We report on the behavior of modal polarization states in a single-frequency, ytterbium-doped, few-mode fiber amplifier. Experimental data show that the polarization of the individual transverse modes depends on the pump power and that the modes tend towards orthogonally polarized states with increasing gain. The observations can be explained by local gain saturation that favors the amplification of differently polarized modes.
© 2008 Optical Society of America
Since its first demonstration ytterbium doped fiber amplifiers  and lasers  have gained much interest, especially as high-power amplifiers for many applications ranging from material processing  to gravitational-wave detection . Large mode area (LMA) fibers are used to avoid high power density in the core and related nonlinear effects. However, LMA fibers often support the propagation of transverse higher-order modes (HOMs), which decrease the quality of the transverse beam profile resulting in a non-diffraction-limited beam (M2>1) . Moreover, each mode can have a different polarization state, which decreases the degree of polarization of the total beam . Therefore, besides the M2-value the polarization extinction ratio is often used as an additional measure of the beam quality .
In this paper we report that HOMs in a single-frequency fiber amplifier not just can have different polarization states, but even prefer them at high pump powers. For the first time to the best of our knowledge, we have measured the polarization states as well as the power fraction of individual modes of a fiber amplifier in dependence on the pump power. The experimental setup of the fiber amplifier and the mode-selective polarization analysis is presented in section 2. First we studied the case where two modes propagate in the fiber, where the two modes were almost orthogonally polarized at high pump powers (section 3). In section 4 the reasons for this behavior are discussed considering local gain saturation. Finally, in section 5 we discuss first measurements of the polarization behavior for the case that three or more modes propagate in the fiber.
2. Experimental setup
A sketch of the experimental setup is shown in Fig. 1. In part (a) the setup of the few-mode fiber amplifier and in part (b) the components that are used to perform the mode-selective polarization analysis are depicted.
The Yb-doped fiber was seeded with a single frequency non-planar ring oscillator at the wavelength of 1064nm and was protected against back reflections with a Faraday isolator. The amplifier was counter-pumped by a fiber-coupled diode laser at 975nm. The double-clad gain fiber had a length of 2m, an inner core diameter of 25µm and a numerical aperture NA=0.06. Assuming a perfect step index profile the fiber supports the propagation of the transverse fiber modes LP 01, LP 11, LP 21 and LP 02. The fiber modes can exist in an arbitrary rotation, which is commonly described by a superposition of two orthogonal modes, e. g. LP 11 and LP′11, where the latter one is rotated by 90 °. HOMs were excited by definite shifting of the fiber incoupling end in the plane perpendicular to the optical axis.
The mode-selective polarization analysis, which we recently presented for investigations on passive fibers in reference  and which represents a combination of a polarization analysis approach  and a mode-selective resonator , accomplishes the polarization state measurement of individual modes. The polarization analysis is performed by rotating a quarter-wave plate in front of a polarizing beam splitter. The measured dependence of the transmitted intensity on the angle of the quarter-wave plate allows to derive the Stokes parameters, which completely characterize the polarization state . The three-mirror ring resonator shows different resonance lengths for the HOMs, so that by scanning its length individual modes can be selected. With this combination besides the polarization state of individual HOMs also their power fraction can be measured.
3. Power and polarization behavior of two modes
We measured the polarization behavior for different incoupling positions and bendings of the fiber. The most striking behavior was observed when only two modes were excited and the pump power was varied. In order to excite the HOMs we shifted the fiber incoupling end 10 µm relative to the optical axis of the seed beam. The total seed power that was coupled into the core was about 100mW and the pump power was varied between 0 and 12.3W. The coupling efficiency of the pump light was about 70% and the absorption of the pump light was about 90%. The total output power after the amplifier was 2.2W at a pump power of 12.3W.
Mainly the fundamental mode (LP01) and one higher order mode (LP′11) propagated in the fiber, while all other modes together contained less than 10% of the total power. Fig. 2 shows the dependence of the measured relative powers in the LP 01 and the LP′11 mode on the pump power. The error was estimated from repeated measurements to be about 5%. At pump powers between 0 and 5W, where the fiber amplifier was not completely saturated, the LP′11 mode experienced a higher gain and its power fraction increased. Contrary, in the saturation regime of the fiber amplifier (at higher pump powers above 6W) the fundamental mode experienced a higher gain and the power fraction in the fundamental mode recovered up to the initial value.
The corresponding evolution of the Stokes parameters as a function of the applied pump power is depicted on the Poincaré sphere in Fig. 3(a). Each polarization state corresponds to a point on the surface of the sphere. Already without pumping both modes showed different polarization states, which resulted from local variations of the birefringence over the fiber cross section. These variations, which exist due to bending of the fiber or due to imperfections from the manufacturing process, lead to an effective birefringence for each mode . When the pump power was increased, the polarization states tended towards opposite sides on the Poincaré sphere. For a better illustration and interpretation of the orientation of the polarization states to each other we calculated the spherical distance of the two different modes, which is the shortest path between the polarization states on the surface of the sphere and is defined as :
where S A and S B are the normalized Stokes vectors of the two considered modes A and B comprising the vector components S1 to S3. This distance can take values between d=0, which means that both modes have equal polarization states, and d=π, which means that both modes are on opposite sides of the Poincaré sphere and respectively their polarization states are orthogonal. The distance d(LP 01,LP′11) versus the pump power is plotted in Fig. 3(b). For increasing pump powers the distance of both modes on the Poincaré sphere tends towards π, which means that the modes tend towards being orthogonally polarized. The error of the spherical distance σd=±0.15rad was derived from the error of a single polarization state, which was estimated from repeated measurements and corresponds to errors of ±3 ° of the azimuth and ellipticity angles of the polarization ellipse.
In Fig. 4 the evolution of polarization states is illustrated by polarization ellipses of both modes, which are depicted for pump powers of 0W, 6.3W and 12.3W. At a pump power of 12.3W the principal axes of the polarization ellipses are almost perpendicular to each other and their helicities are opposite (left- and right-handed). With the help of a quarter-wave plate, a half-wave plate and a polarizing beam splitter we were able to separate both modes, which independently confirms our derivation from Fig. 3(b) that both modes are within orthogonal polarization states.
The absolute polarization states of both modes were sensitive to the incoupling, slight position changes of the fiber and the room temperature. However, we observed the same tendency towards orthogonal polarization states in all measurements. The spherical distance was always monotonically increasing with the pump power and took values between 2.5 and 3 at maximum pump power.
Mainly two effects influence the output polarization states of the modes: On the one hand, there is an effective birefringence for each mode, which depends on the local index of refraction distribution related to mechanical stress and temperature inside the fiber core and the directly surrounding range of the cladding. This effective birefringence causes different polarization states of both modes even without pumping. On the other hand, modes interact during the amplification nonlinear process as they partially share the same gain profile, so that (differently) polarized modes can experience different gain due to local gain saturation over the fiber cross section. As the spherical distance was monotonically increasing, we conclude that the second effect was dominating at high pump powers. The physical origin of this effect is discussed in the next section.
4. Mode interaction caused by local gain saturation
The observed preference for orthogonal polarizations at high pump powers suggests that the gain depends on the polarization state of the modes. It was just recently shown that transverse spatial hole burning (TSHB) causes different gain factors for HOMs . With the help of simulations it was verified - for equally polarized modes - that HOMs profit from the population inversion that is left by the fundamental mode.
For a qualitative explanation of our observations we take into account - in addition to the mode profiles - the impact of modally different polarization states on the local gain saturation. For simplicity we consider only the extreme cases, that the two modes (LP 01 and LP 11) are either equally or orthogonally polarized to each other.
In order to derive the locally saturated gain over the fiber cross section first the local intensity in the fiber I(x,y) is calculated, which depends on the power, the polarization and the phase of both modes. I(x,y) was calculated by superposing both modes first with equal and second with orthogonal polarizations. The powers were chosen to and in accordance to the powers in the experiment after about 1m of propagation in the fiber at maximum pump power. For maximum impact the phase between the modes was assumed to be ϕ=0. In Fig. 5(a) a cut through the intensity profile I(x,y) is shown. In the case of equal polarization both modes can interfere, so that the intensity distribution is becoming asymmetric, while in the case of orthogonal polarizations the intensity profile is wider and symmetric to the core axis, as only the intensity profiles of the individual modes have to be summed up as no interference can take place.
The gain profile g(x,y) (fig. 5(b)) was calculated using
where g ss is the small-signal gain and I sat the saturation intensity, which was calculated by I sat=P sat/A, where A=(12.5 µm)2π is the core area and P sat the saturation power. For our calculations we used the values P sat=100mW and g ss=40, which we derived from our experimental data.
In the case of parallel polarizations the asymmetric intensity profile only causes gain saturation on one side of the fiber cross section, while in the case of orthogonal polarizations the gain is almost homogenously saturated. Although this is an idealized picture and typically there is a mixture of both polarization orientations observable, it shows that orthogonally polarized modes profit from the gain that is left by the equally polarized modes.
In general, the relative phase ϕ between the modes (which was set to ϕ=0 for simple approximations) changes with propagation related to different effective refractive indices of both modes and . The difference of results in a mode beating of about 650 times along two meters of amplifying fiber. Hence the above described interference contrast is periodically reduced resulting in an effectively reduced net effect by substantially averaging the interference. Nevertheless, still a residual impact of the effect can be observed, indicating that even small values of gain difference can cause the observed behavior of the modes.
In summary, we qualitatively showed that transverse spatial hole burning (TSHB) does not only lead to a modal gain, but also to different gain factors for different polarizations. Orthogonally polarized modes experience higher gain as the TSHB is reduced. In order to quantitatively calculate the influence of this effect a detailed simulation of mode propagation comparable to that of reference  including local gain saturation and the polarization of modes would be necessary and is in preparation in our group.
5. Power and polarization behavior of three modes
The interaction behavior of three or more modes is more complicated, as three modes cannot occupy completely orthogonal polarization states to each other. The measured evolution of the polarization states depended on the initial power fractions and the initial polarization states, which changed with the incoupling and bending of the fiber.
Exemplarily, the results of one measurement, where three modes propagated in the fiber, are shown in Fig. 6. The incoupled seed power was about 100mW, the maximum pump power 18.3W and the maximum output power was 3W. The relative power in the modes over the pump power is shown in Fig. 6(a). The fundamental mode LP 01 contained about half of the total power, while LP 11 and LP′11 contained each about a quarter of the total power and the power contained in all other HOMs was less than 5%. Although the power fraction in all modes is nearly constant for different pump powers, there seems to be a correlation between LP 11 and LP′11. This might be reasonable as a power transfer between these modes is predominant as they have the same propagation constants .
The evolution of the spherical distances of the polarization states as a function of pump power is shown for all combinations of the three modes in Fig. 6(b). The highest increase of the spherical distance was observed for the LP 11 and LP′11 modes, which show almost orthogonal polarization states at maximum pump power. This means that the LP 11 and LP′11 modes tend towards the formation of a doughnut mode. Such a ring-like intensity profile allows an effective exploitation of the gain at the outer section of the core that is not saturated by the LP 01 mode. The spherical distances of the fundamental mode to LP 11 and LP′11 increase less or even decrease. This is a consequence of the limited degrees of freedom of the polarization states, wherefore not all the distances of the modes can be maximized at the same time. However, the sum of all three distances d sum=d(LP 01,LP 11)+d(LP 01,LP′11)+d(LP 11,LP′11) reached values about 6.0±0.2 at pump powers above 14.5W, which is close to the maximum of 2π that can be reached if the polarization states on the Poincaré sphere of all three modes are distributed on a ring, e.g. on the equator.
6. Conclusion and outlook
For the first time, to the best of our knowledge, we measured the power fraction as well as the evolution of the polarization states of individual HOMs propagating in a few-mode fiber amplifier. We observed that transverse modes prefer to take orthogonal polarizations at high gain as far as possible. This observation was qualitatively explained as an effect caused by transverse spatial hole burning which favors the amplification of differently polarized modes. An obvious application is given in the case that only two modes exist: The orthogonality allows simple mode separation with the help of conventional polarization optics, for example in order to clean up the beam to a pure LP 01 mode at high output power.
The reported investigations were partially supported by the Deutsche Forschungsgemeinschaft within the Sonderforschungsbereich 407.
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