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Single-mode annular chirally-coupled core fibers for fiber lasers
Optics Communications 410 (2018) 297–304
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Single-mode annular chirally-coupled core fibers for fiber lasers Haitao Zhang *, He Hao, Linlu He, Mali Gong Center for Photonics and Electronics, State Key Laboratory of Precision Measurement and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Keywords: Annular chirally-coupled core fiber Fiber design Fiber laser Higher-order mode suppression
a b s t r a c t Chirally-coupled core (CCC) fiber can transmit single fundamental mode and effectively suppresses higher-order mode (HOM) propagation, thus improve the beam quality. However, the manufacture of CCC fiber is complicated due to its small side core. To decrease the manufacture difficulty in China, a novel fiber structure is presented, defined as annular chirally-coupled core (ACCC) fiber, replacing the small side core by a larger side annulus. In this paper, we designed the fiber parameters of this new structure, and demonstrated that the new structure has a similar property of single mode with traditional CCC fiber. Helical coordinate system was introduced into the finite element method (FEM) to analyze the mode field in the fiber, and the beam propagation method (BPM) was employed to analyze the influence of the fiber parameters on the mode loss. Based on the result above, the fiber structure was optimized for efficient single-mode transmission, in which the core diameter is 35 μm with beam quality M2 value of 1.04 and an optical to optical conversion efficiency of 84%. In this fiber, fundamental mode propagates in an acceptable loss, while the HOMs decay rapidly. © 2017 Elsevier B.V. All rights reserved.
1. Introduction High-power fiber lasers with diffraction-limited output beams are widely applied in many areas such as industry, medicine, telecommunications, and military [1–3]. Scaling the fibers to larger effective modefield-diameter (MFD) ones helps decrease optical density, suppress the nonlinear effects such as stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS), and as a result, increase the pulse energy and peak power of fiber lasers. For step-index fibers, prominent techniques for effective single-mode operations include matched excitation by particularly ion-doping [4], and higher-order mode (HOM) discrimination by bending [5]. However, these methods usually limit the core size less than 30 μm for diffraction-limited beam quality output. For instance, the beam quality factor M2 -value of the normal stepindex fibers with core size of 35 μm will increase to 2.5 since too densely packed mode to discriminate the fundamental mode and its nearest neighbors. Utilizing microstructures, such as photonic crystal fibers (PCFs) [6] and large pitch fibers (LPFs) [7], very large mode area (VLMA, core diameter beyond 50 μm) fibers are developed for high energy lasers. However, microstructure VLMA fibers suffer monolithically integration difficulty (e.g. micro-structured fiber rods), as well as thermal-induced mode instabilities because the mode constraint ability of discontinuous boundary conditions of the core is much weaker than
the continuous boundary ones [8]. The chirally-coupled core (CCC) fiber with a continuous core boundary combines robust single-mode laser performance in large cores (up to 55 μm demonstrated) [9] and stable mode even for high power laser output, while retaining the handling and packaging benefits associated with single mode fibers. The CCC fiber seems to be one of the most promising approaches for large mode and stable high power laser applications. However, except for the usage rights protections by CCC fiber patents (US7424193), the CCC fibers need eccentric preform for its helix side core. Meanwhile, many Chinese fiber manufacturers have the fiber processing technique with concentric preform, and they prefer the fiber structure based on the concentric preform to avoid the change of the fiber processing. Thus, a novel structure is designed and defined as annular chirally-coupled core (ACCC) fiber. The ACCC structure has an 82 μm-diameter multimode side annulus instead of the 16-μm side core in the common CCC fiber so that ACCC fiber can be drawn from a concentric side core preform instead of eccentric one of CCC. The design principle of CCC fiber is to make the HOMs and the fundamental mode (FM) meet and do not meet the quasi-phasematching (QPM) condition, respectively. This principle greatly improves the discrimination of the HOMs and FM losses [10]. Meanwhile, this principle also requires a small side core with very few modes. The smaller side core could decrease the number of the side modes and avoid
* Corresponding author.
E-mail address: [email protected] (H. Zhang). https://doi.org/10.1016/j.optcom.2017.10.003 Received 1 April 2017; Received in revised form 29 September 2017; Accepted 2 October 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
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Optics Communications 410 (2018) 297–304
the FM matching the side modes. On the other side, a smaller margin between the losses of each mode is also sufficient to obtain the single mode preservation. For instance, in the design of the leakage channel fibers, the principles are 1 dB/m loss of HOMs and 0.1 dB/m for the FM [11], which are sufficient to suppress the propagation of HOMs and ensure negligible fundamental mode transmission loss. Furthermore, due to the mode competition, the margin between the losses of each mode could be even smaller. For the ACCC fiber, the side annulus can support many modes, and each mode of the side annulus can couple with modes in the center. Instead of calculating the QPM condition of all the modes, we present a novel design concept and procedure especially for the gain fiber operating in the fiber laser cavity. First, we used the beam propagation method (BPM) to calculate the difference of each mode’s loss directly. Then, we optimized the ACCC structure parameter values by investigating the laser dynamic mechanism in active ACCC fibers. Using the output beam brightness as a judgment criterion, we could describe the performance of the ACCC fiber laser and ultimately optimized the structure definitely, considering both efficiency and beam quality. In this paper, we firstly presented ACCC structure and explained why this novel structure is more suitable for Chinese manufacturing techniques. We used the finite element method (FEM) to simulate the mode field distributions in ACCC fibers and the BPM to calculate the losses. Based on the simulation results of the fiber mode discrimination for fiber passive propagation process, the rough range of ACCC structure parameter could be preliminarily determined. Then, we calculated the mode competition of the ACCC fiber lasers by exploring the laser rate equations and ultimately optimized the structure based on the brightness criterion. And it was demonstrated this new ACCC structure has the similar single-mode characteristics to common CCC fiber.
Fig. 1. (a) Structure of the main preform for CCC fiber. Numbers in brackets are the actual size of the fiber, and the dash line shows the drilling position for the side core preform. Hatchings show the doped cores in the fiber by MCVD. (b) Concentric side core preform. (c) Eccentric side core preform.
Fig. 2. ACCC fiber structure.
diameter of the center core 𝐷c is 35 μm with NA of 0.06, similar with the common CCC fiber. On the other hand, the ACCC fiber structure has a larger helical side annulus, greatly reducing the difficulty of manufacturing the fiber preforms. In ACCC fiber, the outer diameter of the side annulus 𝐷o could be up to 82 μm (NA0.1), therefore the diameter of side preform is about 12 mm, which is thick enough (much greater than 6 mm limitation) to satisfy the requirements of Chinese manufacturers. The helix pitch 𝛬 of the side annulus, the width of the annulus w, and the inner distance between the two doped cores d affect the fiber characteristics and will be optimized in the following sections. Because of the helix structure of CCC fiber, the cross section is associated with its 𝑧-coordinate value. Herein a helical coordinate system {𝑋, 𝑌 , 𝑍} is introduced for the helix structure [12]
2. Annular chirally-coupled core fibers As described below, there are still technical difficulties for manufacturing CCC fibers in China. The doped fiber core, i.e. high refractive index zone herein, is made by the modified chemical vapor deposition (MCVD) method, which involves the deposition inside a substrate tube for the fiber preform. First, limited by the MCVD technology, the diameter of the deposition zone in the preform is difficult to be larger than 5 mm. Second, the preform is usually very long so that the diameter of the preform should not be thinner than 6 mm, or it would be too fragile for further processing. These two restrictions determine the maximum doped zone and the minimum outside diameter dimension of the preforms. To fabricate the CCC fiber, we need to prepare two fiber preforms, a main preform for signal propagation and a side core preform for HOMs loss dissipation. The main preform needs to be drilled a proper hole to insert the side core preform. If we dope the center signal core to a maximum diameter of 5 mm by MCVD, we can design the whole preform structure as shown in Fig. 1(a), by scaling up the corresponding fiber dimensions in brackets. In the fiber, the 33-μm center core has a numerical aperture (NA) of 0.06, and the side core has a diameter of 16 μm and NA of 0.1. The distance between the two cores, measured from the edge of the center core to the middle of the side core, is 12 μm and scaled up to 1.8 mm for preform structure. If we used the concentric side core preform like Fig. 1(b), the diameter of the side core preform would be less than 3.6 mm, less than 6-mm preform limit and too thin for a long rod. One of the solutions is grinding the side core preform eccentrically, and making the doped core area close to the boundary, as in Fig. 1(c), which can enlarge the possible side core preform greater than 6 mm. This method, called fire-polishing, is applied by the fiber fabrication industry, however still needs time for Chinese manufacturers to realize. If we could magnify the size of the side core and keep the property of the fiber at the same time, we could use the current fiber producing process and reduce the cost. To satisfy the manufacturing requirements in China, a novel fiber structure denoted as ACCC fiber is designed, as shown in Fig. 2. The
⎧𝑋 = 𝑥 ⋅ cos 𝜏𝑧 + 𝑦 ⋅ sin 𝜏𝑧 ⎪ ⎨𝑌 = −𝑥 ⋅ sin 𝜏𝑧 + 𝑦 ⋅ cos 𝜏𝑧 ⎪𝑍 = 𝑧 ⎩
(1)
where 𝜏 = 2𝜋∕𝛬, {𝑥, 𝑦, 𝑧} is the Cartesian coordinate system, and zero point of the 𝑧 axis is set at the position where x and X axes coincide. In the helical coordinate system, the shape of the fiber cross section decouples with Z direction, and the three-dimensional finite element problem degenerates into two-dimension, which decreases the calculation time. However, this coordinate system is not orthogonal, so the Helmholtz equation used in the FEM in the system needs to be further discussed. This equation is given by ∇ × (∇ × 𝑬) − 𝑘20 𝜺h 𝑬 = 0
(2)
where the 𝑘0 is the wave number in a vacuum, and the 𝜺h is the relative permittivity tensor in the helical coordinates, which is constructed by applying Jacobian matrix 𝑱 of the helical coordinate with respect to the Cartesian system 𝜺h (𝑋, 𝑌 , 𝑍) = 𝑱 ⋅ 𝜺c (𝑥, 𝑦) ⋅ 𝑱 T 0 0⎞ ⎛1 0 𝜏𝑌 ⎞ ⎛𝜀1 (𝑋, 𝑌 ) 𝜀6 (𝑋, 𝑌 ) 𝜀5 (𝑋, 𝑌 )⎞ ⎛ 1 = ⎜0 1 −𝜏𝑋 ⎟ ⎜𝜀6 (𝑋, 𝑌 ) 𝜀2 (𝑋, 𝑌 ) 𝜀4 (𝑋, 𝑌 )⎟ ⎜ 0 1 0⎟ . ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝0 0 1 ⎠ ⎝𝜀5 (𝑋, 𝑌 ) 𝜀4 (𝑋, 𝑌 ) 𝜀3 (𝑋, 𝑌 )⎠ ⎝𝜏𝑌 −𝜏𝑋 1⎠ 298
(3)
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Optics Communications 410 (2018) 297–304
For isotropic medium, 𝜺c = diag (𝜀 (𝑋, 𝑌 ) , 𝜀 (𝑋, 𝑌 ) , 𝜀 (𝑋, 𝑌 )), then the helical permittivity tensor can be simplified as ⎛𝑌 2 𝜏 2 + 1 −𝑋𝑌 𝜏 2 𝑌 𝜏 ⎞ 𝜺h (𝑋, 𝑌 , 𝑍) = 𝜀 (𝑋, 𝑌 ) ⋅ ⎜ −𝑋𝑌 𝜏 2 𝑋 2 𝜏 2 + 1 −𝑋𝜏 ⎟ . ⎜ ⎟ −𝑋𝜏 1 ⎠ ⎝ 𝑌𝜏
(4)
In addition, the curl operator of the helical coordinate also differs from the Cartesian system. For arbitrary vector 𝑭 , the curl is expressed as ∇×𝑭 | | 𝒈1 𝒈2 𝒈3 | | | | = | 𝜕∕𝜕𝑋 𝜕∕𝜕𝑌 𝜕∕𝜕𝑍 | ( ) | 1 | |𝐹 − 𝐹 3 𝑌 𝜏 𝐹 2 + 𝐹 3 𝑋𝜏 −𝐹 1 𝑌 𝜏 + 𝐹 2 𝑋𝜏 + 𝐹 3 𝑋 2 𝜏 2 + 𝑌 2 𝜏 2 + 1 | | | {
(5)
}
where 𝒈1 , 𝒈2 , 𝒈3 are the covariant bases of this system, and 𝐹 𝑖 are the decompositions of the vector 𝑭 in these covariant bases. The derivation of Eq. (5) is in the Appendix. Substitute this permittivity Eq. (4) and the curl expression Eq. (5) into the Helmholtz equation (2), then the mode field distribution of the fiber can be obtained by FEM method.
Fig. 3. Range of mode loss varies with the distance d. Inset: mode field of the center LP01 (left) and LP11 (right) mode.
3. Numerical simulation results and discussions
results in our future work. Herein, for theoretical design and evaluation for the first step, the material of the fiber is regarded as isotropic despite of the birefringence. To optimize the ACCC structure, we investigated the mode loss varying with the parameters of the helix pitch 𝛬, the width of the annulus w, and the inner distance between the two doped cores d. First an investigation was performed for the distance d. The wavelength 𝜆 used in the study is 1064 nm. Setting the pitch 𝛬 varying from 2500 μm to 10 000 μm and width w up to 40 μm, the relationship between mode loss range and the distance d was obtained as in Fig. 3. The central LP01 and LP11 mode (shown in the inset of Fig. 3) share the same varying trend. The light fields in the central core and the side annulus interact with each other by evanescent-wave coupling. When the distance d increases, the overlap factor between the core and annulus evanescent fields drops rapidly, thus the loss of all modes falls simultaneously. This trend could be used to control the total loss of the fiber coarsely for the first step. To choose a proper distance, we need to place the two cores close enough to get effective coupling. Meanwhile, to get a usable gain fiber for laser oscillation and amplification, the FM should maintain a low loss, usually less than 1 dB/m [14]. As shown in Fig. 3, at distance of 5.5, 6 and 6.5 μm, the minimum FM losses are respectively 1.01, 0.49 and 0.24 dB/m, and the minimum LP11 losses are 1.23, 0.84, and 0.68 dB/m. The shorter distance 𝑑, the higher loss is. The distance 𝑑 should be larger than 5.5 μm with less than 1dB/m FM loss for laser applications. As an initial selection, the distance of the two cores was chosen as 6 μm, at which the minimum FM mode loss begins to get lower than 0.5 dB/m and has a relative larger initial loss difference (0.35 dB/m) between FM and LP11 than the case in distance 5.5 μm (0.22 dB/m). The selection of distance as 6 μm is also leaving the room of 0.5 μm for manufacturing errors and enough margins for 𝛬 and w optimization in the next step. The mode losses varying with the helix pitch 𝛬 and the annular width w are shown in Fig. 4. The HOM (LP11 in Fig. 4) always has a higher loss ratio than in the center, and the loss difference ranges from 0.2 to 18 dB/m. The variation of the HOM and FM losses differs with pitch changing, as shown in Fig. 4(a). For example, when w is 3.2 μm and 𝛬 increases from 7000 to 7750 μm, the LP01 loss rises from 7.1 to 7.8 dB/m, while the LP11 loss falls from 17 to 15 dB/m. That is because the helix side rotating along the fiber introduces an additional phase difference between the central mode and side mode, and when helix pitch changes, the coupling from the central HOM and FM to side annulus varies respectively. The loss difference of each mode makes it possible to optimize the fiber to get lower FM loss and higher HOM loss.
As we know, the step-index fiber with the same core diameters has an 𝑀 2 value of about 2.5. The design purpose of the fiber is to reduce the 𝑀 2 to less than 1.1 as well as keep low FM loss to get a high efficient laser oscillation, amplification and propagation. At the first step, we need to increase the HOM loss while control the FM loss to a lower level to ensure mode discrimination. The effects of different fiber parameters were discussed on the loss of each mode by using the BPM. Then, based on mode discrimination in laser passive propagation progress, the structure parameter ranges were preliminary designed. Second, the design results from the first step were applied to the fiber laser simulation. Using the mode competition method, we can investigate optical to optical conversion efficiency of the fiber as well as beam quality for further structure optimization. 3.1. Mode discrimination in propagation for ACCC fibers Herein we used FEM to calculate the mode field distribution under different ACCC fiber structures. By utilizing the calculated mode field as the input field of BPM, we got the light propagation loss of each mode. The ideal ACCC structure consists of isotropic medium. However, the practical manufacturing process might induce the strain-induced birefringence, which is related to the cooling and rotating process during the fiber fabrication. Ma et al. utilized a simplified birefringence model for CCC fibers, which assumes that the birefringence region is a single ellipse located between the two core, and a moderate estimate of birefringence is given. Under this assumption, they explained the physical origin of the multitude of the quasi-phase-matching (QPM) resonances [13]. In the CCC fibers, the small side core supports a few modes, which makes it easy to correspondence each pair of side and central modes and benefits to derive the analytical expression of the QPM based on sketchy birefringence estimation [12,13]. While for ACCC structure with a larger annular side core supporting lots of modes, QPM analytical method is not proper any more for it is almost impossible to match propagation constants of the core and numerous side annular modes. The numerical method of the QPM for ACCC with so many annular modes is also a heroic challenge. And until now, the accurate birefringence value or model of such chirally fiber has not been achieved even from the experiment or the theory. The birefringence model for ACCC fiber which might be more complicated and related to the drawing process could be another important assignment. The influence of the birefringence needs further investigation along with the experimental 299
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Optics Communications 410 (2018) 297–304
Fig. 4. (a) Mode losses of LP01 mode and LP11 mode vary with the helix pitch 𝛬. The annulus widths w are shown in the legend. The dash lines mark the 1 dB/m loss of each mode. (b) Mode losses vary with the annular width w of the side annulus. The pitches 𝛬 are shown in the legend.
For the annulus width of about 5.9 μm, the mode loss decreases with the helix pitch 𝛬 rises. When 𝛬 exceeds 7750 μm, the FM loss falls below 1 dB/m, while the LP11 mode loss is still over 2 dB/m, which could be further optimized in Section 3.2. On the other hand, the mode losses oscillate with the annular width w. For clarity, Fig. 4(b) was drawn for the relationship between the loss and the width w. When w increases from 3.2 to 8.6, the mode losses first decrease from the peak (7.85 dB/m for LP01) to the minimum (0.23 dB/m), then rise again to the maximum (8 dB/m). When w continues to rise, the oscillation mitigates. The reason of the oscillation is that the annular width w influences the side annulus mode distribution, proved by the modified FEM using Eq. (4), shown in Fig. 5. The higher w (from 3.2 to 5.9 μm) leads to a looser constraint of the side annulus, which makes less evanescent field of the side mode and less overlap factor between the central and the side mode, and the center mode losses decrease. Meanwhile, when the width increases to a threshold (8.6 μm in Fig. 5), the side HOM of the radial direction appears, which introduces a larger evanescent field area, and makes the center mode losses increase. Fig. 6 shows the central LP11 mode distributions with different widths w. It is obvious that the 5.9 μm case has the less mode coupling than 3.2 and 8.6 μm. This characteristic could also be used to adjust the total loss of the fiber, or compensate the inaccuracy of the distance of the two cores in fabrication. Optimizing these parameters, we could get the proper fiber structure for single FM transmission. The LP01 mode loss lower than 1dB/m is required in our design. According to the data in Fig. 4, valid parameters are annulus width w as about 5.9 μm and helix pitch 𝛬 over 7750 μm, so we chose a range of w from 5 to 6.5 μm, and 𝛬 from 7600 to 12 400 μm for further optimization.
The dynamic mode analysis of the ACCC laser utilizes the typical linear cavity as described schematically in Ref. [15]. We have calculated the loss of each mode (LP01, LP11, LP21 and LP02 in this case) by BPM in Section 3.1. By substituting the mode loss results into the well-known space-dependent and time-independent steady-state rate equations, we can get the signal power of each mode and the pump power varying along the fiber length [16]. [𝑃𝑝+ (𝑧)+𝑃𝑝− (𝑧)]𝜎𝑎𝑝 𝜑𝑝 (𝑥,𝑦)
𝑁2 (𝑥, 𝑦, 𝑧) = 𝑁1 (𝑥, 𝑦, 𝑧)
ℎ𝜈𝑝 [𝑃𝑝+ (𝑧)+𝑃𝑝− (𝑧)]𝜎𝑒𝑝 𝜑𝑝 (𝑥,𝑦) ℎ𝜈𝑝
±
𝑑𝑃𝑝± (𝑧) 𝑑𝑧
{ =
∬𝐴
+
+ 1 𝜏
∑𝑀
[𝑃𝑠𝑖+ (𝑧)+𝑃𝑠𝑖− (𝑧)]𝜎𝑎𝑠 𝜑𝑠𝑖𝑗 (𝑥,𝑦)
𝑖=1
+
∑𝑀 𝑖=1
ℎ𝜈𝑠 [𝑃𝑠𝑖+ (𝑧)+𝑃𝑠𝑖− (𝑧)]𝜎𝑒𝑠 𝜑𝑠𝑖 (𝑥,𝑦)
[ ] 𝜎𝑒𝑝 𝑁2 (𝑥, 𝑦, 𝑧) − 𝜎𝑎𝑝 𝑁1 (𝑥, 𝑦, 𝑧) 𝜑𝑝 (𝑥, 𝑦)𝑑𝑥𝑑𝑦
}
× 𝑃𝑝± (𝑧) − 𝛼𝑝 𝑃𝑝± (𝑧)
±
𝑑𝑃𝑠𝑖± (𝑧) 𝑑𝑧
{ =
∬𝐴𝒋
(6)
ℎ𝜈𝑠
(7)
} [ ] 𝜎𝑒𝑠 𝑁2 (𝑥, 𝑦, 𝑧) − 𝜎𝑎𝑠 𝑁1 (𝑥, 𝑦, 𝑧) 𝜑𝑠𝑖 (𝑥, 𝑦)𝑑𝑥𝑑𝑦
× 𝑃𝑠𝑖± (𝑧) − 𝛼𝑠𝑖 𝑃𝑠𝑖± (𝑧) −
𝑀 ∑
(8) ± 𝑑𝑖𝑗 [𝑝± 𝑠𝑖 (𝑧) − 𝑝𝑠𝑖 (𝑧)].
𝑗=1
In these equations, 𝜈s and 𝜈p are the laser and pump frequencies. 𝑁1 and 𝑁2 are the population densities of the lower and upper lasing levels. 𝑃𝑝± (𝑧) and 𝑃𝑠𝑖± (𝑧) are the pump powers and ith mode powers in the forward and backward directions, respectively. 𝜑p and 𝜑si are the normalized intensity distributions of the pump and ith mode, and 𝛼si is the loss factors of the ith mode. Other parameters used for mode competition are listed in Table 1. The boundary conditions of the fiber is given by𝑃𝑠𝑖− (0) = 𝑅1 𝑃𝑠𝑖+ (0) and 𝑃𝑠𝑖+ (𝐿) = 𝑅2 𝑃𝑠𝑖− (𝐿), and we assign the reflection rate 𝑅1 = 0.99 and 𝑅2 = 0.04. With the boundary conditions and an assumed small initial signal, we can calculate the output power of each mode by iteration of the rate equations (6)–(8). According to the power of each mode and the corresponding effective mode field diameter (MFD), beam quality factor (𝑀 2 ) can be calculated by the method described in Ref. [17]. In fiber laser systems, the brightness of a laser is used to indicate its ability to produce the highest power in the smallest spot and the smallest divergence. For a circular,
3.2. Mode competition in ACCC fiber lasers In addition to the eigen mode field distribution and propagation characteristics, we are more concerned about the mode competition, the high power dynamic mechanism and the laser output performance in the active ACCC fibers. We could further refine the proper ACCC structure based on the primary parameter ranges optimized in Section 3.1. 300
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Optics Communications 410 (2018) 297–304
Fig. 5. (a) Side mode simulated by FEM. (b) Enlargement of (a), with an annulus width of 8.6 μm. (c)Annulus width of 5.9 μm, HOM is restrained. (d) Annulus width of 3.2 μm, area of evanescent field increases. The outer diameter and position of the side annulus stay the same in these cases. (e) Cross-section graph of (b), (c) and (d). Dash line is the outer border of the side annulus.
Fig. 6. Center LP11 mode, annulus width w of (a) 8.6 μm, (b) 5.9 μm and (c) 3.2μm.
Table 1 Parameters in mode competition. Parameter
Value
Parameter
Value
Fiber length (L) Doping concentration (𝑁0 ) Pump power (𝑃𝑝 ) Front reflectivity of the cavity (𝑅1 ) End reflectivity of the cavity (𝑅2 ) Pump wavelength (𝜆p ) Signal wavelength (𝜆s )
2m 4.98 × 1025 m−3 1500 W 0.99 0.04 976 nm 1064 nm
Planck constant (h) Spontaneous lifetime of the upper lasing level (𝜏) Pump loss factor (𝛼𝑝 ) Pump absorption cross section (𝜎ap ) Pump emission cross section (𝜎ep ) Signal absorption cross section (𝜎as ) Signal emission cross section (𝜎es )
6.626 × 10−34 1.0 ms 2.0 × 10−3 m−1 2.34 × 10−24 m2 2.34 × 10−24 m2 6.58 × 10−27 m2 3.11 × 10−25 m2
low-diverging laser source, the brightness B could be expressed as𝐵 = 𝑃out ∕(𝑀 2 𝜆)2 [18], where 𝑃out is the output power, and 𝜆 is the laser wavelength. And we use the brightness to describe the fiber performance and optimize the structure. Fig. 7(a) and (b) show the 𝑀 2 and output power varying with helix pitch 𝛬 and annulus width w, respectively. The 𝑀 2 ranges between 1.35 and 1.04, and reaches the minimum when w is 7.5 μm or 8 μm. However, at these points, the output power is lower than 800 W, which means a high loss and inefficiency. With the 𝛬 increasing, the output power increases generally, while the beam quality deteriorates at the same time. Considering the high efficiency and good beam quality simultaneously, we use the brightness B to judge the parameters, as in Fig. 7(c). When the helix pitch 𝛬 is 10 000 μm, outer diameters 𝐷o is 82 μm, the width of the side annulus w is 6.5 μm, and the distance of two cores d is 6 μm. The loss of LP01 mode is about 0.24 dB/m, while the losses of LP11, LP21 and LP02 are 1.78 dB/m, 13.54 dB/m and 50.60 dB/m. The loss within a unit length is determined by both the loss of the modes in the side core and the coupling strength from
central modes to these side modes. The difference of losses is sufficient to restrain the HOMs in the Yb-doped fiber and keep the FM amplified. At this condition, the fiber structure has the highest output brightness B of 9.47×1014 W ⋅ m−2 , with 𝑀 2 of 1.04 and output power of 1163 W. That proves the ACCC fiber structure has a good output characteristic. The power variation along the fiber is shown in Fig. 8. As in Fig. 8(a), the HOMs can hardly be amplified in this fiber. The highest power of LP11 is lower than 0.07 W, and the LP21 and LP02 mode are even lower. Therefore, the output beam consists of only FM (over 99.99%). Fig. 8(b) shows the pump power and signal power evolutions along the fiber. With the 1500 W pump, the total output is 1262 W, and the conversion efficiency is about 84.1%. 3.3. HOM suppression and leaked power By suppressing HOM, the ACCC fiber enables high power laser output with an effective large area single-mode. However, the mode coupling of HOM to lossy modes is accompanied by a small leaked power into the 301
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Fig. 7. Relationship between helix pitch 𝛬 and (a) 𝑀 2 , (b) output power and (c) brightness of fiber sat different annulus width w.
Fig. 8. Power evolutions along the fiber. (a) Power of each linear mode; (b) Pump power and signal power.
cladding. The leaked power would potentially deteriorate the possible follow-up cladding light stripper and the fused fiber combiner or splitter. The excess heat generated from the leaky power would also limit the fiber laser power scalability. The leaked power is determined by the mode loss of the fiber. Utilizing the data of mode loss and parameters of ACCC in Sec 3.2 and CCC in Ref. [19], we subtract the amplified mode power along the fiber length in ACCC from those in CCC fibers to compare the ability of the anti-loss and heat resistance between these two CCC-like specialty fibers. The seed power of the LP01, LP11, LP21 and LP02 is assumed to be 50 W, 5 W, 0.5 W and 0.1 W, which is ordinary and reasonable for fiber amplifier in reality. As shown in Fig. 9, the output power of LP01 and LP11 mode in ACCC fiber is a little lower than those in CCC fiber. The LP01 mode power difference is 25.8 W, the mode in ACCC fiber is 2.4% lower than CCC fiber. And the difference of LP11 mode is 29.6 W, 31% lower. The other two modes’ power difference are very small, nearly to zero. The total power of ACCC fiber amplifier is 1150.0 W, 4.6% lower
than the 1205.3 W of CCC fiber’s total power. The power margin of ACCC fiber is almost the same as CCC fiber as well as the leaked power burden. The results in Fig. 9 indicate the LP11 margin is dominant. We could simplify the mode competition and power evolution only for LP01 and LP11 modes. Fixing the loss value of LP01 as 0.24 dB/m, we examine the dependence of the leaked power on the loss of the LP11 mode which representing the capacity of LP11 mode suppression. As shown in Fig. 10, the leaked power of LP01 mode is almost the constant, staying at about 106 W because of the fixed LP01 mode loss. The leaked power of LP11 rapidly goes down and converges with LP11 loss beyond 1.5 dB/m. The inflexion point has the loss coefficient 4.5 dB/m and total power 1183 W. The output power of the amplifier converges to 1230 W with loss beyond 20 dB/m. Our designed ACCC structure has a loss of 1.78 dB/m, corresponding with a leaked LP11 power of 4 W and 1200 W is a reasonable trade-off selection. 302
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by the National Natural Science Foundation of China (grant number 61475081). Appendix To derive the gradient, divergence and curl operator of the helical } { coordinate, first we need to get the covariant basis 𝒈𝟏 ,𝒈𝟐 ,𝒈𝟑 and { 1 2 3} contravariant basis 𝒈 ,𝒈 ,𝒈 of this coordinate system. To express the {𝑥, 𝑦, 𝑧} in the Cartesian coordinate system by {𝑋, 𝑌 , 𝑍}, we have ⎧𝑥 = 𝑋 ⋅ cos 𝜏𝑍 − 𝑌 ⋅ sin 𝜏𝑍, ⎪ ⎨𝑦 = 𝑋 ⋅ sin 𝜏𝑍 + 𝑌 ⋅ cos 𝜏𝑍, ⎪𝑧 = 𝑍. ⎩
(9)
Then the position vector 𝑹 of any point (𝑥, 𝑦, 𝑧) is expressed as 𝑹 = 𝑥𝒊 + 𝑦𝒋 + 𝑧𝒌 = (𝑋 ⋅ cos 𝜏𝑍 − 𝑌 ⋅ sin 𝜏𝑍) 𝒊 (10)
+ (𝑋 ⋅ sin 𝜏𝑍 + 𝑌 ⋅ cos 𝜏𝑍) 𝒋 + 𝑍𝒌
Fig. 9. Mode power margin along the fiber length between CCC and ACCC fiber amplifiers.
where{𝒊, 𝒋, 𝒌} is a set of standard basis in the Cartesian coordinate system. Now we have the covariant basis of the helical coordinate. ⎧𝒈 = 𝜕𝑹 = 𝒊 cos 𝜏𝑍 + 𝒋 sin 𝜏𝑍, ⎪ 1 𝜕𝑋 ⎪ 𝜕𝑹 (11) = −𝒊 sin 𝜏𝑍 + 𝒋 cos 𝜏𝑍, ⎨𝒈 2 = 𝜕𝑌 ⎪ 𝜕𝑹 ⎪𝒈 = ⎩ 3 𝜕𝑍 = −𝒊𝑦𝜏 + 𝒋𝑥𝜏 + 𝒌. From the set of the covariant basis, we can get the metric tensor {𝑔𝑖𝑗 } and its determinant {
1 0 −𝑌 𝜏 ⎞ } ⎛ ⎟ 𝑔𝑖𝑗 = ⎜ 0 1 𝑋𝜏 ⎜ ⎟ ⎝−𝑌 𝜏 𝑋𝜏 𝑋 2 𝜏 2 + 𝑌 2 𝜏 2 + 1⎠
(12)
{ } 𝑔 = det 𝑔𝑖𝑗 = 1.
Fig. 10. Leaked power and total power of ACCC fiber.
(13)
Then the triple product is [ ] √ 𝒈1 𝒈2 𝒈3 = 𝑔 ≡ 1.
To reduce the leaked power as well as achieve enough HOM suppression, we could select the NA of the outer cladding to a lower value, such as changing NA from 0.46 to 0.22. The heat generation of the fiber might be reduced on the condition that the higher brightness pumping diode lasers are utilized.
(14)
In generic skew coordinates, this triple product varies with the coordinate value. Having a constant triple product, the difficulty in calculating is reduced. Now the contravariant basis is expressed as [20] ⎧𝒈1 = [𝒈2 × 𝒈3] = 𝒊 cos 𝜏𝑍 + 𝒋 sin 𝜏𝑍 + 𝒌𝜏𝑌 ⎪ 𝒈1 𝒈2 𝒈3 ⎪ ⎪𝒈2 = 𝒈3 × 𝒈1 = −𝒊 sin 𝜏𝑍 + 𝒋 cos 𝜏𝑍 − 𝒌𝜏𝑋 [ ] ⎨ 𝒈1 𝒈2 𝒈3 ⎪ 𝒈1 × 𝒈2 ⎪ 3 ] = 𝒌. ⎪𝒈 = [ 𝒈1 𝒈2 𝒈3 ⎩
4. Conclusion In this paper, we introduced a novel ACCC fiber structure to satisfy the requirements of Chinese manufacturers. With the FEM analysis method and helical coordinates, the mode fields in the fiber were simulated. The BPM was used to find the mode propagation condition and loss varying with different fiber parameters. Finally, we completed the fiber design by the mode dynamic competition method. Using the output brightness as the judgment criterion, we refined the ACCC fiber parameters, i.e. the helix pitch as 10 000 μm, outer diameters and the width of the side annulus as 82 μm and 6.5 μm, and the distance of two cores as 6 μm. At these parameters, this fiber structure has good conversion efficiency (about 84%) and beam quality (𝑀 2 = 1.04), alike to typical CCC fibers.
(15)
In a general coordinate, whether orthogonal or not, the gradient, divergence and curl operators can be expressed as [21] ∇𝑓 = 𝒈1
𝜕𝑓 𝜕𝑓 𝜕𝑓 + 𝒈2 + 𝒈3 𝜕𝑋 𝜕𝑌 𝜕𝑍
∇⋅𝑭 =𝑔
Acknowledgments
−1∕2
( (√ 1 ) 𝜕 𝑔𝐹 ⋅
𝜕𝑋
(16) 𝜕
+
(√ 2 ) 𝑔𝐹 𝜕𝑌
| 𝒈 𝒈2 𝒈3 || | 1 | | ∇ × 𝑭 = 𝑔 −1∕2 | 𝜕∕𝜕𝑋 𝜕∕𝜕𝑌 𝜕∕𝜕𝑍 | . | | |𝑭 ⋅ 𝒈1 𝑭 ⋅ 𝒈2 𝑭 ⋅ 𝒈3 | | |
The authors would like to thank Dr. Xiuquan Ma from School of Mechanical Science and Engineering, Huazhong University of Science & Technology. The discussion with him gave us inspiration, which is essential to the completion of this paper. This work was supported
𝜕 +
(√
𝑔𝐹 3
𝜕𝑍
)) (17)
(18)
To simplify the calculation, we need the decomposition of the vector 𝑭 in covariant basis, i.e. 𝐹 𝑖 . Form the knowledge in tensor analysis, we 303
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have 𝑭 ⋅ 𝒈𝑖 = 𝐹𝑖 = 𝑔𝑚𝑖 𝐹 𝑖 . Put this equation and (14) into (17) and (18), we have ∇𝑓 = 𝒈1
𝜕𝑓 𝜕𝑓 𝜕𝑓 + 𝒈2 + 𝒈3 𝜕𝑋 𝜕𝑌 𝜕𝑍
(19)
∇⋅𝑭 =
𝜕𝐹 1 𝜕𝐹 2 𝜕𝐹 3 + + 𝜕𝑋 𝜕𝑌 𝜕𝑍
(20)
[7] F. Jansen, F. Stutzki, A. Liem, C. Jauregu, J. Limpert, A. Tünnermann, High power Q-switched fiber laser system delivering 22mj pulse energy with excellent beam quality, in: Lasers, Sources, and Related Photonic Devices, Optical Society of America, San Diego, California, 2012. [8] E. Coscelli, R. Dauliat, F. Poli, D. Darwich, A. Cucinotta, S. Selleri, K. Schuster, A. Benoit, R. Jamier, P. Roy, F. Salin, Analysis of the modal content into large-modearea photonic crystal fibers under heat load, IEEE J. Sel. Top. Quantum Electron. 22 (2) (2016) 323–330. [9] C. Zhu, I.N. Hu, X. Ma, A. Galvanauskas, Single mode 9.1mJ and 10ns pulses from 55 μm core Yb-doped CCC fiber MOPA, in: CLEO: 2013, Optical Society of America, San Jose, California, 2013. [10] C. Liu, G. Chang, N. Litchinitser, A. Galvanauskas, D. Guertin, N. Jabobson, K. Tankala, Effectively single-mode chirally-coupled core fiber, in: Advanced SolidState Photonics, Optical Society of America, Vancouver, 2007. [11] L. Dong, X. Li, J. Peng, Leakage channel optical fibers with large effective area, 24 (8) (2007) 1689–1697. [12] X. Ma, C. Liu, G. Chang, A. Galvanauskas, Angular-momentum coupled optical waves in chirally-coupled-core fibers, Opt. Express 19 (27) (2011) 26515–26528. [13] X. Ma, Understanding and controlling angular momentum coupled optical waves in chirally-coupled-core (CCC) fibers. 2011, The University of Michigan. [14] F. Jansen, F. Stutzki, H.J. Otto, M. Baumgartl, C. Jauregui, J. Limpert, A. Tünnermann, The influence of index-depressions in core-pumped Yb-doped large pitch fibers, Opt. Express 18 (26) (2010) 26834–26842. [15] D. Chen, H. Zhang, Numerical analysis of mode competition and selection in Ybdoped multicore fiber lasers, 2014, 928508. [16] I. Kelson, A.A. Hardy, Strongly pumped fiber lasers, IEEE J. Quantum Electron. 34 (9) (1998) 1570–1577. [17] H. Zhang, D. Chen, H. Ren, M. Gong, A novel method of evaluating large mode area fiber design by brightness factor, Chin. Phys. B 24 (2) (2015) 170–175. [18] J. Seurin, R.V. Leeuwen, P. Pradhan, High-brightness pump sources using 2D VCSEL arrays, 7615 (4) (2010) 276-276. [19] C.H. Liu, G. Chang, N. Litchinitser, A. Galvanauskas, D. Guertin, N. Jabobson, K. Tankala, Effectively single-mode chirally-coupled core fiber, Adv. Solid-State Photonics (2007). [20] M.W. McCall, Electromagnetic analysis of complex media via helical coordinates, Optics & Photonics 2005, International Society for Optics and Photonics, 2005. [21] P.L. Khare, Divergence and curl in nonorthogonal curvilinear coordinates, Amer. J. Phys. 38 (7) (1970) 915.
∇×𝑭 | | 𝒈1 𝒈2 𝒈3 | | | | = | 𝜕∕𝜕𝑋 𝜕∕𝜕𝑌 𝜕∕𝜕𝑍 |. ( 2 2 )| | 1 3 2 3 1 2 3 2 2 |𝐹 − 𝐹 𝑌 𝜏 𝐹 + 𝐹 𝑋𝜏 −𝐹 𝑌 𝜏 + 𝐹 𝑋𝜏 + 𝐹 𝑋 𝜏 + 𝑌 𝜏 + 1 | | | (21) The gradient and divergence in helical coordinate is identical with Cartesian coordinate, however, the curl is quite different from original. References [1] S.T. Hendow, R. Romero, S.A. Shakir, P.T. Guerreiro, Percussion drilling of metals using bursts of nanosecond pulses, Opt. Express 19 (11) (2011) 10221–10231. [2] Y. Li, K. Guan, Z. Hu, Y. Chen, An optical fiber lateral displacement measurement method and experiments based on reflective grating panel, Sensors 16 (6) (2016) 808. [3] Y. Li, K. Guan, Z. Hu, Fiber optic displacement measurement model based on finite reflective surface, Opt. Laser Technol. 84 (2016) 32–39. [4] D.P. Shepherd, J.I. Mackenzie, R.J. Beach, T. Bhutta, Spatial dopant profiles for transverse-mode selection in multimode waveguides, J. Opt. Soc. Amer. B 19 (7) (2002) 1539–1543. [5] J.P. Koplow, D.A.V. Kliner, L. Goldberg, Single-mode operation of a coiled multimode fiber amplifier, Opt. Lett. 25 (7) (2000) 442–444. [6] F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, A. Tünnermann, 26 mJ, 130 W Q-switched fiber-laser system with near-diffraction-limited beam quality, Opt. Lett. 37 (6) (2012) 1073–1075.
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Single-mode annular chirally-coupled core fibers for fiber lasers Haitao Zhang *, He Hao, Linlu He, Mali Gong Center for Photonics and Electronics, State Key Laboratory of Precision Measurement and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Keywords: Annular chirally-coupled core fiber Fiber design Fiber laser Higher-order mode suppression
a b s t r a c t Chirally-coupled core (CCC) fiber can transmit single fundamental mode and effectively suppresses higher-order mode (HOM) propagation, thus improve the beam quality. However, the manufacture of CCC fiber is complicated due to its small side core. To decrease the manufacture difficulty in China, a novel fiber structure is presented, defined as annular chirally-coupled core (ACCC) fiber, replacing the small side core by a larger side annulus. In this paper, we designed the fiber parameters of this new structure, and demonstrated that the new structure has a similar property of single mode with traditional CCC fiber. Helical coordinate system was introduced into the finite element method (FEM) to analyze the mode field in the fiber, and the beam propagation method (BPM) was employed to analyze the influence of the fiber parameters on the mode loss. Based on the result above, the fiber structure was optimized for efficient single-mode transmission, in which the core diameter is 35 μm with beam quality M2 value of 1.04 and an optical to optical conversion efficiency of 84%. In this fiber, fundamental mode propagates in an acceptable loss, while the HOMs decay rapidly. © 2017 Elsevier B.V. All rights reserved.
1. Introduction High-power fiber lasers with diffraction-limited output beams are widely applied in many areas such as industry, medicine, telecommunications, and military [1–3]. Scaling the fibers to larger effective modefield-diameter (MFD) ones helps decrease optical density, suppress the nonlinear effects such as stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS), and as a result, increase the pulse energy and peak power of fiber lasers. For step-index fibers, prominent techniques for effective single-mode operations include matched excitation by particularly ion-doping [4], and higher-order mode (HOM) discrimination by bending [5]. However, these methods usually limit the core size less than 30 μm for diffraction-limited beam quality output. For instance, the beam quality factor M2 -value of the normal stepindex fibers with core size of 35 μm will increase to 2.5 since too densely packed mode to discriminate the fundamental mode and its nearest neighbors. Utilizing microstructures, such as photonic crystal fibers (PCFs) [6] and large pitch fibers (LPFs) [7], very large mode area (VLMA, core diameter beyond 50 μm) fibers are developed for high energy lasers. However, microstructure VLMA fibers suffer monolithically integration difficulty (e.g. micro-structured fiber rods), as well as thermal-induced mode instabilities because the mode constraint ability of discontinuous boundary conditions of the core is much weaker than
the continuous boundary ones [8]. The chirally-coupled core (CCC) fiber with a continuous core boundary combines robust single-mode laser performance in large cores (up to 55 μm demonstrated) [9] and stable mode even for high power laser output, while retaining the handling and packaging benefits associated with single mode fibers. The CCC fiber seems to be one of the most promising approaches for large mode and stable high power laser applications. However, except for the usage rights protections by CCC fiber patents (US7424193), the CCC fibers need eccentric preform for its helix side core. Meanwhile, many Chinese fiber manufacturers have the fiber processing technique with concentric preform, and they prefer the fiber structure based on the concentric preform to avoid the change of the fiber processing. Thus, a novel structure is designed and defined as annular chirally-coupled core (ACCC) fiber. The ACCC structure has an 82 μm-diameter multimode side annulus instead of the 16-μm side core in the common CCC fiber so that ACCC fiber can be drawn from a concentric side core preform instead of eccentric one of CCC. The design principle of CCC fiber is to make the HOMs and the fundamental mode (FM) meet and do not meet the quasi-phasematching (QPM) condition, respectively. This principle greatly improves the discrimination of the HOMs and FM losses [10]. Meanwhile, this principle also requires a small side core with very few modes. The smaller side core could decrease the number of the side modes and avoid
* Corresponding author.
E-mail address: [email protected] (H. Zhang). https://doi.org/10.1016/j.optcom.2017.10.003 Received 1 April 2017; Received in revised form 29 September 2017; Accepted 2 October 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
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Optics Communications 410 (2018) 297–304
the FM matching the side modes. On the other side, a smaller margin between the losses of each mode is also sufficient to obtain the single mode preservation. For instance, in the design of the leakage channel fibers, the principles are 1 dB/m loss of HOMs and 0.1 dB/m for the FM [11], which are sufficient to suppress the propagation of HOMs and ensure negligible fundamental mode transmission loss. Furthermore, due to the mode competition, the margin between the losses of each mode could be even smaller. For the ACCC fiber, the side annulus can support many modes, and each mode of the side annulus can couple with modes in the center. Instead of calculating the QPM condition of all the modes, we present a novel design concept and procedure especially for the gain fiber operating in the fiber laser cavity. First, we used the beam propagation method (BPM) to calculate the difference of each mode’s loss directly. Then, we optimized the ACCC structure parameter values by investigating the laser dynamic mechanism in active ACCC fibers. Using the output beam brightness as a judgment criterion, we could describe the performance of the ACCC fiber laser and ultimately optimized the structure definitely, considering both efficiency and beam quality. In this paper, we firstly presented ACCC structure and explained why this novel structure is more suitable for Chinese manufacturing techniques. We used the finite element method (FEM) to simulate the mode field distributions in ACCC fibers and the BPM to calculate the losses. Based on the simulation results of the fiber mode discrimination for fiber passive propagation process, the rough range of ACCC structure parameter could be preliminarily determined. Then, we calculated the mode competition of the ACCC fiber lasers by exploring the laser rate equations and ultimately optimized the structure based on the brightness criterion. And it was demonstrated this new ACCC structure has the similar single-mode characteristics to common CCC fiber.
Fig. 1. (a) Structure of the main preform for CCC fiber. Numbers in brackets are the actual size of the fiber, and the dash line shows the drilling position for the side core preform. Hatchings show the doped cores in the fiber by MCVD. (b) Concentric side core preform. (c) Eccentric side core preform.
Fig. 2. ACCC fiber structure.
diameter of the center core 𝐷c is 35 μm with NA of 0.06, similar with the common CCC fiber. On the other hand, the ACCC fiber structure has a larger helical side annulus, greatly reducing the difficulty of manufacturing the fiber preforms. In ACCC fiber, the outer diameter of the side annulus 𝐷o could be up to 82 μm (NA0.1), therefore the diameter of side preform is about 12 mm, which is thick enough (much greater than 6 mm limitation) to satisfy the requirements of Chinese manufacturers. The helix pitch 𝛬 of the side annulus, the width of the annulus w, and the inner distance between the two doped cores d affect the fiber characteristics and will be optimized in the following sections. Because of the helix structure of CCC fiber, the cross section is associated with its 𝑧-coordinate value. Herein a helical coordinate system {𝑋, 𝑌 , 𝑍} is introduced for the helix structure [12]
2. Annular chirally-coupled core fibers As described below, there are still technical difficulties for manufacturing CCC fibers in China. The doped fiber core, i.e. high refractive index zone herein, is made by the modified chemical vapor deposition (MCVD) method, which involves the deposition inside a substrate tube for the fiber preform. First, limited by the MCVD technology, the diameter of the deposition zone in the preform is difficult to be larger than 5 mm. Second, the preform is usually very long so that the diameter of the preform should not be thinner than 6 mm, or it would be too fragile for further processing. These two restrictions determine the maximum doped zone and the minimum outside diameter dimension of the preforms. To fabricate the CCC fiber, we need to prepare two fiber preforms, a main preform for signal propagation and a side core preform for HOMs loss dissipation. The main preform needs to be drilled a proper hole to insert the side core preform. If we dope the center signal core to a maximum diameter of 5 mm by MCVD, we can design the whole preform structure as shown in Fig. 1(a), by scaling up the corresponding fiber dimensions in brackets. In the fiber, the 33-μm center core has a numerical aperture (NA) of 0.06, and the side core has a diameter of 16 μm and NA of 0.1. The distance between the two cores, measured from the edge of the center core to the middle of the side core, is 12 μm and scaled up to 1.8 mm for preform structure. If we used the concentric side core preform like Fig. 1(b), the diameter of the side core preform would be less than 3.6 mm, less than 6-mm preform limit and too thin for a long rod. One of the solutions is grinding the side core preform eccentrically, and making the doped core area close to the boundary, as in Fig. 1(c), which can enlarge the possible side core preform greater than 6 mm. This method, called fire-polishing, is applied by the fiber fabrication industry, however still needs time for Chinese manufacturers to realize. If we could magnify the size of the side core and keep the property of the fiber at the same time, we could use the current fiber producing process and reduce the cost. To satisfy the manufacturing requirements in China, a novel fiber structure denoted as ACCC fiber is designed, as shown in Fig. 2. The
⎧𝑋 = 𝑥 ⋅ cos 𝜏𝑧 + 𝑦 ⋅ sin 𝜏𝑧 ⎪ ⎨𝑌 = −𝑥 ⋅ sin 𝜏𝑧 + 𝑦 ⋅ cos 𝜏𝑧 ⎪𝑍 = 𝑧 ⎩
(1)
where 𝜏 = 2𝜋∕𝛬, {𝑥, 𝑦, 𝑧} is the Cartesian coordinate system, and zero point of the 𝑧 axis is set at the position where x and X axes coincide. In the helical coordinate system, the shape of the fiber cross section decouples with Z direction, and the three-dimensional finite element problem degenerates into two-dimension, which decreases the calculation time. However, this coordinate system is not orthogonal, so the Helmholtz equation used in the FEM in the system needs to be further discussed. This equation is given by ∇ × (∇ × 𝑬) − 𝑘20 𝜺h 𝑬 = 0
(2)
where the 𝑘0 is the wave number in a vacuum, and the 𝜺h is the relative permittivity tensor in the helical coordinates, which is constructed by applying Jacobian matrix 𝑱 of the helical coordinate with respect to the Cartesian system 𝜺h (𝑋, 𝑌 , 𝑍) = 𝑱 ⋅ 𝜺c (𝑥, 𝑦) ⋅ 𝑱 T 0 0⎞ ⎛1 0 𝜏𝑌 ⎞ ⎛𝜀1 (𝑋, 𝑌 ) 𝜀6 (𝑋, 𝑌 ) 𝜀5 (𝑋, 𝑌 )⎞ ⎛ 1 = ⎜0 1 −𝜏𝑋 ⎟ ⎜𝜀6 (𝑋, 𝑌 ) 𝜀2 (𝑋, 𝑌 ) 𝜀4 (𝑋, 𝑌 )⎟ ⎜ 0 1 0⎟ . ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝0 0 1 ⎠ ⎝𝜀5 (𝑋, 𝑌 ) 𝜀4 (𝑋, 𝑌 ) 𝜀3 (𝑋, 𝑌 )⎠ ⎝𝜏𝑌 −𝜏𝑋 1⎠ 298
(3)
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For isotropic medium, 𝜺c = diag (𝜀 (𝑋, 𝑌 ) , 𝜀 (𝑋, 𝑌 ) , 𝜀 (𝑋, 𝑌 )), then the helical permittivity tensor can be simplified as ⎛𝑌 2 𝜏 2 + 1 −𝑋𝑌 𝜏 2 𝑌 𝜏 ⎞ 𝜺h (𝑋, 𝑌 , 𝑍) = 𝜀 (𝑋, 𝑌 ) ⋅ ⎜ −𝑋𝑌 𝜏 2 𝑋 2 𝜏 2 + 1 −𝑋𝜏 ⎟ . ⎜ ⎟ −𝑋𝜏 1 ⎠ ⎝ 𝑌𝜏
(4)
In addition, the curl operator of the helical coordinate also differs from the Cartesian system. For arbitrary vector 𝑭 , the curl is expressed as ∇×𝑭 | | 𝒈1 𝒈2 𝒈3 | | | | = | 𝜕∕𝜕𝑋 𝜕∕𝜕𝑌 𝜕∕𝜕𝑍 | ( ) | 1 | |𝐹 − 𝐹 3 𝑌 𝜏 𝐹 2 + 𝐹 3 𝑋𝜏 −𝐹 1 𝑌 𝜏 + 𝐹 2 𝑋𝜏 + 𝐹 3 𝑋 2 𝜏 2 + 𝑌 2 𝜏 2 + 1 | | | {
(5)
}
where 𝒈1 , 𝒈2 , 𝒈3 are the covariant bases of this system, and 𝐹 𝑖 are the decompositions of the vector 𝑭 in these covariant bases. The derivation of Eq. (5) is in the Appendix. Substitute this permittivity Eq. (4) and the curl expression Eq. (5) into the Helmholtz equation (2), then the mode field distribution of the fiber can be obtained by FEM method.
Fig. 3. Range of mode loss varies with the distance d. Inset: mode field of the center LP01 (left) and LP11 (right) mode.
3. Numerical simulation results and discussions
results in our future work. Herein, for theoretical design and evaluation for the first step, the material of the fiber is regarded as isotropic despite of the birefringence. To optimize the ACCC structure, we investigated the mode loss varying with the parameters of the helix pitch 𝛬, the width of the annulus w, and the inner distance between the two doped cores d. First an investigation was performed for the distance d. The wavelength 𝜆 used in the study is 1064 nm. Setting the pitch 𝛬 varying from 2500 μm to 10 000 μm and width w up to 40 μm, the relationship between mode loss range and the distance d was obtained as in Fig. 3. The central LP01 and LP11 mode (shown in the inset of Fig. 3) share the same varying trend. The light fields in the central core and the side annulus interact with each other by evanescent-wave coupling. When the distance d increases, the overlap factor between the core and annulus evanescent fields drops rapidly, thus the loss of all modes falls simultaneously. This trend could be used to control the total loss of the fiber coarsely for the first step. To choose a proper distance, we need to place the two cores close enough to get effective coupling. Meanwhile, to get a usable gain fiber for laser oscillation and amplification, the FM should maintain a low loss, usually less than 1 dB/m [14]. As shown in Fig. 3, at distance of 5.5, 6 and 6.5 μm, the minimum FM losses are respectively 1.01, 0.49 and 0.24 dB/m, and the minimum LP11 losses are 1.23, 0.84, and 0.68 dB/m. The shorter distance 𝑑, the higher loss is. The distance 𝑑 should be larger than 5.5 μm with less than 1dB/m FM loss for laser applications. As an initial selection, the distance of the two cores was chosen as 6 μm, at which the minimum FM mode loss begins to get lower than 0.5 dB/m and has a relative larger initial loss difference (0.35 dB/m) between FM and LP11 than the case in distance 5.5 μm (0.22 dB/m). The selection of distance as 6 μm is also leaving the room of 0.5 μm for manufacturing errors and enough margins for 𝛬 and w optimization in the next step. The mode losses varying with the helix pitch 𝛬 and the annular width w are shown in Fig. 4. The HOM (LP11 in Fig. 4) always has a higher loss ratio than in the center, and the loss difference ranges from 0.2 to 18 dB/m. The variation of the HOM and FM losses differs with pitch changing, as shown in Fig. 4(a). For example, when w is 3.2 μm and 𝛬 increases from 7000 to 7750 μm, the LP01 loss rises from 7.1 to 7.8 dB/m, while the LP11 loss falls from 17 to 15 dB/m. That is because the helix side rotating along the fiber introduces an additional phase difference between the central mode and side mode, and when helix pitch changes, the coupling from the central HOM and FM to side annulus varies respectively. The loss difference of each mode makes it possible to optimize the fiber to get lower FM loss and higher HOM loss.
As we know, the step-index fiber with the same core diameters has an 𝑀 2 value of about 2.5. The design purpose of the fiber is to reduce the 𝑀 2 to less than 1.1 as well as keep low FM loss to get a high efficient laser oscillation, amplification and propagation. At the first step, we need to increase the HOM loss while control the FM loss to a lower level to ensure mode discrimination. The effects of different fiber parameters were discussed on the loss of each mode by using the BPM. Then, based on mode discrimination in laser passive propagation progress, the structure parameter ranges were preliminary designed. Second, the design results from the first step were applied to the fiber laser simulation. Using the mode competition method, we can investigate optical to optical conversion efficiency of the fiber as well as beam quality for further structure optimization. 3.1. Mode discrimination in propagation for ACCC fibers Herein we used FEM to calculate the mode field distribution under different ACCC fiber structures. By utilizing the calculated mode field as the input field of BPM, we got the light propagation loss of each mode. The ideal ACCC structure consists of isotropic medium. However, the practical manufacturing process might induce the strain-induced birefringence, which is related to the cooling and rotating process during the fiber fabrication. Ma et al. utilized a simplified birefringence model for CCC fibers, which assumes that the birefringence region is a single ellipse located between the two core, and a moderate estimate of birefringence is given. Under this assumption, they explained the physical origin of the multitude of the quasi-phase-matching (QPM) resonances [13]. In the CCC fibers, the small side core supports a few modes, which makes it easy to correspondence each pair of side and central modes and benefits to derive the analytical expression of the QPM based on sketchy birefringence estimation [12,13]. While for ACCC structure with a larger annular side core supporting lots of modes, QPM analytical method is not proper any more for it is almost impossible to match propagation constants of the core and numerous side annular modes. The numerical method of the QPM for ACCC with so many annular modes is also a heroic challenge. And until now, the accurate birefringence value or model of such chirally fiber has not been achieved even from the experiment or the theory. The birefringence model for ACCC fiber which might be more complicated and related to the drawing process could be another important assignment. The influence of the birefringence needs further investigation along with the experimental 299
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Fig. 4. (a) Mode losses of LP01 mode and LP11 mode vary with the helix pitch 𝛬. The annulus widths w are shown in the legend. The dash lines mark the 1 dB/m loss of each mode. (b) Mode losses vary with the annular width w of the side annulus. The pitches 𝛬 are shown in the legend.
For the annulus width of about 5.9 μm, the mode loss decreases with the helix pitch 𝛬 rises. When 𝛬 exceeds 7750 μm, the FM loss falls below 1 dB/m, while the LP11 mode loss is still over 2 dB/m, which could be further optimized in Section 3.2. On the other hand, the mode losses oscillate with the annular width w. For clarity, Fig. 4(b) was drawn for the relationship between the loss and the width w. When w increases from 3.2 to 8.6, the mode losses first decrease from the peak (7.85 dB/m for LP01) to the minimum (0.23 dB/m), then rise again to the maximum (8 dB/m). When w continues to rise, the oscillation mitigates. The reason of the oscillation is that the annular width w influences the side annulus mode distribution, proved by the modified FEM using Eq. (4), shown in Fig. 5. The higher w (from 3.2 to 5.9 μm) leads to a looser constraint of the side annulus, which makes less evanescent field of the side mode and less overlap factor between the central and the side mode, and the center mode losses decrease. Meanwhile, when the width increases to a threshold (8.6 μm in Fig. 5), the side HOM of the radial direction appears, which introduces a larger evanescent field area, and makes the center mode losses increase. Fig. 6 shows the central LP11 mode distributions with different widths w. It is obvious that the 5.9 μm case has the less mode coupling than 3.2 and 8.6 μm. This characteristic could also be used to adjust the total loss of the fiber, or compensate the inaccuracy of the distance of the two cores in fabrication. Optimizing these parameters, we could get the proper fiber structure for single FM transmission. The LP01 mode loss lower than 1dB/m is required in our design. According to the data in Fig. 4, valid parameters are annulus width w as about 5.9 μm and helix pitch 𝛬 over 7750 μm, so we chose a range of w from 5 to 6.5 μm, and 𝛬 from 7600 to 12 400 μm for further optimization.
The dynamic mode analysis of the ACCC laser utilizes the typical linear cavity as described schematically in Ref. [15]. We have calculated the loss of each mode (LP01, LP11, LP21 and LP02 in this case) by BPM in Section 3.1. By substituting the mode loss results into the well-known space-dependent and time-independent steady-state rate equations, we can get the signal power of each mode and the pump power varying along the fiber length [16]. [𝑃𝑝+ (𝑧)+𝑃𝑝− (𝑧)]𝜎𝑎𝑝 𝜑𝑝 (𝑥,𝑦)
𝑁2 (𝑥, 𝑦, 𝑧) = 𝑁1 (𝑥, 𝑦, 𝑧)
ℎ𝜈𝑝 [𝑃𝑝+ (𝑧)+𝑃𝑝− (𝑧)]𝜎𝑒𝑝 𝜑𝑝 (𝑥,𝑦) ℎ𝜈𝑝
±
𝑑𝑃𝑝± (𝑧) 𝑑𝑧
{ =
∬𝐴
+
+ 1 𝜏
∑𝑀
[𝑃𝑠𝑖+ (𝑧)+𝑃𝑠𝑖− (𝑧)]𝜎𝑎𝑠 𝜑𝑠𝑖𝑗 (𝑥,𝑦)
𝑖=1
+
∑𝑀 𝑖=1
ℎ𝜈𝑠 [𝑃𝑠𝑖+ (𝑧)+𝑃𝑠𝑖− (𝑧)]𝜎𝑒𝑠 𝜑𝑠𝑖 (𝑥,𝑦)
[ ] 𝜎𝑒𝑝 𝑁2 (𝑥, 𝑦, 𝑧) − 𝜎𝑎𝑝 𝑁1 (𝑥, 𝑦, 𝑧) 𝜑𝑝 (𝑥, 𝑦)𝑑𝑥𝑑𝑦
}
× 𝑃𝑝± (𝑧) − 𝛼𝑝 𝑃𝑝± (𝑧)
±
𝑑𝑃𝑠𝑖± (𝑧) 𝑑𝑧
{ =
∬𝐴𝒋
(6)
ℎ𝜈𝑠
(7)
} [ ] 𝜎𝑒𝑠 𝑁2 (𝑥, 𝑦, 𝑧) − 𝜎𝑎𝑠 𝑁1 (𝑥, 𝑦, 𝑧) 𝜑𝑠𝑖 (𝑥, 𝑦)𝑑𝑥𝑑𝑦
× 𝑃𝑠𝑖± (𝑧) − 𝛼𝑠𝑖 𝑃𝑠𝑖± (𝑧) −
𝑀 ∑
(8) ± 𝑑𝑖𝑗 [𝑝± 𝑠𝑖 (𝑧) − 𝑝𝑠𝑖 (𝑧)].
𝑗=1
In these equations, 𝜈s and 𝜈p are the laser and pump frequencies. 𝑁1 and 𝑁2 are the population densities of the lower and upper lasing levels. 𝑃𝑝± (𝑧) and 𝑃𝑠𝑖± (𝑧) are the pump powers and ith mode powers in the forward and backward directions, respectively. 𝜑p and 𝜑si are the normalized intensity distributions of the pump and ith mode, and 𝛼si is the loss factors of the ith mode. Other parameters used for mode competition are listed in Table 1. The boundary conditions of the fiber is given by𝑃𝑠𝑖− (0) = 𝑅1 𝑃𝑠𝑖+ (0) and 𝑃𝑠𝑖+ (𝐿) = 𝑅2 𝑃𝑠𝑖− (𝐿), and we assign the reflection rate 𝑅1 = 0.99 and 𝑅2 = 0.04. With the boundary conditions and an assumed small initial signal, we can calculate the output power of each mode by iteration of the rate equations (6)–(8). According to the power of each mode and the corresponding effective mode field diameter (MFD), beam quality factor (𝑀 2 ) can be calculated by the method described in Ref. [17]. In fiber laser systems, the brightness of a laser is used to indicate its ability to produce the highest power in the smallest spot and the smallest divergence. For a circular,
3.2. Mode competition in ACCC fiber lasers In addition to the eigen mode field distribution and propagation characteristics, we are more concerned about the mode competition, the high power dynamic mechanism and the laser output performance in the active ACCC fibers. We could further refine the proper ACCC structure based on the primary parameter ranges optimized in Section 3.1. 300
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Fig. 5. (a) Side mode simulated by FEM. (b) Enlargement of (a), with an annulus width of 8.6 μm. (c)Annulus width of 5.9 μm, HOM is restrained. (d) Annulus width of 3.2 μm, area of evanescent field increases. The outer diameter and position of the side annulus stay the same in these cases. (e) Cross-section graph of (b), (c) and (d). Dash line is the outer border of the side annulus.
Fig. 6. Center LP11 mode, annulus width w of (a) 8.6 μm, (b) 5.9 μm and (c) 3.2μm.
Table 1 Parameters in mode competition. Parameter
Value
Parameter
Value
Fiber length (L) Doping concentration (𝑁0 ) Pump power (𝑃𝑝 ) Front reflectivity of the cavity (𝑅1 ) End reflectivity of the cavity (𝑅2 ) Pump wavelength (𝜆p ) Signal wavelength (𝜆s )
2m 4.98 × 1025 m−3 1500 W 0.99 0.04 976 nm 1064 nm
Planck constant (h) Spontaneous lifetime of the upper lasing level (𝜏) Pump loss factor (𝛼𝑝 ) Pump absorption cross section (𝜎ap ) Pump emission cross section (𝜎ep ) Signal absorption cross section (𝜎as ) Signal emission cross section (𝜎es )
6.626 × 10−34 1.0 ms 2.0 × 10−3 m−1 2.34 × 10−24 m2 2.34 × 10−24 m2 6.58 × 10−27 m2 3.11 × 10−25 m2
low-diverging laser source, the brightness B could be expressed as𝐵 = 𝑃out ∕(𝑀 2 𝜆)2 [18], where 𝑃out is the output power, and 𝜆 is the laser wavelength. And we use the brightness to describe the fiber performance and optimize the structure. Fig. 7(a) and (b) show the 𝑀 2 and output power varying with helix pitch 𝛬 and annulus width w, respectively. The 𝑀 2 ranges between 1.35 and 1.04, and reaches the minimum when w is 7.5 μm or 8 μm. However, at these points, the output power is lower than 800 W, which means a high loss and inefficiency. With the 𝛬 increasing, the output power increases generally, while the beam quality deteriorates at the same time. Considering the high efficiency and good beam quality simultaneously, we use the brightness B to judge the parameters, as in Fig. 7(c). When the helix pitch 𝛬 is 10 000 μm, outer diameters 𝐷o is 82 μm, the width of the side annulus w is 6.5 μm, and the distance of two cores d is 6 μm. The loss of LP01 mode is about 0.24 dB/m, while the losses of LP11, LP21 and LP02 are 1.78 dB/m, 13.54 dB/m and 50.60 dB/m. The loss within a unit length is determined by both the loss of the modes in the side core and the coupling strength from
central modes to these side modes. The difference of losses is sufficient to restrain the HOMs in the Yb-doped fiber and keep the FM amplified. At this condition, the fiber structure has the highest output brightness B of 9.47×1014 W ⋅ m−2 , with 𝑀 2 of 1.04 and output power of 1163 W. That proves the ACCC fiber structure has a good output characteristic. The power variation along the fiber is shown in Fig. 8. As in Fig. 8(a), the HOMs can hardly be amplified in this fiber. The highest power of LP11 is lower than 0.07 W, and the LP21 and LP02 mode are even lower. Therefore, the output beam consists of only FM (over 99.99%). Fig. 8(b) shows the pump power and signal power evolutions along the fiber. With the 1500 W pump, the total output is 1262 W, and the conversion efficiency is about 84.1%. 3.3. HOM suppression and leaked power By suppressing HOM, the ACCC fiber enables high power laser output with an effective large area single-mode. However, the mode coupling of HOM to lossy modes is accompanied by a small leaked power into the 301
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Fig. 7. Relationship between helix pitch 𝛬 and (a) 𝑀 2 , (b) output power and (c) brightness of fiber sat different annulus width w.
Fig. 8. Power evolutions along the fiber. (a) Power of each linear mode; (b) Pump power and signal power.
cladding. The leaked power would potentially deteriorate the possible follow-up cladding light stripper and the fused fiber combiner or splitter. The excess heat generated from the leaky power would also limit the fiber laser power scalability. The leaked power is determined by the mode loss of the fiber. Utilizing the data of mode loss and parameters of ACCC in Sec 3.2 and CCC in Ref. [19], we subtract the amplified mode power along the fiber length in ACCC from those in CCC fibers to compare the ability of the anti-loss and heat resistance between these two CCC-like specialty fibers. The seed power of the LP01, LP11, LP21 and LP02 is assumed to be 50 W, 5 W, 0.5 W and 0.1 W, which is ordinary and reasonable for fiber amplifier in reality. As shown in Fig. 9, the output power of LP01 and LP11 mode in ACCC fiber is a little lower than those in CCC fiber. The LP01 mode power difference is 25.8 W, the mode in ACCC fiber is 2.4% lower than CCC fiber. And the difference of LP11 mode is 29.6 W, 31% lower. The other two modes’ power difference are very small, nearly to zero. The total power of ACCC fiber amplifier is 1150.0 W, 4.6% lower
than the 1205.3 W of CCC fiber’s total power. The power margin of ACCC fiber is almost the same as CCC fiber as well as the leaked power burden. The results in Fig. 9 indicate the LP11 margin is dominant. We could simplify the mode competition and power evolution only for LP01 and LP11 modes. Fixing the loss value of LP01 as 0.24 dB/m, we examine the dependence of the leaked power on the loss of the LP11 mode which representing the capacity of LP11 mode suppression. As shown in Fig. 10, the leaked power of LP01 mode is almost the constant, staying at about 106 W because of the fixed LP01 mode loss. The leaked power of LP11 rapidly goes down and converges with LP11 loss beyond 1.5 dB/m. The inflexion point has the loss coefficient 4.5 dB/m and total power 1183 W. The output power of the amplifier converges to 1230 W with loss beyond 20 dB/m. Our designed ACCC structure has a loss of 1.78 dB/m, corresponding with a leaked LP11 power of 4 W and 1200 W is a reasonable trade-off selection. 302
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by the National Natural Science Foundation of China (grant number 61475081). Appendix To derive the gradient, divergence and curl operator of the helical } { coordinate, first we need to get the covariant basis 𝒈𝟏 ,𝒈𝟐 ,𝒈𝟑 and { 1 2 3} contravariant basis 𝒈 ,𝒈 ,𝒈 of this coordinate system. To express the {𝑥, 𝑦, 𝑧} in the Cartesian coordinate system by {𝑋, 𝑌 , 𝑍}, we have ⎧𝑥 = 𝑋 ⋅ cos 𝜏𝑍 − 𝑌 ⋅ sin 𝜏𝑍, ⎪ ⎨𝑦 = 𝑋 ⋅ sin 𝜏𝑍 + 𝑌 ⋅ cos 𝜏𝑍, ⎪𝑧 = 𝑍. ⎩
(9)
Then the position vector 𝑹 of any point (𝑥, 𝑦, 𝑧) is expressed as 𝑹 = 𝑥𝒊 + 𝑦𝒋 + 𝑧𝒌 = (𝑋 ⋅ cos 𝜏𝑍 − 𝑌 ⋅ sin 𝜏𝑍) 𝒊 (10)
+ (𝑋 ⋅ sin 𝜏𝑍 + 𝑌 ⋅ cos 𝜏𝑍) 𝒋 + 𝑍𝒌
Fig. 9. Mode power margin along the fiber length between CCC and ACCC fiber amplifiers.
where{𝒊, 𝒋, 𝒌} is a set of standard basis in the Cartesian coordinate system. Now we have the covariant basis of the helical coordinate. ⎧𝒈 = 𝜕𝑹 = 𝒊 cos 𝜏𝑍 + 𝒋 sin 𝜏𝑍, ⎪ 1 𝜕𝑋 ⎪ 𝜕𝑹 (11) = −𝒊 sin 𝜏𝑍 + 𝒋 cos 𝜏𝑍, ⎨𝒈 2 = 𝜕𝑌 ⎪ 𝜕𝑹 ⎪𝒈 = ⎩ 3 𝜕𝑍 = −𝒊𝑦𝜏 + 𝒋𝑥𝜏 + 𝒌. From the set of the covariant basis, we can get the metric tensor {𝑔𝑖𝑗 } and its determinant {
1 0 −𝑌 𝜏 ⎞ } ⎛ ⎟ 𝑔𝑖𝑗 = ⎜ 0 1 𝑋𝜏 ⎜ ⎟ ⎝−𝑌 𝜏 𝑋𝜏 𝑋 2 𝜏 2 + 𝑌 2 𝜏 2 + 1⎠
(12)
{ } 𝑔 = det 𝑔𝑖𝑗 = 1.
Fig. 10. Leaked power and total power of ACCC fiber.
(13)
Then the triple product is [ ] √ 𝒈1 𝒈2 𝒈3 = 𝑔 ≡ 1.
To reduce the leaked power as well as achieve enough HOM suppression, we could select the NA of the outer cladding to a lower value, such as changing NA from 0.46 to 0.22. The heat generation of the fiber might be reduced on the condition that the higher brightness pumping diode lasers are utilized.
(14)
In generic skew coordinates, this triple product varies with the coordinate value. Having a constant triple product, the difficulty in calculating is reduced. Now the contravariant basis is expressed as [20] ⎧𝒈1 = [𝒈2 × 𝒈3] = 𝒊 cos 𝜏𝑍 + 𝒋 sin 𝜏𝑍 + 𝒌𝜏𝑌 ⎪ 𝒈1 𝒈2 𝒈3 ⎪ ⎪𝒈2 = 𝒈3 × 𝒈1 = −𝒊 sin 𝜏𝑍 + 𝒋 cos 𝜏𝑍 − 𝒌𝜏𝑋 [ ] ⎨ 𝒈1 𝒈2 𝒈3 ⎪ 𝒈1 × 𝒈2 ⎪ 3 ] = 𝒌. ⎪𝒈 = [ 𝒈1 𝒈2 𝒈3 ⎩
4. Conclusion In this paper, we introduced a novel ACCC fiber structure to satisfy the requirements of Chinese manufacturers. With the FEM analysis method and helical coordinates, the mode fields in the fiber were simulated. The BPM was used to find the mode propagation condition and loss varying with different fiber parameters. Finally, we completed the fiber design by the mode dynamic competition method. Using the output brightness as the judgment criterion, we refined the ACCC fiber parameters, i.e. the helix pitch as 10 000 μm, outer diameters and the width of the side annulus as 82 μm and 6.5 μm, and the distance of two cores as 6 μm. At these parameters, this fiber structure has good conversion efficiency (about 84%) and beam quality (𝑀 2 = 1.04), alike to typical CCC fibers.
(15)
In a general coordinate, whether orthogonal or not, the gradient, divergence and curl operators can be expressed as [21] ∇𝑓 = 𝒈1
𝜕𝑓 𝜕𝑓 𝜕𝑓 + 𝒈2 + 𝒈3 𝜕𝑋 𝜕𝑌 𝜕𝑍
∇⋅𝑭 =𝑔
Acknowledgments
−1∕2
( (√ 1 ) 𝜕 𝑔𝐹 ⋅
𝜕𝑋
(16) 𝜕
+
(√ 2 ) 𝑔𝐹 𝜕𝑌
| 𝒈 𝒈2 𝒈3 || | 1 | | ∇ × 𝑭 = 𝑔 −1∕2 | 𝜕∕𝜕𝑋 𝜕∕𝜕𝑌 𝜕∕𝜕𝑍 | . | | |𝑭 ⋅ 𝒈1 𝑭 ⋅ 𝒈2 𝑭 ⋅ 𝒈3 | | |
The authors would like to thank Dr. Xiuquan Ma from School of Mechanical Science and Engineering, Huazhong University of Science & Technology. The discussion with him gave us inspiration, which is essential to the completion of this paper. This work was supported
𝜕 +
(√
𝑔𝐹 3
𝜕𝑍
)) (17)
(18)
To simplify the calculation, we need the decomposition of the vector 𝑭 in covariant basis, i.e. 𝐹 𝑖 . Form the knowledge in tensor analysis, we 303
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have 𝑭 ⋅ 𝒈𝑖 = 𝐹𝑖 = 𝑔𝑚𝑖 𝐹 𝑖 . Put this equation and (14) into (17) and (18), we have ∇𝑓 = 𝒈1
𝜕𝑓 𝜕𝑓 𝜕𝑓 + 𝒈2 + 𝒈3 𝜕𝑋 𝜕𝑌 𝜕𝑍
(19)
∇⋅𝑭 =
𝜕𝐹 1 𝜕𝐹 2 𝜕𝐹 3 + + 𝜕𝑋 𝜕𝑌 𝜕𝑍
(20)
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∇×𝑭 | | 𝒈1 𝒈2 𝒈3 | | | | = | 𝜕∕𝜕𝑋 𝜕∕𝜕𝑌 𝜕∕𝜕𝑍 |. ( 2 2 )| | 1 3 2 3 1 2 3 2 2 |𝐹 − 𝐹 𝑌 𝜏 𝐹 + 𝐹 𝑋𝜏 −𝐹 𝑌 𝜏 + 𝐹 𝑋𝜏 + 𝐹 𝑋 𝜏 + 𝑌 𝜏 + 1 | | | (21) The gradient and divergence in helical coordinate is identical with Cartesian coordinate, however, the curl is quite different from original. References [1] S.T. Hendow, R. Romero, S.A. Shakir, P.T. Guerreiro, Percussion drilling of metals using bursts of nanosecond pulses, Opt. Express 19 (11) (2011) 10221–10231. [2] Y. Li, K. Guan, Z. Hu, Y. Chen, An optical fiber lateral displacement measurement method and experiments based on reflective grating panel, Sensors 16 (6) (2016) 808. [3] Y. Li, K. Guan, Z. Hu, Fiber optic displacement measurement model based on finite reflective surface, Opt. Laser Technol. 84 (2016) 32–39. [4] D.P. Shepherd, J.I. Mackenzie, R.J. Beach, T. Bhutta, Spatial dopant profiles for transverse-mode selection in multimode waveguides, J. Opt. Soc. Amer. B 19 (7) (2002) 1539–1543. [5] J.P. Koplow, D.A.V. Kliner, L. Goldberg, Single-mode operation of a coiled multimode fiber amplifier, Opt. Lett. 25 (7) (2000) 442–444. [6] F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, A. Tünnermann, 26 mJ, 130 W Q-switched fiber-laser system with near-diffraction-limited beam quality, Opt. Lett. 37 (6) (2012) 1073–1075.
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