## Abstract

An influence of backward reflection on spatio-temporal instability of the fundamental mode in Yb^{3+}-doped few-mode polarization maintaining fiber amplifiers with a core diameter of 10 μm was studied experimentally and theoretically. The mode instability threshold was registered to decrease dramatically in the presence of a backward reflection of the signal from the output fiber end; an increase of the signal bandwidth or input power resulted in the increase of the threshold. Numerical simulation revealed a self-consistent growth of the higher-order mode LP_{11} and a traveling index grating accompanying the population grating induced by the mode interference field (due to different polarizability of the excited and unexcited Yb^{3+} ions). The presence of the backward-propagating wave resulted in four-wave mixing on the common index grating induced by the interference field of pairs of the fundamental LP_{01} and LP_{11} modes.

© 2016 Optical Society of America

## 1. Introduction

Mode instability (MI) in Yb-doped fiber amplifiers attracted great interest in the last years due to its limiting effect for the power scaling. The MI effect was referred to a nonlinear transformation of power from the intense fundamental mode LP_{01} to the initially-weak higher-order modes [1,2]. It was shown that this effect results in the decrease of an output beam quality, randomization of temporal dynamics and damage of fiber splicer. The MI was observed in large mode area fibers, in photonic crystal fibers at pump powers ranging from several hundred watts to kilowatt, and in few-mode fibers with a relatively small core (10 μm) and high Yb dopant concentration at the pump power of just few watts [3]. The physical mechanism of the MI was explained by the beat of different transverse modes, which results in the periodic intensity modulation along the fiber and subsequently in the formation of the long-period refractive index gratings (RIGs). These gratings, in turn, enable energy transfer between the transverse modes due to the mode coupling [1–6]. Two mechanisms of formation of RIGs caused by population inversion (due to polarizability difference, Δ*p*, of the excited and unexcited Yb^{3+} ions) and temperature distributions induced by the modes interference field were discussed as the main reason for the MI. The thermal gratings were found to be the main mechanism of the MI in the large mode area fibers [4–6]. According to the vice version the RIGs engendered by the polarizability difference are the dominant mechanism for the low-threshold MI effect in the few-mode fibers with the 10-μm core diameter [3].

This paper is devoted to the detailed experimental investigation and theoretical modeling of the MI effect in Yb-doped few-mode fibers with core diameter of 10 um in the presence of a backward propagating wave (due to a backward reflection on the fiber end).

## 2. Experimental setup and measurement results

The polarization maintaining Yb-doped fiber (PANDA) was used as an amplifier of a linearly polarized CW signal at 1064 nm (Fig. 1). As the input signal, we used a radiation of an amplified single-frequency laser with the output power 1-500 mW and a bandwidth of Δλ < 0.04 nm or a fiber laser with a bandwidth varied from 0.14 to 0.5 nm. The seed radiation was injected into the 6 meters long fiber amplifier after passing through the optical isolator. The passive loss level of the active fiber was measured to be less than 20 dB/km at 1.15 µm. The experimental parameters are summarized in Table 1.

An image of the active fiber end and the beam profile (from the “Mode Visualization Plane”) were registered using a CMOS-camera (Fig. 2(a-d)). The amplified beam became unstable when the pump power exceeded a critical level: a transformation of the fundamental mode into the LP_{11} mode was clearly observed (compare Fig. 2(b) and 2(c,d)). The mode instability power threshold (MIPT) was determined as the maximum output power registered by power meter PM1. The MIPT was measured to be less in a perpendicular cleaved fiber amplifier (<1 W) in comparison with an angle-cleaved active fiber (~14 W). Changing the amplifier pump direction (from counter- to co-propagating to the signal) resulted in a small difference (~10%) of the MIPT at the instrumental uncertainty level.

In the main experimental series, input and output of the few-mode active fiber were spliced to the single-mode fibers. This introduces a selectivity of the fundamental mode in the few-mode active fiber and allows the measurement of the output power without high order modes using the power meter PM1. A photodiode near the spliced area at the amplifier output was used to detect the intensity of the high order modes. The backward reflection (BR) from the output passive fiber end (with variable angled cleave) provided the power controllable backward-propagating wave, which was measured using a fiber coupler by power meter PM2. The slope efficiency of the amplifier was excellent (> 80%) below MIPT, and dropped abruptly at a power level above MIPT (Fig. 2(e)). MIPT was also registered by sharp increase of the photodiode signal and appearance of the signal oscillations with modulation depth of 20-40% (Fig. 2(f)).

MIPT dependence on the input signal power for several bandwidths is close to linear on a logarithmic scale (Fig. 3(a)). Its extrapolation shows the possibility of a significant increase of MIPT at strong input signal with wide bandwidth. However, in the experiments we had chosen the input signal parameters (power 43 mW, bandwidth 0.14 and 0.25 nm) which provide a relatively small MIPT (~10 W) to eliminate induced stress of the isolator. The strong MIPT decrease was found even at low level of BR (Fig. 3(b)). MIPT dependence on the backward reflection coefficient was also linear on the logarithmic scale.

## 3. Numerical modeling of MI in the presence of a backward-reflected wave

In order to analyze the MI, the Yb-doped polarization maintaining phosphosilicate double-clad fiber was modeled as the composite cylinder (with a small diameter comparing to the length of a fiber) consisting of an Yb^{3+}-doped core, a silica-glass pump cladding and a polymer cladding with the parameters similar to the experimental ones (Tables 1 and 2). The fiber was side-pumped by an additional transporting fiber in the GT-wave configuration.

Proposed model assumed the existence of the fundamental mode LP_{01} and an initially small seed of the second mode LP_{11} in the fiber amplifier. The counter-propagating waves (in the forward direction with symbol “+”, and in the backward direction with symbol “-“) were taken into consideration. The complex amplitudes of the linear-polarized modes were described by the expressions:

*ω*and ${k}_{0,1}$are the frequency and propagation constants of the modes,

*t*is time,

*z*is the coordinate along the fiber,

*r*is the transverse coordinate of the fiber,

*φ*is the polar angle,

*Ω*is the frequency shift of the Stokes and anti-Stokes components of the LP

_{11}mode with complex amplitudes ${A}_{1}^{\pm ,s}$and${A}_{1}^{\pm ,as}$, respectively; ${\psi}_{0,1}$ are the radial distributions of the modes: ${\psi}_{0,1}(r)={\u0421}_{0,1}{J}_{0,1}({u}_{0,1}r/{r}_{0})/{J}_{0,1}({u}_{0,1})$ if

*r*≤

*r*

_{0}, and ${\psi}_{0,1}(r)={\u0421}_{0,1}{K}_{0,1}({w}_{0,1}r/{r}_{0})/{K}_{0,1}({w}_{0,1})$, if

*r*>

*r*

_{0},

*C*

_{0,1}are the normalizing constants,

*J*

_{i}and

*K*

_{i}are the Bessel and Macdonald functions of order

*i*, the constants ${u}_{0,1}^{\pm}$ and ${w}_{0,1}^{\pm}$ are defined by the Eqs.:${u}_{0,1}^{2}={r}_{0}^{2}({n}_{0}^{2}({2\pi /\lambda )}^{2}-{k}_{0,1}^{2}),{w}_{0,1}^{2}={r}_{0}^{2}({k}_{0,1}^{2}-{({n}_{0}-\Delta n)}^{2}({2\pi /\lambda )}^{2})$ (these parameters depend on the signal wavelength,

*λ*), and linked by the characteristic Eqs.:${u}_{0,1}{J}_{1,2}({u}_{0,1})/{J}_{0,1}({u}_{0,1})={w}_{0,1}{K}_{1,2}({w}_{0,1})/{K}_{0,1}({w}_{0,1})$ [11,12]. The mode spatial structures were assumed to be unchangeable due to nonlinear self-action and mode interaction.

The system of Eqs. for the complex amplitudes of the quasi-monochromatic modes in weakly guiding approximation (after averaging across the fiber for the each mode) was as follows:

*υ*

_{0,1}is the mode speeds, $q={k}_{0}-{k}_{1}\left(\left|q/{k}_{0}\right|<<1\right)$, $s={k}_{0}+{k}_{1}$

*,$\u3008\mathrm{...}\u3009=\frac{1}{\pi {r}_{0}^{2}}{\displaystyle {\int}_{0}^{\infty}{\displaystyle {\int}_{0}^{2\pi}\mathrm{...}rdrd\varphi}}$*, $\u3008{\psi}_{0,1}^{2}\u3009=1$ (from the normalization),

*δN*

_{ex}is the population change of the exited state

^{2}

*F*

_{5/2}, $\beta =\frac{8{\pi}^{2}}{{\lambda}_{s}{n}_{0}}{F}_{L}^{2}\frac{\Delta p}{{\sigma}_{em}^{s}+{\sigma}_{ab}^{s}}$, ${F}_{L}=\frac{{n}_{0}^{2}+2}{3}$ is the Lorentz local-field factor.

The expression (6) assumed the refractive index change due to the temperature change *δT* (the first summand) and the population change (the summand with the *β* parameter) due to Δ*p* in the fiber core [8,9,13]. We neglected any other nonlinear effects (stimulated Brillouin and Raman scatterings, Kerr nonlinearity and others). To bolster this argument, the increment of the stimulated backward Brillouin scattering was estimated to be < 1.

The excited-state population change was described by the following Eq.:

*S*= π

_{cl}*r*

_{1}

^{2},

*P*

_{p}and

*ν*

_{p}are the pump power and frequency, respectively;

*I*

_{s}is total signal intensity (including both intensities of all modes propagating in the both directions and their interference field).

The temperature distribution inside the core was described by the following Eq.:

*hν*

_{T}is the energy of nonradiative transitions between sublevels of the ground state

^{2}

*F*

_{7/2}after spontaneous emission.

The full equation system was completed by equations for the pumping power in the active fiber *P*_{p} and in the auxiliary fiber *P*_{ax} (in the case of GT-wave fiber [14]):

*γ*is the transformation coefficient of the pump from the auxiliary to the active fiber.

The boundary conditions assumed the existence of an initial signal in the fundamental mode LP_{01} and an initially small seed of the LP_{11} mode on the input of the fiber amplifier (*z* = 0), and the reflection of the modes on the output boundary (*z* = *L*): ${A}_{0,1}^{-}(z=L)=\sqrt{{R}_{0,1}}{A}_{0,1}^{+}(z=L)$, where *R*_{0,1} are the reflection coefficients. In the simulations (similar to the experiments) the signal had been switched on before the pump was switched on (the signal and pump switch-on times *t*_{s} and *t*_{p}, and the delay time *t*_{d} are presented in the Table 1). The ratio of power of the LP_{01} and LP_{11} modes at the input of the fiber amplifier was varied from 40 to 10^{4}. The initial perturbations of the temperature and population were assumed to be zero. To simplify the Eq. system the procedure of transformation of the higher-order correlators to the lower-order correlators (described in our previous paper [3]) was used. The final system of the partial derivative Eqs. (with *t* and *z* fluents) was solved using predictor-corrected method for the coordinate and time steps.

Considering the theory of the grating formation, the LP_{01} and LP_{11} mode interference field induces the population gratings and RIGs: the long-period RIGs are induced by interference of the co-propagating modes, and the short-period gratings are induced by the counter-propagating mode interference. Both the long-period and short-period RIGs can provide the nonlinear four-wave mixing and energy transfer, similar to the bulk media [15,16]. In the experimental conditions the signal bandwidth, Δ*ν,* was wide enough, and the signal coherence length was less than the fiber amplifier length v* _{0,1}*/Δ

*ν*<< 2

*L*. Hence, the interaction of the counter-propagating waves due to the short-period RIGs was negligibly small (we neglected the short-period RIGs for the simulations).

The first calculation series were made for the medium-narrow bandwidth (v_{0,1}/2*L* << Δ*ν* < *L ^{−1}(1/*v

*v*

_{0}- 1/*), when the LP*

_{1})^{−1}_{01}and LP

_{11}mode walk-off time on the fiber length

*L*is less than the signal coherence time. In that case the partial derivatives with respect to time in the left side of the Eqs. (3)-(5) were neglected.

The numerical modeling showed that the LP_{11} mode can have a bigger gain than the gain of the LP_{01} mode (Figs. 4(a) and (b)), and this effect is stronger in comparison with the fiber amplifier without BR. The additional gain of the LP_{11} mode is explained by the nonlinear energy transfer from the fundamental mode due to scattering on the common long-period dynamic RIGs accompanying the population gratings induced by the interference field components: $({A}_{0}^{+}{A}_{1}^{+as*}+{A}_{0}^{-*}{A}_{1}^{-s}){\psi}_{0}{\psi}_{1}exp(-i\Omega t-iqz+i\phi )$). The positive feedback between growth of the RIG amplitude and the LP_{11} mode amplitude is determined by an appropriate phase shift (depended on the frequency detuning, Ω). The forward propagating anti-Stokes and backward propagating Stokes components of the LP_{11} mode have the stronger gain (indicating their cooperative contribution to RIG), than another wave pair. The RIG caused by Δ*p* was found to grow (along the fiber amplifier and in time at the output) much stronger than the thermal RIG (Fig. 4(c) and (d)).

In our experiments with the fiber-core diameter of ~10 μm the domination of the electronic RIG over the thermal grating can be explained by rather fast erasing of the thermal RIG due to thermal conductivity across the fiber core. The increase of the core diameter (for the fibers with 30-μm core diameter, for example) will result in an increase of the steady-state thermal grating amplitude, but the electronic grating amplitude decreases due to a gain saturation. The similar nonlinear effect of the four-wave interaction by the competing electronic and thermal RIGs accompanying the dynamic population gratings is known to exist in the bulk crystal amplifiers and “self-starting” lasers [17,18].

The LP_{01}–mode gain was found to strongly depend on the BR coefficient: an increase of the reflection coefficient resulted in an increase of the higher mode increment, and the MIPT was determined as the maximum of the LP_{01}-mode output power before roll over (Fig. 5(a)). The calculated MIPT grew with increase of the LP_{01} mode input power and with decrease of the LP_{11} mode seed (Fig. 5(b)). The MIPT grew also with decrease of the core diameter and NA. The relative gain of the LP_{11} mode had a maximum vs the frequency shift *Ω* (~few kHz) at a low pump power. At the higher pump power this dependence had a number of maxima, in that case the output power starts to oscillate (Fig. 6(a)). The chaotic pulsations of all interacting waves were calculated at the bigger pump power (Fig. 6(b)).

For the broadband input signal when the mode walk-off time in the amplifier is more than the coherence time (Δ*ν* > *L ^{−1}(1/*v

*v*

_{0}- 1/*) the partial derivative in time in Eqs. (3)-(5) has to be taken into consideration. The counter-propagating broadband LP*

_{1})^{−1}_{01}and LP

_{11}transverse modes were assumed to consist of a number of the longitudinal modes. Numerical calculation of Eqs. for the complex amplitudes of each longitudinal mode, and Eqs. for the pump, population, temperature and their gratings showed the nonlinear power transformation from the LP

_{01}mode to the anti-Stokes-shifted LP

_{11}mode. The additional nonlinear gain of the LP

_{11}mode was found to depend on the signal bandwidth Δ

*ν*, and the signal and pump powers. The output signal power in the LP

_{01}mode increased up to a “threshold” (Fig. 7), and the MIPT was found to grow with increase of Δ

*ν*and the mode-power ratio at the fiber input.

## 4. Discussion

The numerical results for dependence of the MIPT on the reflection coefficient, the input signal power and the signal bandwidth are in good qualitative accord with the experimental results (compare Fig. 2(e) and Figs. 5(a) and 7(a), Fig. 3(b) and Fig. 5(b)). However, the equal reflection coefficient *R*_{0} = *R*_{1} corresponded only to the perpendicular cleave of the fiber (with the anti-reflecting coating, for example). Moreover, the BR coefficient of the LP_{11} mode was negligibly small due to its illumination out of the core on the splice with the single mode fiber on the amplifier output.

The numerical simulation for different BR coefficients (*R*_{0} ≠ *R*_{1}) showed that the reflection coefficient of the narrow-band LP_{11} mode (*R*_{1}) doesn’t strongly affect the MIPT at the fixed *R*_{0} coefficient. The numerical simulation of MI was also made for two counter-propagating mutually-incoherent LP_{01} modes without any reflection (*R*_{1} = *R*_{0} = 0), but boundary conditions at both amplifier ends assumed the existence of the LP_{01} mode and the LP_{11} mode seeds. The MIPT of two mutually-incoherent counter-propagating waves was found to be the same as the MIPT in the amplifier with BR, when the BR power of the LP_{01} mode was equal to the power of the backward-propagating mode at the amplifier boundary *z* = *L* (compare the green and violet curves on Fig. 8). The independence of the MIPT of the narrow-band signal on the LP_{11} mode reflection coefficient *R*_{1} is explained by the mutual RIG formation of the forward-propagating anti-Stokes and backward propagating Stokes waves (interfering with the fundamental modes) assuming that the reflection of the Stokes and anti-Stokes waves occurs independently.

## 5. Summary

The experiments showed the strong influence of the backward reflection on the spatio-temporal instability of the fundamental mode in the few-mode Yb^{3+}-doped fiber amplifier with the core diameter of 10 μm: the MIPT decreases with increase of the reflection coefficient, and decrease of the signal input power and frequency bandwidth. The numerical simulation indicates the nonlinear interaction of the fundamental LP_{01} and higher-order LP_{11} modes (with anti-Stokes and Stokes frequency shift) by mutual scattering (four-wave mixing) on the common long-period RIGs induced in the fiber by the interference of the counter-propagating mode pairs. The well-known results of four-wave mixing and instabilities of the counter-propagating waves in bulk media indicate that the MIPT decreases in presence of the backward propagating waves for variety of the fiber core diameters due to optical nonlinearities of different origins (electronic, thermal or strictional nonlinearities) [15,16,19,20].

## Acknowledgments

This work was supported in part by the program of Russian Academy of Sciences “Novel nonlinear-optical materials, structures and methods for development of laser systems with unique characteristics”.

## References and links

**1. **C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express **19**(4), 3258–3271 (2011). [CrossRef] [PubMed]

**2. **T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express **19**(14), 13218–13224 (2011). [CrossRef] [PubMed]

**3. **M. Kuznetsov, O. Vershinin, V. Tyrtyshnyy, and O. Antipov, “Low-threshold mode instability in Yb^{3+}-doped few-mode fiber amplifiers,” Opt. Express **22**(24), 29714–29725 (2014). [CrossRef] [PubMed]

**4. **A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express **19**(11), 10180–10192 (2011). [CrossRef] [PubMed]

**5. **C. Jauregui, T. Eidam, H.-J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express **20**(12), 12912–12925 (2012). [CrossRef] [PubMed]

**6. **B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express **20**(10), 11407–11422 (2012). [CrossRef] [PubMed]

**7. **M. Melkumov, I. Bufetov, K. Kravtsov, A. Shubin, and E. Dianov, *Cross Sections of Absorption and Stimulated Emission of Yb ^{3+} Ions in Silica Fibers Doped with P2O5 and Al2O3* (FORC, Moscow, 2004).

**8. **M. S. Kuznetsov, O. L. Antipov, A. A. Fotiadi, and P. Mégret, “Electronic and thermal refractive index changes in Ytterbium-doped fiber amplifiers,” Opt. Express **21**(19), 22374–22388 (2013). [CrossRef] [PubMed]

**9. **A. Fotiadi, O. Antipov, M. Kuznetsov, and P. Mégret, “Refractive index changes in rare earth-doped optical fibers and their applications in all-fiber coherent beam combinig,” in *Coherent Laser Beam Combining,* A. Brignon, ed. (John Wiley & Sons, 2013), chap. 7, pp. 193 – 230.

**10. **M. Bass, E. Van Stryland, D. Williams, and W. Wolfe, *Handbook for Optics*, 2nd ed. (MGH, 1995).

**11. **H.-G. Unger, *Planar Optical Waveguides and Fibres* (Oxford University Press, 1977).

**12. **A. W. Snyder and J. D. Low, *Optical Waveguide Theory* (Chapman and Hall, 1983).

**13. **A. A. Fotiadi, O. L. Antipov, and P. Mégret, “Resonantly induced refractive index changes in Yb-doped fibers: the origin, properties and application for all-fiber coherent beam combining,” in *Frontiers in Guided Wave Opticsand Optoelectronics*, B. Pal, ed. (Intec, 2010), pp. 209–234.

**14. **C. Codemard, K. Yla-Jarkko, J. Singleton, P. W. Turner, I. Godfrey, S.-U. Alam, J. Nolssson, J. Sahu, and A. B. Grudinin, in *Proceeding of European Conference on Optical Communication* (*ECOC'2002*, Copenhagen, Denmark, 2002), PD1.6.

**15. **S. N. Vlasov and V. I. Talanov, *Wave Self-Focusing* (IAP RAS, 1997).

**16. **O. L. Antipov, “Instability of counterpropagating homogeneous laser beams in media with a local slow-response nonlinearity,” Sov. J. Quantum Electron. **22**(1), 88–90 (1992). [CrossRef]

**17. **O. L. Antipov, A. S. Kuzhelev, V. A. Vorob’yov, and A. P. Zinov’ev, “Pulse repetitive Nd: YAG laser with distributed feedback by self-induced population grating,” Opt. Commun. **152**(4–6), 313–318 (1998). [CrossRef]

**18. **O. Antipov, A. Kuzhelev, and D. Chausov, “Formation of dynamic cavity in a self-starting high-average-power Nd:YAG laser oscillator,” Opt. Express **5**(12), 286–291 (1999). [CrossRef] [PubMed]

**19. **V. I. Bespalov and G. A. Pasmanik, *Nonlinear Optics and Adaptive Laser Systems* (Izdatel'stvo Nauka, 1986).

**20. **J. R. Ackerman and P. S. Lebow, “Observation and compensation of frequency detuning in high-reflectivity Brillouin enhanced four-wave mixing,” J. Opt. Soc. Am. B **8**(5), 1028–1039 (1991). [CrossRef]