of the amplified beam with

*σ*=2

*d*. The M

^{2}of the beam is calculated to be 1.3. The beam is unambiguously identified the inphase fundamental supermode of the 19-core array. We then vary fiber length up to 20 m, and very little beam quality degradation is observed.

In order to see the effect of Gaussian beam waist on the output beam quality, *σ* is varied and the M^{2} vs *σ* is shown in Fig. 4. It is shown that the beam quality (M^{2} factor) is good as long as the beam waist is not too small (>1.5*d*). The beam waist *σ*=2*d* is shown to be optimal.

When the input beam is different from Gaussian, the output beam quality may degrade. To show this, we use an input with arbitrary amplitude and phase for each core and the output beam is shown in Fig. 5. The calculated M^{2} factor is 4.15, which shows a mix of different modes.

## 3. Experiment

The Yb-doped 19-core fiber is custom-made by Nufern, East Granby, CT. A picture of the fiber facet is shown in Fig. 6(a). The inner-clad has a D-shape with flat width 625 µm, round diameter 720 µm and NA of 0.48. The flat side of the inner-clad is used for side-pumping as described below. The outer-clad is a polymer coating. The 19 cores are located in the middle of the inner-clad, and their configuration is shown in Fig. 6(b). The designed core diameter is 7 µm, distance between adjacent cores is 10.5 µm, NA is 0.067, V value is around 1.4, and doping N_{0}=7×10^{19} cm^{-3}. It is seen that the fabrication is less than perfect, especially for the outer-ring of cores. The fiber has about 2.5 dB/m absorption at wavelength 976 nm.

The experimental setup is shown in Fig. 7. We use a Q-switched 5-m long single-core single transverse-mode Yb-doped fiber laser as the master oscillator (upper box in Fig. 6). The single-core fiber is a standard Nufern LMA-YDF-10/400 with core diameter of 10 µm, inner-clad diameter 400 µm, and core NA 0.075. This seed fiber laser offers a beam with M^{2} very close to 1. The seed laser is pumped using a 976 nm laser diode (LD), and is Q-switched using an acousto-optic (AO) cell. The gold-coated grating at the right works as the mirror and also stabilizes the wavelength^{6}. The seed is focused by a lens and coupled to the input of the 19-core amplifier (lower box of Fig. 7). The 19-core fiber is side-pumped close to the right end by a 976 nm Jenoptic LD using our proprietary side-pumping technology [13]. The fiber tip of the LD pigtail makes close contact with the flat side of the D-shape and 90% coupling efficiency is achieved.

The Q-switched seed is operated at a pulse repetition frequency (PRF) of 5 kHz. The coupling to a 5.8 m 19-core amplifier is carefully done to maximize the output power. A CCD camera is then used to measure the near field and far field of the amplified beam. The measured near field and far field are shown in Figs. 8(a) and 8(b) respectively. They should be compared with Fig. 3, and their similarity to the in-phase supermode’s near and far fields is obvious. Differences include: (1) for the near field, the intensities of the cores belonging to the same ring are not very uniform, and (2) for the far field, there is a small side peak. Considering the irregularities due to fabrication (Fig. 6), these imperfections are understandable.

We further measure the M^{2} factor of the amplified beam. Two lenses are used to shape the beam, and the 2-sigma radius (*W*) of the resulting beam is measured at 10 positions (triangular signs in Fig. 9). Then the following curve is used to fit the measured beam radius and obtain the M^{2} factor11

where *W _{0}* is the radius at the waist position

*z*. From Fig. 9, M

_{0}^{2}of 1.5 is obtained. This is close to the calculated in-phase mode M

^{2}factor and far from that of any higher order supermode. Therefore, the fundamental mode operation is proved.

The typical seed and amplified pulse shapes and spectrums are shown in Fig. 10. The two pulse shapes are not measured at the same time, so they are different due to the shift of the Q-switched oscillator. No relaxation oscillation is observed in the amplified signal. Also, no obvious backscattering is observed for output average power up to 3.5 W. From these facts, SBS is not appreciable in our experiment. The large effective mode area of the 19-core fiber may be the reason for this. The spectra are measured using an optical spectral analyzer (Anrisu MS9710B) with resolution of 0.07 nm. The sub-peaks of the spectra are due to multi-longitudinal-mode of the Q-switched oscillator.

Finally, we present the gain characteristics of the amplifier. The loss (about 0.25 dB/m at 1042 nm wavelength) is known from Nufern data sheet. We can deduce the seed pulse energy/power that is coupled into the amplifier by measuring the output from the amplifier without pump. The pulse energy is measured following the method used in Ref. [2] using a Newport detector Model 818-IR with a risetime around 2 µs. Because the risetime of the detector is much longer than the pulse width, the measured signal is the integration of the actual pulse, and the peak of the measured signal is proportional to pulse energy. On the other hand, the contribution due to ASE is small for this rise time. The detector is first characterized at high PRF, at which the ASE is negligible and the pulse energy can be inferred from average power measured by a thermal detector. Then the detector can be used to measure pulse energy at other PRFs. Keeping PRF at 5 kHz, with the coupled seed pulse energy around 6 µJ, the energy gain and amplified energy as functions of pump power is shown in Fig. 11. A 90% coupling efficiency of the side-pump is used. It is seen that energy gain of about 20 dB and pulse energy more than 0.65 mJ are achieved with good beam quality described above.

## 4. Summary and discussion

Using a MOFA structure, we demonstrate that the output from a phase locked 19-core Yb-doped fiber amplifier is operating in the in-phase fundamental supermode with good beam quality. We consider this as one step further in high power/energy fiber amplifier technology. We first theoretically show that, with a Gaussian beam as seed, the amplified beam from the 19-core fiber consists of mainly in-phase mode based on ICMT with gain. Experimentally, we use a Q-switched single transverse-mode fiber laser as seed and a custom-made 19-core fiber as amplifier to demonstrate the in-phase mode operation. This is proved by the measured near field, far field, and M^{2} factor (around 1.5). The measured results are in good agreement with with the theory. SBS is not appreciable due to the large effective mode area. A pulse energy gain of 20 dB (5.8 m long fiber) and pulse energy up to 0.65 mJ are obtained.

As shown in the paper, the fiber is not yet perfect, and this needs to be improved. Whether the scaling-up to 37-core will still preserve the in-phase mode or not under the conditions described in this paper needs further investigation. Finally, the mode competition mechanism in a multi-core fiber laser needs to be further studied.

## Acknowledgments

This work is supported in part by MDA SBIR contract # N00178-04-C-3077.

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