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Abstract
We demonstrate a novel approach to enhance the mode stability through increased effective index difference (Δneff) between the higher-order modes (LP18, LP09 and LP19) of a multimode fiber. Fibers with large diameters have bigger effective mode areas (Aeff) and can be useful for high power lasers and amplifiers. However, a large mode area (LMA) results in an increased number of modes that can be more susceptible to mode coupling. The modal effective index difference (Δneff) strongly correlates with mode stability and this increases as the modal order (m) increases. We report here that the mode spacing between the higher order modes can be further enhanced by introducing doped concentric rings in the core. In our work, we have shown a more than 35% increase in the mode spacing between the higher order modes by optimizing the doping profile of a LMA fiber. The proposed design technique is also scalable and can be applied to improve the mode spacing between different higher order modes and their neighboring antisymmetric modes, as necessary.
© 2017 Optical Society of America
1. Introduction
High power lasers and amplifiers are necessary for many applications in the field of science and technology. Recent research shows remarkable increase in the output power of fiber lasers [1,2]. Compared to solid state lasers, fiber lasers are more compact, lightweight and flexible. Various approaches have been considered to increase the power of fiber based lasers and amplifiers. However, an increase in power brings the need to mitigate for non-linearities in the fiber such as Brillouin and Raman scattering, four-wave mixing (FWM) and self-phase modulation [3,4]. Mitigating these non-linear effect require large mode area (LMA) fibers in which optical mode is less intense. Therefore, much of the recent research focuses on LMA fibers for the high-power fiber lasers [5]. Multimode fibers (MMF) offer large mode areas and high beam quality [6] but the existence of many modes can result in random mixing between the desired mode and unwanted modes. However, recent research shows that higher order modes (HOM) have the ability to provide for more stable operation because the signal stability increases with an increase in the modal order [5], [7,8]. A number of approaches have been presented in recent years to use HOM for high power fiber lasers and amplifiers. One of the approaches involves the coupling of light from the fundamental mode to a single desired HOM using fiber Bragg gratings and then to propagate light in the higher order mode amplifying fiber [9]. These techniques provide considerably larger modal areas as well as more stable operation as described above. Moreover, for a given effective area, HOMs are also less sensitive to area reduction than the fundamental mode when bending [10,11]. At the same time, compared to the fundamental mode, higher order modes are less prone to mode coupling as with the increase in the modal order (m) the difference in effective index (Δneff) between a given higher mode (LP0m) and its nearest antisymmetric mode (LP1m) also increases [12].
In this paper, we have proposed a novel approach to increase the Δneff that would result in a resilience to mode-mixing and more stable signal propagation with the advantage of significantly larger effective mode area (Aeff) by using HOMs. Annular rings with doping of increased or reduced refractive index are used at particular radial locations inside the core as shown in Fig. 1, such that the effective refractive index (neff) of a desired mode is increased or reduced [13]. To demonstrate our concept, we have considered a MMF of numerical aperture (NA=0.22) with core radius (Rcore=25 μm) and refractive index (ncore=1.457). Similarly, cladding radius and refractive index are taken as (Rclad=62.5 μm) and (nclad=1.4403), respectively. For this study the central operating wavelength of (λ=1.05 μm) is chosen.
2. Theory
2.1. Modal solutions
Use of the higher order mode (HOM) of a multimode fiber (MMF) shows several advantages which includes mode area scaling to control laser high power and dispersion management for ultra-short pulses. Modal instabilities in HOMs are a common problem arising due to external perturbations, such as bending and fabrication imperfections. According to the symmetry rule, bend perturbation is odd in nature and direct coupling arises between even and odd order modes. Among the many modes guided by a MMF, sometimes a desired dominant mode (LP0m) may transfer its energy to its nearest anti-symmetric (LP1(m+1) and LP1(m−1)) modes on both sides [12]. A low effective index difference (Δneff) between adjacent modes enhances the modal energy transfer which results actual modal energy loss along with the interference effects. This could result in inter-mode mixing in a MMF. Although a lower order (lower value of m) mode may be easier to excite, a higher value of m gives larger modal separation (Δneff) values, so a compromise is needed. However, for a given value of m, if the modal separation to the nearest anti-symmetric modes can be increased, this would be a more preferable design. Here, a novel MMF design with several concentric material strips at strategically located positions is proposed to increase the ) in desired modes, such as, LP18, LP09 and LP19 as a high Δneff would restrict unwanted modal energy transfer due to external perturbations. Instead of using a perturbation approach, we have used a rigorous full vectorial H-field based finite element method (FEM) [14,15] to find the modal solutions of our proposed MMF design. The FEM is one of the most numerically efficient and accurate approaches to obtain the modal solutions of an optical waveguide, to calculate the propagation constant (β), effective index (neff =β/ko, where wavenumber ko is given by 2π/λ) and Aeff.
In order to increase the modal solution accuracy, the available two-fold symmetry of the fiber is exploited and only one-quarter of the structure is simulated [16,17]. This allows more dense mesh distribution in the quarter structure of the fiber instead of distributing the same mesh over the whole structure. The polar mesh [18] discretization is also used, which can accommodate the discretized elements more efficiently at the circular boundaries, which can provide more accurate results compared to the mesh distribution based on the Cartesian coordinate system [19]. It is well known that the simulation accuracy of the FEM is highly dependent on the number of discretized elements used. Variation of the effective index (neff) with the number of mesh elements (N) is shown in Fig. 2 by a solid black line for the higher order LP09 mode. It can be observed that initially as mesh density increases the neff also increases rapidly and then asymptotically settles to a constant value. It should be noted that accuracy is up to the 3rd decimal place when the number of elements used is, N=7×104 and it increases to the 5th decimal place when N=5 × 105. Here N, is the number of triangular elements used to represent a quarter of the MMF. A powerful Aitken’s extrapolation technique is used to test the convergence of the modal solutions [20,21]. Three successive neff values for corresponding mesh divisions with a geometric ratio are used in the Eq. (1).
Using Eq. 1 the extrapolated values of are shown in the Fig. 2 by the red-dashed line. Aitken’s values are calculated for the mesh values N=7.66 × 104, 3.07 × 105, and 1.23 × 106 increased in fixed geometric ratio yielding the neff values 1.4460355, 1.4461157, and 1.4461358, respectively. It should be noted that in each solution the mesh density is two times (no. of the elements is four times) that of the previous and thus the geometrical mesh ratio is kept constant. From these values a more accurate extrapolated value of 1.4461398 is obtained. Similarly, Fig. 2 clearly shows the convergence of the extrapolated results and raw FEM results. As the trend of neff with increasing N for different modes is similar, so the accuracy of Δneff between two modes with increasing N is also greater.
Fig. 2 Variation of neff of the LP09 mode with the mesh number (N) and convergence realized with the Aitken extrapolation technique.
2.2. Mode stability
As previously discussed, higher order modes (HOM) provide the unique benefit of increased stability due to a greater mode spacing (Δneff) compared with that of the fundamental modes (LP01) [22]. In this paper, we propose a unique approach that can reduce mode mixing between the higher order LP18, LP09 and LP19 modes, that could alternatively only be possible by considering a much higher modal order. For our design, we used LP09 as the central propagation mode and modified the refractive index profile of the fiber such that the mode stability between symmetric (LP09) and the two nearest antisymmetric (LP18 and LP19) modes is increased. A measure of the modal stability can be defined using the effective index differences as, S1=Δneff (LP18 − LP09) and S2=Δneff (LP09 − LP19). However, if required any other HOM of interest can also be selected and using the same concept its mode stability can be enhanced. For this structure, the effective indices of the LP18, LP09, and LP19 are calculated as 1.447317, 1.4461104, and 1.4448554, respectively, yielding S1=0.0012066 and S2=0.001255. The variation of the dominant Hy field profile of the LP18, LP09 and LP19 modes is shown in Fig. 3 by red, black and blue lines, respectively. Hy field contours of the LP18, LP09 and LP19 modes are also shown Fig. 3 as insets. It can be observed that the field profile of LP09 shown by a black line, has the highest magnitude at the center (r =0) of the fiber and eight zero values along the radial direction. The antisymmetric LP18 and LP19 modes shown with red and blue lines, respectively, have zero field values at the center (r = 0) of the fiber. The dominant Hy fields of the LP18 and LP19 modes have eight and nine zero field values along the radial direction, respectively. Table 1 shows all the radial locations where these modes have zero field values. For example, LP18 has its first three zero field locations at r =0, 3.78 and 6.966 μm. Using these zero crossings we have identified multiple locations that are suitable for doping such that the mode stability (S1 and S2) can be increased.
Fig. 3 Variations of Hy fields of the LP18, LP09, and LP19 modes along the r-axis of MMF, contour field profiles in inset and the key points of interest are also shown.
Table 1. Zero crossing locations of field profiles of the LP18, LP09 and LP19 modes along r-axis (μm).
Some specific points, A, B, C, D, E and F are selected as shown in Fig. 3, where either LP18 or LP19 has zero crossing. The reason for choosing these points is that we want to have less of an effect on one of the mode and have more of an effect on the other two though using strips of different doping. The modal field values at these points are also given in Table 2. For example at point A, the Hy field value of LP19 is zero, whereas, the field values of LP18 and LP09 modes are 0.06507 and 0.08321, respectively. As a result any change of refractive index doping at point A will have an almost negligible effect on the LP19 mode and comparatively more of an effect on the LP09 than the LP18 mode. However, at point F, where the Hy field value of LP18 mode is zero and these values for LP09 and LP19 modes are 0.04658 and 0.05879, respectively. Hence, at point F, doping will have no effect on the LP18 but will have more of an effect on LP19 than the LP09 mode. This will result in an increase in effective index both of these modes while keeping the effective index of the LP18 mode unchanged, and a suitable selection can result in an increase in the modal stability. In order to increase the Δneff between these modes a circular strip of 0.3 μm width is considered that can have an increased or reduced refractive index by Δn. Here, we have taken Δn =0.0167, which is also the difference between the core and cladding refractive indices. However, different values of Δn in these strips can be chosen according to the required level of stability between modes.
Table 2. Field values of LP18, LP09 and LP19 at A, B, C, D, E and F points.
3. Numerical results
Table 3 shows the effect of doping a single individual layer at the above mentioned six positions along the r-axis. The second column of Table 3 shows the original Δneff between the modes without any doping. It can be noted that the Δneff between LP09 and LP19 is slightly higher than the Δneff between LP18 and LP09. The values of +Δn or −Δn on particular points are chosen such that the effect on the central mode, in our case LP09, is negligible or can be compensated with another doping layer where Δn is chosen with an opposite sign to the first point. For example, in layer A (at r=6.193 μm), +Δn is chosen to increase the stability, S1 between LP09 and LP19, but unfortunately this reduces the stability, S2 between LP18 and LP09. On the other hand a reduction of refractive index in layer B increases modal separation S2, but reduces that of S1. However, an increase of refractive index in layer E and reduction in layer F enhances both the S1 and S2 modal stabilities. The same doping approach is considered at all six positions with either +Δn or −Δn as shown in Table 3. Here, two approaches can be considered;
- Using single layer doping to increase the stability between modes.
- Using the combination of two or more layers to increase the stability.
Table 3. Individual strip doping effect on Δneff at points A, B, C, D, E, and F.
In Table 4 the above described approaches are shown with the percentage increase in Δneff (S1 and S2). The percentage increase is calculated with respect to the original Δneff between the modes as shown in column two. Here, three different options are suggested depending upon the required increase in the Δneff. It can be seen that with a single layer of −Δn doping at point F, S1 and S2 are increased by 16% and 14%, respectively. However, using two layers (at E & F points) the stabilities S1 and S2 between modes can be increased by 20% and 23%, respectively. It should be noted that the Δn doping at points E and F are taken as positive and negative, respectively. For further enhancement, three layers can be doped simultaneously at points C, D, and E which results in an increase of 35% and 38% for Δneff (LP18 − LP09) and Δneff (LP09 − LP19), respectively. Hence, our proposed design results in increased modal spacing between the higher order modes LP18, LP09 and LP19, thus providing for more stable and mode-mixing resistant operation. Figure 4 shows the refractive index profile for three layer doping at C, D and E points. Here, Δn at points C and E are taken as positive such that it increases the local refractive index from 1.457 to 1.4737. Whereas, at point D, Δn is taken as negative resulting in the local index being equal to that of the cladding. The combination is chosen such that Δneff (LP18 − LP09) and Δneff (LP09 − LP19) have an almost equal increase. Here, all three annular strips are centered at points C, D and E and have an equal width of 0.3 μm. The LP09 mode is considered as a central propagation mode in our design. Hence next, the effect of three layers of doping at points C, D, and E points on the field profile of LP09 is studied. Figure 5 shows the Hy field profile of the LP09 mode before and after doping. The black line shows the undoped field profile whereas the dotted blue line represents the field profile of LP09 after doping these three layers at C, D, and E. It can be observed that until the appearance of the doped strips the field profile was almost unchanged, however beyond these strips, the field value is reduced compared to that of the original undoped fiber.
Table 4. Percentage increase in the Δneff using individual and combination approach.
Fig. 5 Variations in the Hy fields of LP09 modes along the r-axis of the undoped fiber and the fiber with C, D, and E layers doped. The contour field profiles are also shown inset.
The Δn =0.0167 value used for the circular strips C, D, and E is equivalent to the refractive index difference between core and cladding of the fiber but of necessary other Δn values or even unequal values can be used for a particular design. To observe the effect of increased or reduced value of Δn, we have halved the refractive index difference as, ±Δn =0.00835 and found that the modal stabilities S1 and S2 reduces to 18% and 19%, respectively. However, when refractive index difference is increased to double; as ±Δn =0.0334, the modal stabilities S2 increases to 74% but S1 increased only slightly to 41%, as field profiles were distorted significantly. However, it can be stated that a significant enhancement in the modal stability can be achieved by this approach.
Although we have focused on the enhancement of mode spacing between LP09 and its neighboring antisymmetric LP18 and LP19 modes, however, our approach is scalable and can be applied for any higher order modes. As an example, for the enhancement of mode spacing of LP08 mode and its neighboring antisymmetric LP17 and LP18 modes, we have carried out additional simulations after introducing the similar annular rings. Similar six zero crossing points, A′ to F′ for LP17 or LP18 are identified as A′=6.928, B′=7.856, C′=10.138, D′=11.518, E′=13.279, F′=15.098 (in μm). Numerical simulations were carried out with strips width of 0.3 μm and ±Δn =0.0167 introduced at these points. Table 5 shows the increase in Δneff (LP17 − LP08) and Δneff (LP08 − LP18) with single layer (F′) or multiple layers (E′, F′ and (C′, D′, E′) with perturbed annular strips. It can be observed that the stability is increased to 46% for the LP08 mode when three annular layers at (C′, D′, E′) points are used. This confirms that the concept presented here can be applied to any higher order modes, as necessary.
Table 5. Percentage increase in the Δneff of LP08 mode and its neighboring antisymmetric modes using individual and combination of two or three strips doping.
3.1. Fabrication tolerance of strips width
Here, we demonstrate the effect of possible variations in doping that can occur during the fabrication process. As the combined doping of the three layers at points C, D, and E could be more sensitive to fabrication tolerances than the two or single layer doping we will consider the three layers (C, D and E) case for further investigation. The effects of a change in the layer width (w) from 0.3 μm to a higher or lower value are shown in Fig. 6. As discussed earlier, with w=0.3 μm the stabilities S1 and S2 between the modes are 35% and 38%, respectively. With an increase in the width from w=0.3 μm to w=0.4 μm, the stabilities S1 and S2 further increase to 38% and 51%, respectively. This is because the area of the doped layer is increased when the width is changed from 0.3 μm to 0.4 μm thus increasing its overall effect. However, as the width deceases to w=0.20 μm, the stability improvement reduces but still it remains above 25%. Hence, in our proposed design the modal stability improvement will remain at least 25% larger for a width change of ±0.1 μm.
3.2. Fabrication tolerance of strips center location
In our proposed design three layers are centered at points C=9.058 μm, D=10.138 μm and E=11.867 μm and during fiber drawing these positions may change. Figure 7 shows the effect of a change in the location of the doped layer’s center points. It can be seen that the stability is highly dependent on the specified locations in Fig. 3. When there is no shift in the location of the strip’s centers, the stability improvement is larger than 35%. With a tolerance of ±0.05μm the stability improvement still remains above 27% but as the shift is increased to 0.1μm the S1 improvement drops to 20% while S2 improvement is increased to 51%. Moreover, when the annular strips shift is −0.1 μm, the modal stability S1 improvment increases to 46% but that of S2 decreases to 27%. For comparison the spacing between the modes before introduction of the C, D and E layers is also shown by two horizontal lines.
Fig. 7 Effect on the Δneff of a variation in the position of C, D, and E layers from center location.
3.3. Fabrication tolerance of wavelength change
The above analysis is carried out at the center wavelength of λ=1.05 μm. However, when the operating wavelength changes from this value the stability between modes can also change. To observe the impact of wavelength change on the stability between the modes, we varied the operating wavelengths and this is shown in Fig. 8. It can be observed that the stability between the LP18, LP09 and LP19 modes increases with an increase in the wavelength. It should be noted that the refractive index of a material is also dependent on the wavelength. In our analyses we have used the core and clad refractive index values as ncore=1.457 and nclad=1.4403 respectively at λ=1.05 μm. However, if required the effect of refractive index variation with the wavelength can also be included. It should be noted that without doping at the C, D, and E layers, the stability increases almost linearly with an increase in the wavelength. The increased wavelength reduces mode confinement and effective index values and this also increases the separation between the modal index values. A similar effect is noticed after the introduction of doping at layers C, D and E layers. When the center wavelength is λ=1.05 μm, the modal stability values are S1=0.0016254 and S2=0.0017357, which are improvements of 35% and 38% from their undoped values, respectively. However, it can be noted that with a change in the wavelength from λ=1.05 μm to λ=0.85 μm the stability difference after doping shows a similar trend to the undoped case. The modal stability values at λ=0.85 μm reduced to S1=0.0010095 and S2=0.0014572, however this reflects a 25% and 74%, improvement from the standard fiber. Similar behavior is observed when the center wavelength is increased to λ=1.25 μm and modal stability increases to S1=0.002199 and S2=0.0019263, and these reflect improvements 35% and 25% increase from the standard fiber. The modal stability still remains 25% higher for the wavelength range from λ=0.85 μm to λ=1.25 μm, and hence the proposed design results in a sufficiently improved stability between the competing modes and is also expected to provide for large effective mode areas Aeff because of operating in HOMs [12].
4. Conclusions
We have proposed a novel design approach, which has been validated by rigorous numerical analysis, to improve the stability between the modes can be increased by more than 35%. Increasing the Δneff between modes results in a more stable and mode-mixing resistant operation and thus allows scalability of power in laser applications. It is also shown here that the design is reasonably stable to possible fabrication tolerance such as the position and width of the doped annular sections.
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