General analysis of the mode interaction in multimode fiber is presented in this paper. By taking local gain into consideration, the general coupled mode equations in multimode active fiber are deduced in the model and the effect of various factors can be analyzed based on the general coupled mode equations. Analytical expression of the beam quality factor is deduced for the optical field emerging from the multimode active fiber. The evolution of the mode power and M2 factor along the fiber are analyzed by numerical evaluations.
©2012 Optical Society of America
Fiber lasers have widely application potential due to their advantages in good beam quality, high optical to optical efficiency and superior thermal-optical properties. Nevertheless, with the increased output power, thermo-optical effects, surface damage and nonlinear effects will arise inevitably that constraints further power scaling of the fiber lasers. The master oscillator power amplifier (MOPA) system offers an effective solution to the problem, which boost the output power of the master oscillator by employing cascaded structure configuration. For the sake of obtaining good beam quality of the output laser beam, single-mode active fibers are always employed in the master oscillator [1–3]. Meanwhile, multimode active fibers with large mode area (LMA) have been utilized to decrease the high power density within the core in the amplifier sections [2, 3]. Therefore, the examination of transverse mode behavior is very important to get the knowledge of the gain mechanism in the multimode active fiber and how the beam quality and polarization characteristics evolve in the MOPA system. In general, the mode interaction in the multimode active fiber is resulted from various factors including local gain, temperature and bend. In the previous papers, local gain is mainly taken into account when analyzing the transverse mode interaction [4–9].To the best of our knowledge, there is hardly a comprehensive model suitable to analyze the mode interaction induced by the factors mentioned above. Therefore, a general analysis of mode interaction in the multimode active fiber is required.
In this paper, we will present a general model to investigate the mode interaction in the multimode active fiber under different factors. General coupled mode equations are deduced in the model and various factors are analyzed. Moreover, formulation of the beam quality factor is deduced in the paper. On the base of this model, the evolution of the mode power and M2 factor along the fiber are investigated by numerical simulations.
2. Models and theoretical analysis
This model presented here starts with the unconjugated reciprocity theorem and describes the propagation of the light along the optical waveguide with nonuniformity that varies with distance z along the waveguide, i.e. the waveguide is no longer translationally invariant.
The perturbation method is available in the nonuniform waveguide provided that the perturbation-induced refractive index changes are slight perturbations of the uniform waveguide, and this model is based on the statement mentioned above. Hence, the description here can employ the complete set of bound eigenmodes of the unperturbed fiber and leads to a set of linear, coupled mode equations for the nonuniform waveguide.
2.1 The general coupled mode equation
Assume that the waveguide is anisotropic; hence the Maxwell equations for a source-free and anisotropic medium are:
According to the two-dimensional form of the divergence theorem:
Assume that is the refractive index distribution of the unperturbed waveguide while represents the refractive index distribution of the slightly perturbed waveguide; hence and represent the electromagnetic fields in the perturbed waveguide. The perturbation theory suggests that the transverse components of and can be expressed as the linear combination of bound eigenmodes and radiation modes in the unperturbed waveguide :11]. On the other hand, the radiation modes experience gain much less than the bound egienmodes in the multimode active fiber. For these reasons we neglect the radiation modes in Eq. (8). Then:
Here the backward-propagating modes are not taken into account due to emphasis on the fiber amplifier.
From the Maxwell equations, the longitudinal component Ez in the perturbed waveguide is given by:
In the same way, due that are the bound eigenmodes in the unperturbed waveguide, so:
In the same way, .
We choose as the kth backward-propagating bound eigenmodes in the unperturbed waveguide, so:
Considering the property of waveguide, hence:
Applying the orthogonality conditions of the bound eigenmodes in the unperturbed waveguide, so:
Hence, mode interaction within the perturbed optical waveguides has been converted to a series of only z-dependent ordinary differential equations which can be solved using the forth-order Runge-Kutta method. Compared with the BPM, this model reduces the amount of calculation significantly, and describes the mode interaction intuitionisticly.
2.2 Local gain
In the active fiber, due to gain saturation, the gain distribution is nonuniform within the core of fiber. For the Tm-doped fiber laser and amplifier emitting at near 2μm, the formulation of gain in the fiber is given by :
Nonuniformity in gain distribution leads to the interaction among different bound eigenmodes, and this impact on the mode coupling can be treated as a change of refractive index within the core. Assume that the local gain distribution is isotropic so that the impact on the refractive index can be equivalent with the same imaginary part for every orientation, and hence the effective refractive index can be given by:
Virtually it is available to employ the mode coupling theory to describe the thermally induced mode interaction in the waveguide provided that the thermally induced changes of refractive index can be treated as slight perturbations for the uniform waveguide. Hence we can utilize the coupled mode equations deduced in Section 2.1 to depict the thermally induced modal coupling.13, 14].
Bends along the fiber lead to both the birefringence and spatial mode coupling. In general, the approach available to deal with the mode interaction caused by curvature in the fiber is the multi-section model . In the multi-section model, the fiber in each section is assumed to be in the same plane, and the fibers in different sections have different refractive index principal axes, leading to the polarization-dependent mode coupling.
In each section, deviation of the fiber from its perfect geometry due to the curvature can be described by the change of the refractive index distribution. Assume that bends only occur in x-z plane, and then the refractive index changes of the anisotropic media in each section can be given by :
And the total impacts on the mode coupling within the fiber can be treated as a slight perturbation of the uniform fiber under the condition mentioned at the beginning of Section 2.
Meanwhile, bend-induced rotations of the refractive-index principal axes between the adjacent sections lead to the polarization-dependent mode coupling. Assume that the principal axes between the adjacent sections are rotated by an angle, and hence the transverse components of the electric field in the frontal section relative to the principal axes of the next section are expressed by:
At the junction between the adjacent sections, axis rotation leads to energy transform between the bound and leaky modes, and the remaining energy is redistributed on the bound modes within the next section. So the mode coefficients in the next section are given by:
2.5 Beam quality factor
The beam quality factor of the laser beam emerging from the multimode active fiber can be calculated provided that mode distribution at the output end of the fiber is known. As follows, the beam factor for the step-index multimode active fiber is calculated. The results thus obtained can easily be extended to the general case.
Considering the weak-guidance approximation, there are a series of linearly polarized (LP) eigenmodes which constitute a group of complete orthogonal basis for the steady fields in the step-index fiber. Due to the degeneracy in x-y polarizations, the transverse components of the electrical field can be decomposed into the linear combination of the corresponding polarized LP eigenmodes, respectively:
Hence the M2 factor can be calculated as follows:
3. Numerical simulation
In this section, we simulated the mode interaction in Tm-doped multimode fiber amplifiers under various factors numerically on the base of the model presented above. The laser wavelength is 1950nm. The fiber is assumed to be the double-clad fiber, and the radius of core and inner cladding are about 10μm and 200μm, respectively. The numerical aperture (NA) of the core is about 0.1. So the normalized frequency is about 3 and hence there are LP01 and LP11 modes in the multimode fiber while each mode has two orthogonal orientations. Considering the degeneracy of different azimuthal functions, hence there are LP011x, LP011y, LP111x, LP111y, LP110x and LP110y modes in the fiber. The beat length Lβ01-11 between the LP01 and LP11 mode is about 0.0011m. Assume that the initial signal power injected into the amplifier is 0.01W, and different input modes possess the same power ratio with respect to the total signal power.
3.1 Local gain
The mode interaction in the multimode Tm-doped active fiber with different doping distribution is investigated here. The injected pump power is assumed to be 20W. Different doping profiles are listed in Fig. 1(a) and the corresponding dependences of mode power and M2 factor on the length of fiber are depicted in Fig. 1(b)-1(d).
As is shown in Fig. 1(a), there are three kinds of doping distributions . The first two belong to flat doping where Г is the doping confinement factor defined as the ratio of the maximum gain radius to the core radius, and the third one is parabolic doping which satisfies N(r) = N(1-r2), r<1, where r is the normalized radius and N is assumed to be 2.43х1026 which is the same with those of the first two.
Figure 1(b)-1(d) presents the corresponding three results of mode power and M2 factor. When the doping distribution is flat doping with Г = 0.5, the LP11 mode is suppressed significantly by the fundamental mode because the fundamental mode experiences larger gain than higher-order modes. As a result, the beam quality is improved with the increase of power of the fundamental mode. The same results are also presented when the doping distribution is parabolic doping. Inspired by this, various kinds of beam can be obtained provided that the corresponding doping distribution is designed reasonably.
Actually the phase difference between the LP01 and LP11 modes achieves self-imaging  within a range of the beat length. Therefore the M2 factor should oscillate with a period of the beat length along the fiber. However, the oscillation of M2 factor cannot be presented subtly in the Fig. 1(b)-1(d) due that the sampling step is much larger than the beat length. When the sampling step is far less than the beat length, the accurate evolution of mode power and M2 factor for flat doping with Г = 1 are depicted in Fig. 2 , as well as other doping distributions.
As is shown in Fig. 2, energy transformation among different modes within a range of the beat length is quite weak due to the small pump power and hence the oscillation amplitude of M2 factor is about 0.05 and not prominent.
Generally speaking, the thermal effect becomes prominent as the pump power P increases or different cooling methods are employed leading to different heat transfer coefficients hc. Impacts of different pump powers and heat transfer coefficients on mode interaction are discussed in this section. The fiber parameters are the same with those in section 3.1 while the doping profile is the flat doping with Г = 1, and the simulated results of mode power and M2 factor are shown in Fig. 3 .
As is shown in Fig. 3(a)-3(d), when water-cooling is employed, the thermal effect is suppressed effectively and the thermal-induced change of refractive index is weak so that mode interaction among different modes is not strong. However, when no cooling is employed, as the pump power increases, the perturbation of refractive index caused by the thermal effect becomes larger, and leads to stronger energy transformation among different modes, as well as the M2 factor. The figures show that the period of mode interaction is about 1mm, which is equal to the beat length Lβ01-11. Because there are only two different propagation constants among the total LP modes, the phase distribution of the field achieves self-imaging after experiencing the beat length. Hence, the thermally induced refractive index change shows periodicity with a period of the beat length Lβ01-11, which can be interpreted as a series of micro wedges of higher refractive index . The wedge transfers a small fraction of the light from the fundamental mode to the LP11 mode. If there are more LP modes with different propagation constants within the fiber, the periodicity of the thermally induced index change will change approximately as the least common multiple of all the different beat lengths within the fiber. In order to verify the viewpoint mentioned above, we assume that there are LP01, LP11, LP21and LP02 modes within the fiber without consideration of the mode orientation. The beat lengths of any two modes are listed in Table 1 . These modes occupy the same power fraction and the other parameters remain unchanged. The corresponding results are depicted in Fig. 4 .
As is shown in Table 1, the least common multiple of all the different beat lengths within the fiber is about the beat length Lβ21-02. The curve of mode power (left) in Fig. 4 depicts that the period of mode interaction caused by the thermally induced index change is about 0.0057m, which is equal to the beat length Lβ21-02, the least common multiple of all the different beat lengths within the fiber. That is, the phase distribution of the field within the fiber achieves self-imaging after propagating through a length of Lβ21-02.
3.3 Micro bends
Assume that the orientation of the micro bends within the fiber, as is shown in Fig. 5 , is parallel to LP111 mode and satisfies f(z) = Adsin(k’z) along the fiber where Ad is the bend amplitude and is the spatial frequency of the micro bends. It is assumed to neglect the stress-induced birefringence due to the curvature. Therefore, without loss of generality, it is reasonable to consider the bound modes with one polarization orientation only. Impacts of different bend amplitudes and spatial frequencies of the micro bends on mode interaction are discussed in this section. The fiber parameters are the same with those in section 3.1 while the doping profile is the flat doping with Г = 1. Figure 6 presents the simulated results for different bend amplitudes with the same frequency.
As is shown in Fig. 6(a), when the bend amplitude Ad is twentieth of the radius of core, the oscillation period of mode power and M2 factor is about 0.02m, which is two-tenfold compared with the beat length. Likewise, the similar result is presented in Fig. 6(b), where Ad is a tenth of the radius of core. Moreover, by numerical simulations, the same result also applies with other bend amplitudes. It suggests that the spatial period of mode interaction within the fiber, in which energy transfer among different modes completes a cycle, cuts as the bend amplitude of micro bends increases. This phenomenon can be explained by an in-depth study on Eq. (20). Considering the micro bends within the fiber, so the in Eq. (20) can be given by:
Simplifying the equation above, then
After integrating, the z-dependent item of the first section in Ckj, as well as other sections, presents as iαf(z), where
Hence there should be an oscillating item exp(iαf(z)) within the mode coefficients bj(z) and bk(z). When f(z) = Adsin(k’z), this item is expressed as:Figure 7 shows the simulated results of mode power and M2 factor for different spatial frequencies while maintaining the same bend amplitude.
It suggests that only when the spatial frequency of micro bends is equal to the beat frequency between the LP01 and LP11 modes, the mode interaction with each other becomes prominent and hence the M2 factor oscillates in magnitude. Therefore, this conclusion can be generalized into the fiber supporting more bound modes. That is, mode interaction between a pair of modes is more prominent than that of others provided that the beat frequency between them is closest to the spatial frequency of micro bends.
In this paper, we have built a model to investigate the mode interaction in the multimode active fiber under different factors. The general coupled mode equations have been deduced in the model and various factors have been analyzed based on the general coupled mode equations. Moreover, the analytical expression of the beam quality factor has been calculated for the optical field in the multimode active fiber. On the base of this model, the evolution of the mode power and the M2 factor along the fiber have been analyzed by numerical simulations and the simulation results are consistent with the physical insights well. Hence, the model in the paper can deal with the problem of mode interaction under various factors and provide instructive suggestions when designing the fiber lasers and amplifiers.
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