Bend loss of the first three modes of leakage channel fibers with various designs are studied using finite element method. It is found that very low bend loss at small bend radius can be achieved with large d/Λ. It is also found that best differential mode loss is achieved at large bend radius. It is further found that 2nd order mode loss becomes 9.1 times of fundamental mode loss at small bend radius for the bend orientation where the bending plane crosses centers of two holes and independent of d/Λ. Bend loss dependence of bend orientations are also studied. Excellent agreement between experiment and simulation is obtained.
© 2008 Optical Society of America
A leakage channel fiber has a core formed by a ring of few large air holes [1, 2]. Such fiber can be designed to have a low confinement loss for fundamental mode while providing a high confinement loss for all higher order modes. This built-in mode filtering is distributed along the length of fiber and is not based on any resonant effect as previous approaches [3, 4]. This makes mode filtering of leakage channel fibers robust and more tolerant of process variations during manufacturing and use. Another unique feature of leakage channel fiber is that, along with robust fundamental mode propagation, it can also provide much improved bend loss performance at comparable core diameters than that provided by photonic crystal fibers . In a first demonstration few years ago, a passive leakage channel fiber was shown to robustly propagate fundamental mode with an effective area of 1417 µm2 . Moreover it can be bent down to 6.7 cm bend radius without significant loss. Since then, robust singe mode operation in ytterbium-doped leakage channel fibers with an effective area of 3160 µm2 has been demonstrated . Even at such large effective area, the leakage channel fiber can be bent down to a radius of 15 cm without significant excessive loss.
Fiber lasers have been a major commercial success in last few years, driven by demand for robust, low maintenance, and low cost laser systems. At this point, maximum peak power of a fiber laser is limited by fiber nonlinear effects. There are strong interests to further scale up peak power levels available from an optical fiber. This is currently achieved mostly by increasing core diameter of a conventional fiber to operate in the multimode regime. Coiling is typically used to reduce higher order mode propagation so that effective single mode operation is achieved . It has been pointed out recently that coiling can significantly reduce effective mode area of an optical fiber  to the point that very little overall increase of effective area is achieved in the coiled fiber.
One major advantage of leakage channel fibers is that large propagation loss of higher order modes comes from fiber design. Effective mode filtering can be achieved without using any coiling or bends. In previous works [1, 2], differential mode loss of straight leakage channel fibers has been studied in details. Bend loss of fundamental mode of leakage channel fibers is also studied using an approximate bend loss formula in . There is still a need to verify the accuracy of the approximate analysis. There has been no study of bend loss of higher order modes in leakage channel fibers. A study of differential mode loss in leakage channel fibers is essential for the understanding and optimization of leakage channel fibers. Considering the urgent practical needs of these large core leakage channel fibers to expand peak power limit of fiber lasers and amplifiers, a study in this area is urgent and critical.
2. Setting up finite element method for bend loss study
Finite element method (FEM) has been used in a number of previous work for the analysis of bend loss of optical waveguide and holey fibers with appropriate choice of perfect matched layers (PML) [9, 10]. We will start with determining appropriate PML to use in our case. Figure 1 shows the geometry used for this study. Only half of the geometry is necessary due to the mirror symmetry of the bent fiber. Λ and d are defined as hole spacing and hole diameter respectively in Fig. 1. Core diameter is defined as 2ρ=2Λ-d. PML is defined by horizontal width Lx, vertical width Ly and thickness Lpml.
In order to determine appropriate PML parameters to use, we analyzed a structure with d=12.68 µm and Λ=37.68 µm. This corresponds to a core diameter 2ρ of 50 µm and d/Λ of 0.673. Given the fiber parameters mentioned above, complex mode refractive index of the first three modes obtained with Multipole method are 1.443926-i*1.167×10-9, 1.443815-i3.076*10-8 and 1.443815-i*3.355*10-8 at 1.048 µm wavelength. These values are considered to be very accurate due to the high accuracy of Multipole method. Since the real parts of the FEM simulation converge quickly to the exact numbers, PML optimization is focused on only the imaginary parts of the complex mode refractive index and the Multipole results are used to benchmark the results.
PML is set up to absorb all outgoing radiation with negligible reflections. It needs to have sufficient dimension and absorption. We choose to use the following imaginary dielectric constant inside PML:
npml is the local refractive index within PML and b is absorption coefficient to be optimized. With a mesh size of 1 µm, it is determined that Lpml needs to be at least 15 µm, 10 µm and 5 µm for b of 9.8, 15 and 20 respectively so that relative error of the imaginary part of mode refractive index of the first three modes is small and stable. It is also determined that a mesh size of 1 µm is sufficient for an relative error of less than 1% for the imaginary part of the mode refractive index of the fundamental mode and an error of less than 2% for the imaginary part of the refractive index of the 2nd and 3rd modes. It is further determined that the choice of Lx and Ly does not matter. Fiber bending is simulated with well known conformal mapping.
PML optimization for bent fibers is also performed. In this case we used a fiber with d/Λ=0.65 and 2ρ=50 µm. Simulation was done for b=20 and mesh size of 1 µm at bent radii of R=1 m and 0.2 m. Since we do not have an accurate value to benchmark in these cases (Multipole method cannot handle bent fiber), we monitored the regime where imaginary part of mode refractive index is stable before it start changing rapidly. It is determined that Lpml needs to be larger than 5 µm and 2 µm for R=1 m and 0.2 m respectively to allow imaginary part of the mode refractive index of the first two modes to stay in the stable regime. In the following analysis, we used b=20 and a mesh size of 1 µm. Lpml=10 µm is used for the side away from the center of the bend and Lpml=20 µm is used for all other sides. Wavelength is at 1.0482 µm for all the simulations. Refractive index of air in the holes is 1.
3. Bend loss simulation of leakage channel fibers
Simulated first three modes are shown in Fig. 2 in a bent LCF with d/Λ=0.65 and 2ρ=50 µm. Fundamental mode, second and third mode of bending radius R=11 cm are given (from left to right) in Fig. 2(a) for bend orientation AA, i.e. bend plane intersecting two holes, and in Fig. 2(b) for bend orientation BB, i.e. bend plane intersecting center of glass regions between holes. One general feature is that mode profile moves away from the center of the bend (located to the left of the plot) as a result of an increase of refractive index. In general, the third mode has similar loss to that of the second mode with AA bend and has higher loss than that of the second mode in BB bend. In the following analysis, third mode is simulated but not shown. Real part of the mode refractive index shows a general increase towards small bend radius.
Fundamental mode and 2nd mode bend loss versus bend radius is summarized in Fig. 3 under both AA and BB bend for fibers with d/Λ=0.65, 0.7, 0.75, 0.8 and 0.85. The fiber has a core diameter 2ρ=50 µm. Confinement loss for all modes increases with a decrease of bend radius. Rate of increase of confinement loss also increase towards small bends. At bend radius around 1 m, there is very little difference between AA and BB bend for all modes studied. For the fundamental mode, loss for BB bend is slightly higher than that for AA bend at small bend radius. For the 2nd mode, loss for the BB bend can be significantly less than that for the AA bend at small bend radius. To our surprise, the 2nd mode can even have less loss than fundamental mode under BB bend at very small bend radius. Luckily, this only happens at very small bend radius where fundamental mode loss is very high, i.e. where we would not use the fiber. In any case, the reduction of differential mode loss between the 2nd mode and fundamental mode under BB orientation towards small bend radius needs to be considered when designing a LCF. In a practical fiber without deliberate control of bend orientations, it is realistic to expect that sections of fibers will have AA bend and sections of fiber will have BB bend. The lower bend loss of the 2nd mode under BB orientation can effectively reduce overall differential mode loss between the 2nd mode and fundamental mode. From a designer’s perspective, higher differential mode loss has to be designed in a fiber without deliberate bend orientation controls to counter this reduction of differential mode loss under BB orientation. A further point to note is that all bend loss is significantly reduced for larger d/Λ. For very small bend radius operation, large d/Λ is preferred. Differential mode loss is, however, largest at large bend radius, a point to be further discussed later. Deliberate control of bend orientation may have to be implemented to operate at very small bend radius. A further point to note is that all bend loss is significantly reduced for larger d/Λ. For very small bend radius operation, large d/Λ is preferred. Differential mode loss is, however, largest at large bend radius, a point to be further discussed later.
The data in Fig. 3 is plotted in Fig. 4 as a plot for the 2nd mode loss versus the fundamental mode loss for d/Λ=0.65, 0.7, 0.75 and 0.8. This is done for better visualization of differential mode loss. Curves for AA orientation are plotted as solid line with filled symbols, while curves for BB orientation are plotted as dotted line and unfilled symbols. For each d/Λ, lower left part of the curve corresponds to large bend radius and top right part small bend radius. There are a few general points to note. The first is that curves for AA and BB orientations essentially overlap at large bend radius (bottom left part of the curves), as we pointed out earlier in the last section, and they deviate towards small bend radius (top right part of the curves). The curve for AA orientation is higher than the curve for BB orientation. The second point is that each curves for AA orientation falls onto a trend line represented by α2nd=9.1*αFM at small bend radius, i.e. second order mode loss is 9.1 times fundamental mode loss when both are expressed in dBs. A third point to note is that, at large bend radius, bottom left part of each curves, 2nd order mode loss is higher than that of the corresponding point on trend line. This is the preferred regime to operate for each d/Λ as high differential loss can be obtained at this part of the curves, α2nd≈25*αFM when straight. If desired device length, acceptable fundamental mode loss and core diameter are known, d/Λ can be chosen to give the desired fundamental mode loss over the device length. A larger 2nd order mode loss to fundamental mode loss ratio is, therefore, essential to give a higher differential mode loss over the device length. The fourth point to note is that each curve for BB orientation is substantially lower than corresponding curves for AA orientation at small bend radius (top right part of curves). This indicates that 2nd order mode loss for BB orientation is lower than corresponding loss for AA orientation for the same fundamental mode loss, a point we made earlier. Each curves for BB orientation is essentially parallel to each other at small bend radius, indicating that they will increase by a same factor when fundamental mode loss increases by a factor of 10.
In a previous study of straight LCFs , it has been shown that LCFs can be designed for a wide range of core diameters. We have studied bend loss of a LCF with 30µm core diameter and d/Λ=0.7 in this work to further understand bend loss dependence on core diameters. The data is plotted in Fig. 5 as 2nd mode loss versus fundamental mode loss for both AA and BB orientations for the 30µm core LCF as well the 50µm LCF with d/Λ=0.7. The higher 2nd order mode loss of the 30µm core LCF than that of the 50µm core LCF at large bend radius, i.e. small fundamental mode loss, is clearly seen for both AA and BB orientations. For BB orientation, 2nd mode loss of the 30µm LCF is also higher than that of the 50µm LCF for small bend radius, i.e. large fundamental mode loss, while, for AA orientation, the 2nd mode loss of the 30µm core LCF converges with that of the 50µm core LCF at very small bend radius. In any case, the increase of 2nd mode loss for the 30µm LCF is helpful, but not significant, proving that the differential loss has weak core diameter dependence and much larger core diameter is possible with appropriate LCF designs.
4. Experimental result
Comparison between FEM simulations and experimental results is conducted in this section. The bend loss of a fabricated fiber with four large holes and two small holes was measured for both AA and BB directions. To ensure bending orientations, we took the images of the end face of the fiber cross section at the output stage and measure the transmitted power. The cross-sectional photo of the fiber is shown in the inset of Fig. 6. The geometry information is obtained by boundary extraction from the photo to give a large hole spacing ΛA 52 µm, a large hole diameter dA 45.4 µm, a small hole separation ΛB 52 µm and a small hole diameter dB 39 µm. In addition, we notice that there is a small angle deviation of 2° between adjacent large holes as shown in Fig. 6, which may be due to fabrication variations. This fiber with aforementioned parameters is then simulated using FEM. The measured loss for AA and BB orientations are plotted as circles and squares respectively; the simulated data for each orientation are shown as solid and dashed lines. The simulation fits the measured data reasonably well.
Lower bend loss can be achieved by increasing relative hole diameter, i.e. use larger d/Λ. This approach can indeed allow low fundamental mode loss at very small bend radius. For a LCF with d/Λ=0.85, fundamental bend loss as low as 0.1dB/m can be achieved for a bend radius of 4cm. At this small bend radius, 2nd order mode can be estimated by the trend line given by α2nd=9.1*αFM. As shown in Fig. 4, relatively higher 2nd order mode loss is obtained at larger bend radius, α2nd≈25*αFM when stright. In many practical situations, higher differential mode loss is preferred. It is consequently better to work at large bend radius. Additional differential mode loss increase can be achieved by optimization of rare-earth doping profile so that it overlaps spatially better with the fundamental mode than the 2nd order mode. Further differential mode loss can be gained by resonantly coupling out the 2nd order mode as demonstrated in a theoretical study in  at the expense of narrow wavelength range of operation and much stringent fabrication tolerances. There is an additional reason for working at larger bend radius. This is due to that mode distortion at small bend radius leads to a reduction of effective area of the optical fiber . In our simulation of effective mode area of LCFs with a core diameter of 2ρ=50 µm and d/Λ=0.7, 0.75, 0.8 and 0.85, the reduction of effective area of the 50 µm core diameter LCFs at small bend radius is dramatic. This effective area reduction has a weak dependence on d/Λ. To be able to maintain 90% of the maximum effective area for the LCFs with 50 µm core diameter, bend radius needs to be kept above ~0.4 m. At R=0.2 m, effective area is reduced to ~75% of the maximum.
Bend loss of the 2nd order mode, i.e. HE01+HE21, is studied in details in Figs. 3 and 4. We have also studied bend loss of the 3rd mode, i.e. HE01-HE21, and found it to be higher than that of the 2nd mode with very similar dependence on bends. It is not presented here. In our previous study of straight fibers , even higher order modes were studied but found to be of much higher losses.
In this study of bending effect, we have focused on 50µm core LCFs with various d/Λ. In a previous study of straight LCFs , we have found that core diameters exceeding 100µm are possible with appropriate designs of LCFs. In addition, LCFs have weak wavelength dependence, i.e. capable of broad band operation. This weak wavelength dependence also translates into ease of fabrication due to the relaxed tolerance.
In convention large mode area fibers based on multimode approach , small coils are required to provide significant differential mode loss for single mode operation. This can reduce the effective area of coiled fiber to the point that no overall gain in effective area is achieved. On the other hand, LCF can be designed to provide maximum differential mode loss in straight fibers and large differential mode loss at large coil diameters. This would provide a real practice solution for large effective area fibers.
We have studied bend performance of LCFs for both fundamental modes and higher order modes. It is found that low bend loss of all modes can be achieved with large d/Λ. It is further found that highest differential mode loss is achieved at large bend radius. At small bend radius, 2nd mode loss becomes 9.1 times of the fundamental mode loss for AA orientation. Although fundamental mode has slightly smaller bend loss for AA orientation than that for BB orientation, 2nd order mode has much smaller bend loss for BB orientation than that for AA orientation.
References and links
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