Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2024; 8(1): 45-55
Published online February 25, 2024 https://doi.org/10.3807/COPP.2024.8.1.45
Copyright © Optical Society of Korea.
Jing Gao^{1}, Linbo Zhang^{1}, Dongdong Jiao^{1}, Guanjun Xu^{1,2} , Xue Deng^{1}, Qi Zang^{1}, Honglei Yang^{3}, Ruifang Dong^{1,2} , Tao Liu^{1,2} , Shougang Zhang^{1,2}
Corresponding author: ^{*}xuguanjun@ntsc.ac.cn, ORCID 0000-0003-1108-0459
^{**}taoliu@ntsc.ac.cn, ORCID 0000-0001-6706-8980
^{***}dongruifang@ntsc.ac.cn, ORCID 0000-0003-2353-1819
^{†}These authors contributed equally to this paper.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
A fiber spool with ultra-low vibration sensitivity has been demonstrated for the ultra-narrow-linewidth fiber-stabilized laser by the multi-object orthogonal experimental design method, which can achieve the optimization object and analysis of influence levels without extensive computation. According to a test of 4 levels and 4 factors, an L_{16} (4^{4}) orthogonal table is established to design orthogonal experiments. The vibration sensitivities along the axial and radial directions and the normalized sums of the vibration sensitivities are determined as single objects and comprehensive objects, respectively. We adopt the range analysis of object values to obtain the influence levels of the four design parameters on the single objects and the comprehensive object. The optimal parameter combinations are determined by both methods of comprehensive balance and evaluation. Based on the corresponding fractional frequency stability of ultra-narrow-linewidth fiber-stabilized lasers, we obtain the final optimal parameter combination A3B1C2D1, which can achieve the fiber spool with vibration sensitivities of 10^{−12}/g magnitude. This work is the first time to use an orthogonal experimental design method to optimize the vibration sensitivities of fiber spools, providing an approach to design the fiber spool with ultra-low vibration sensitivity.
Keywords: Fiber spool, Optimization, Orthogonal experimental design, Ultra-narrow-linewidth fiber-stabilized laser, Vibration sensitivity
OCIS codes: (020.0020) Atomic and molecular physics; (060.0060) Fiber optics and optical communications; (060.2310) Fiber optics; (140.3425) Laser stabilization; (140.3510) Lasers, fiber
The fiber spool with ultra-low vibration sensitivity provides important applications including optical clocks, high-precision spectroscopy, gravitational redshift measurement, ultra-narrow-linewidth lasers, relativistic tests, gravitational wave detection, and coherent communication [1–6]. For example, ultra-narrow-linewidth lasers are often achieved by using the fiber method based on a fiber spool with ultra-low vibration sensitivity [7–20], where a fiber serves as a reference for the laser frequency. So the stability of the optical length of the fiber primarily determines the fractional frequency stability of ultra-narrow-linewidth fiber-stabilized lasers [7–20]. However, the stability of the fiber length is mainly defined by the geometric stability of the fiber winding surface of the fiber spool [15–20].
The main factors affecting the geometric stability of the fiber winding surface of fiber spools [15–20] include vibration, temperature, and thermal noise. In general, the index to assess the impact of vibration on the geometric stability of the fiber winding surface is the vibration sensitivity of fiber spools [15–19]. To reduce this impact, the finite element method plays an important role [15, 16, 18, 19]. Benefiting from this, the vibration sensitivity of fiber spools is reduced from 10^{−9}/g to 10^{−11}/g orders of magnitude by the passive method [15–19]. Furthermore, the fractional frequency stability of state-of-the-art laboratory ultra-narrow-linewidth fiber-stabilized laser approaches an order of magnitude of 10^{−15} at 1 s [15, 16, 18, 19].
To design the fiber spool with ultra-low vibration sensitivity, a large number of numerical simulations are often required to obtain a better combination of parameters, and it is difficult to compare the influence levels of parameters on vibration sensitivity under the same conditions. In addition, with the improvement of ultra-narrow-linewidth fiber-stabilized laser performance requirements, it has changed from mainly focusing on vibration sensitivity in the vertical direction (primary vibration direction) in the past to paying attention to the three directions (x-axis, y-axis, and z-axis). In this context, it is apparent that it is hard to obtain the optimal parameters combination with ultra-low vibration sensitivity for the three directions only by the usual single-factor and single-object analysis computation, as well as the common observation method. The orthogonal experimental design method [21] can significantly reduce the number of tests with no reduction in test feasibility. The range analysis can be used to obtain the level of influence of each factor on the object and to determine the optimal combination of parameters required for multi-object. The orthogonal experimental design method can be utilized to optimize the design parameters of the fiber spool to achieve ultra-low vibration sensitivities in the three directions, but to our knowledge, no correlated investigations have been reported.
In this work, on the basis of a multi-object orthogonal experimental design approach, we optimize the geometrical parameters of a self-designed fiber spool with a diameter of 120 mm to achieve the minimum effect induced by vibration. Quasi-static mechanical finite element model is developed to simulate and calculate the vibration sensitivity in accordance with the vibration characteristics of the fiber spool. The L_{16} (4^{4}) orthogonal table is used to design the orthogonal experiment based on a test of 4 levels and 4 factors. The vibration sensitivities in axial and radial directions are determined as single-object values, and the normalized sums of the vibration sensitivities are the comprehensive object values. According to the range analysis, the influence levels of the four geometrical parameters on the object values are analyzed and discussed. The two methods including comprehensive balance and comprehensive evaluation are adopted to achieve the optimal parameter combination. We determine the final optimal parameter combination A3B1C2D1 based on the corresponding fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser. This work provides a method of vibration sensitivities optimization for the fiber spools.
This work is structured as follows: The design of a self-designed fiber spool with a 120 mm diameter is presented in Section 2. An orthogonal experiment is designed in Section 3 including test indexes, design variables, orthogonal experimental table, and test results. According to a comprehensive balance method and a comprehensive evaluation method, we provide optimization of vibration sensitivity in Section 4. The final optimization results are determined in Section 4.3. The conclusions are summarized in Section 5.
A fiber spool with a 120 mm diameter is designed in this work. We present the geometric model of the fiber spool in Fig. 1. In general, the fiber spool is a cylindrical rotating body. The fiber spool mainly includes the fixed surface, the fiber winding surface, and the central hole for the winding of optical fibers. The z-axis is the axial direction, which is the central rotation axis of the fiber spool. The radial direction is the x-axis and y-axis. The fiber winding surface’s height of the fiber spool is 40 mm. The thickness of the bottom fixing surface is designed to be 8 mm. The distance between the fixed surface and the fiber winding surface is 15 mm. To facilitate the winding of optical fibers, the central hole with a diameter of 15 mm is drilled along the z-axis. The fiber spool is made of titanium, which is relatively easy to machine. The specific stiffness of titanium is relatively high, which is beneficial for reducing the vibration sensitivity of the fiber spool [17]. In addition, the fiber spool made of titanium is not sensitive to temperature fluctuation [17]. L_{1}, H_{1}, L_{2}, and L_{3} are the design parameters used to achieve the ultra-low vibration sensitivity of the fiber spool.
Generally, vibrations from the environment can cause changes in the optical properties of the optical fiber by deforming the fiber spool, such as frequency noise, optical power fluctuation, and polarization changes [15–19, 22]. The vibration acceleration in the environment causes mechanical deformation of the fiber spool and thereby leads to the variation in fiber length. Taking ultra-narrow-linewidth fiber-stabilized lasers as an example, this variation of fiber length is then converted into laser frequency noise [15–19, 22]. Vibration-induced deformation of the fiber spool alters the fiber length and stress, which subsequently affects the fiber density and bending. These changes ultimately lead to modifications in the refractive index and total reflection, resulting in variations in optical power [23, 24]. Ideally, the polarization state of a laser beam in an optical fiber remains unchanged. However, vibrations from the environment can be transmitted through the fiber spool to the optical fiber, resulting in mechanical stress, bending, and non-circularity variations. When a laser beam enters the fiber, it leads to fluctuations in the polarization state of the beam [25, 26].
For ultra-narrow-linewidth fiber-stabilized lasers, the optical fibers on the fiber spool are mainly bare fibers, including fiber core (diameter about 9 μm), cladding (diameter about 125 μm), soft coating (diameter about 190 μm), hard coating (diameter about 250 μm), but not the fiber jacket [12–20]. In this work, the optical fibers we study are bare fibers.
A quasi-static mechanical model is adopted to analyze the vibration sensitivities of the cylindrical fiber spool. This analytical approach is widely used in the vibration sensitivities analysis of the optical reference cavity [27–40] and fiber spool [15–19, 22]. When analyzing the vibration sensitivity of the fiber spool, low Fourier frequencies are considered crucial, as higher-frequency components can be effectively attenuated using traditional vibration isolation methods [15–18, 22, 28]. Therefore, this work mainly investigates the effect of vibrations with Fourier frequencies below 100 Hz. Based on the definition of the vibration sensitivity of the fiber spool [15–19, 22], the vibration sensitivity calculation method is developed. In this work, we consider the fiber spool as a single elastic body in the finite element analysis model. Simulation deformations of the fiber spool are within the elastic limit of the material. When the fiber spool is loaded by vibration acceleration, elastic deformations of the fiber spool cause the fiber winding surface to change, and then vary the fiber length. The fiber spool is vibrated at a 1 g acceleration to determine the vibration sensitivity. In this work, we take Titanium as an example to study the vibration sensitivities of the fiber spool. Theoretically, the deformation of the fiber spool is proportional to the specific stiffness (the ratio of elastic modulus to density) of the material [17, 18, 22]. For ultra-narrow-linewidth fiber-stabilized lasers, Titanium is widely used for the fiber spool. The main reasons include Titanium being relatively easy to machine, specific stiffness (about 2.3 × 10^{7} m^{2}/s^{2}) being relatively large, and not being sensitive to temperature fluctuation, which is more than 10 K during the measurement [17, 18, 22]. In the simulation model, the mechanical characteristics of titanium [17] are used, which are shown in Table 1. A half model is adopted in the simulation of vibration sensitivities along the axial and radial direction to save simulation analysis time. Tetrahedral elements are used to mesh the fiber spool’s geometry. Each tetrahedral element has four nodes, and its side length is approximately 1.5 mm.
TABLE 1 Material properties of the fiber spool
Material Properties | Elastic Modulus (GPa) | Poisson Ratio (%) | Density (kg/m^{3}) |
---|---|---|---|
Titanium | 105 | 0.37 | 4,500 |
For the fiber spool, vibration sensitivities along axial and radial directions are the important indexes. For the identical material and shape of the fiber spool, various geometric parameters can have different effects on the vibration sensitivity of the fiber spool. Vibration sensitivity ultimately affects the fractional frequency stability of ultra-narrow-linewidth fiber-stabilized lasers. In this work, we select the vibration sensitivities along axial (S_{z}) and radial directions (S_{x} and S_{y}) and the normalized sums (S_{1} and S_{2}) of the vibration sensitivities as single-object values and comprehensive object values, respectively.
In this work, the main geometric parameters that affect the vibration sensitivities of the fiber spool include L_{1}, H_{1}, L_{2}, and L_{3}, which at different levels influence the vibration sensitivity. The choice of the range of design variables is very important in the optimization process of these parameters, which not only affects whether the selected range has optimal or second-optimal solutions, but also influences the search efficiency of the optimization process. To achieve the minimum vibration sensitivity, we adopt the range of design variables in Table 2. There are four design variables, including L_{1}, H_{1}, L_{2}, and L_{3}, and each variable is divided into four levels. In Table 2, A, B, C, and D represent L_{1}, H_{1}, L_{2}, and L_{3}, respectively.
TABLE 2 Factors and levels of orthogonal test
Level | Factors (mm) | |||
---|---|---|---|---|
A (L_{1}) | B (H_{1}) | C (L_{2}) | D (L_{3}) | |
1 | 30.4 | 8.0 | 14.6 | 18 |
2 | 30.6 | 8.2 | 14.8 | 20 |
3 | 30.8 | 8.4 | 15.0 | 22 |
4 | 31.0 | 8.6 | 15.2 | 24 |
Generally, the combination of all the different test conditions is known as a comprehensive test [21]. This work is a test of 4 factors and 4 levels. There are 256 different test combinations to be performed individually in a comprehensive test, which is apparently time-consuming and unnecessary. The L_{16} (4^{4}) orthogonal table is chosen to design an orthogonal experiment requiring only 16 tests, approximately 94% less than the comprehensive test. In the test, we get the vibration sensitivity values for various parameter combinations. Table 3 provides the test scheme and test results. The absolute values of vibration sensitivities along axial and radial directions are represented by S_{z}, S_{x} (S_{y}). S_{1} and S_{2} denote the normalized sums of vibration sensitivities, which are also the comprehensive object values. In general, the vibration in the axial direction is primary in the laboratory environment [41–45]. Consequently, we assume that S_{z} accounts for 80% and 70%, and S_{x} and S_{y} account for 20% and 30% in the normalization process, respectively.
TABLE 3 Orthogonal table of experiment L_{16} (4^{4}) and test data
Level | A | B | C | D | S_{z} | S_{x} (S_{y}) | S_{1} | S_{2} |
---|---|---|---|---|---|---|---|---|
L_{1}/mm | H_{1}/mm | L_{2}/mm | L_{3}/mm | 10−11/g | 10−12/g | - | - | |
1 | 30.4 | 8.0 | 14.6 | 18 | 6.8 | 13 | 30.2 | 27.5 |
2 | 30.4 | 8.2 | 14.8 | 20 | 10.6 | 50.3 | 51.7 | 49.5 |
3 | 30.4 | 8.4 | 15.0 | 22 | 13.3 | 0.1 | 56.37 | 49.3 |
4 | 30.4 | 8.6 | 15.2 | 24 | 18.8 | 0.1 | 80.0 | 70.0 |
5 | 30.6 | 8.0 | 14.8 | 22 | 5.7 | 60.4 | 32.0 | 33.2 |
6 | 30.6 | 8.2 | 14.6 | 24 | 14.6 | 23 | 65.1 | 58.9 |
7 | 30.6 | 8.4 | 15.2 | 18 | 7.3 | 85.4 | 42.3 | 44.4 |
8 | 30.6 | 8.6 | 15.0 | 20 | 7.9 | 13.9 | 35.0 | 31.8 |
9 | 30.8 | 8.0 | 15.0 | 24 | 2.5 | 117.6 | 26.1 | 33.0 |
10 | 30.8 | 8.2 | 15.2 | 22 | 5 | 15.6 | 22.8 | 21.3 |
11 | 30.8 | 8.4 | 14.6 | 20 | 5.1 | 145.1 | 41.1 | 48.4 |
12 | 30.8 | 8.6 | 14.8 | 18 | 0.2 | 45.4 | 6.2 | 9.4 |
13 | 31.0 | 8.0 | 15.2 | 20 | 4.1 | 0.7 | 16.9 | 14.8 |
14 | 31.0 | 8.2 | 15.0 | 18 | 8.8 | 66.6 | 46.2 | 46.1 |
15 | 31.0 | 8.4 | 14.8 | 24 | 3.3 | 40.6 | 18.9 | 20.0 |
16 | 31.0 | 8.6 | 14.6 | 22 | 3.1 | 121.2 | 29.2 | 36.0 |
We use the two methods including a comprehensive balance method and a comprehensive evaluation method to achieve the optimal parameter combination of ultra-low vibration sensitivities. As shown in Fig. 2, the optimization process is as follows: Firstly, according to range analysis, we can get the optimal combinations of two single objects, which are both vibration sensitivities along axial and radial directions; And then a comprehensive balance method is adopted to determine the global optimal combination. Secondly, the optimal parameters’ combinations are achieved based on range analysis of the normalization sums S_{1} and S_{2}. When the optimal parameter combination schemes determined by the comprehensive balanced method and comprehensive evaluation method are selected, the final optimal parameter combination scheme is determined by the comparison of the corresponding fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser.
According to a single index, the vibration sensitivities along the axial and radial directions are simulated and analyzed. The optimal combination of four design parameters including L_{1}, H_{1}, L_{2}, and L_{3} is determined for each single index. Based on the importance of the vibration sensitivity indexes, as well as the major and minor factors in each index, we finally identify the overall optimal combination of parameters. In this work, the range approach is adopted to analyze and obtain the relationship between four factors and indexes.
According to the results of the range analysis of the vibration sensitivity along the axial direction, we can obtain the influence of four parameters on the vibration sensitivity S_{z} of the fiber spool. The results of the ranges analysis of the vibration sensitivity S_{z} are shown in Table 4 and Fig. 3. L_{1} and L_{3} have the greatest and least influence on vibration sensitivity S_{z}, respectively. The order of the degree of influence on vibration sensitivity S_{z} is A > B > C > D. In Table 4, V_{1-1}, V_{2-1}, V_{3-1}, and V_{4-1} represent the sums of the vibration sensitivity along the axial direction for four different levels, respectively. V_{1-1-M}, V_{2-1-M}, V_{3-1-M}, and V_{4-1-M} denote the means of V_{1-1}, V_{2-1}, V_{3-1}, and V_{4-1}, respectively. R_{1} denotes the range of the vibration sensitivity along the axial direction. Theoretically, the smaller the vibration sensitivity S_{z} of the fiber spool, the better the fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser. The optimized combination of four geometrical parameters is determined as A3B1C2D1, that is, L_{1}, H_{1}, L_{2}, and L_{3} are 30.8 mm, 8.0 mm, 14.8 mm, and 18 mm, respectively.
TABLE 4 Results of the ranges analysis of S_{z}
Ranges Analysis | Factors (mm) | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Sums | V_{1-1} | 49.5 | 19.1 | 29.6 | 23.1 |
V_{2-1} | 35.5 | 39.0 | 19.8 | 27.7 | |
V_{3-1} | 12.8 | 29.0 | 32.5 | 27.1 | |
V_{4-1} | 19.3 | 30.0 | 35.2 | 39.2 | |
Means | V_{1-1-M} | 12.4 | 4.8 | 7.4 | 5.8 |
V_{2-1-M} | 8.9 | 9.8 | 5.0 | 6.9 | |
V_{3-1-M} | 3.2 | 7.3 | 8.1 | 6.8 | |
V_{4-1-M} | 4.8 | 7.5 | 8.8 | 9.8 | |
Ranges | R_{1} | 9.2 | 5.0 | 3.8 | 3 |
The influence of four design parameters on the vibration sensitivity S_{x} (S_{y}) along the radial direction and the vibration sensitivity range’s analysis are shown in Table 5 and Fig. 3. In Table 5, we use V_{1-2}, V_{2-2}, V_{3-2}, and V_{4-2} to denote the sums of the vibration sensitivity S_{x} (S_{y}) for four levels, respectively. V_{1-2-M}, V_{2-2-M}, V_{3-2-M}, and V_{4-2-M} are adopted to express the means of V_{1-2}, V_{2-2}, V_{3-2}, and V_{4-2}. The range of the vibration sensitivity S_{x} (S_{y}) is denoted as R_{2}. The order of the degree of influence on vibration sensitivity S_{x} (S_{y}) is A > C > B > D. which is different from the vibration sensitivity S_{z} along the axial direction. Similar to the vibration sensitivity S_{z}, L_{1}, and L_{3} have the greatest and least influence on vibration sensitivity S_{x} (S_{y}) along the radial direction, respectively. In theory, the ultra-narrow-linewidth fiber-stabilized laser requires that the vibration sensitivity S_{x} (S_{y}) of the fiber spool is as low as possible. Based on this, we determine A1B2C4D4 as the optimal combination of four design parameters, which means that L_{1}, H_{1}, L_{2}, and L_{3} are 30.4 mm, 8.2 mm, 15.2 mm, and 24 mm, respectively.
TABLE 5 Results of the ranges analysis of S_{x} (S_{y})
Ranges Analysis | Factors (mm) | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Sums | V_{1-2} | 63.5 | 191.7 | 302.3 | 210.4 |
V_{2-2} | 182.7 | 155.5 | 196.7 | 210 | |
V_{3-2} | 323.7 | 271.2 | 198.2 | 197.3 | |
V_{4-2} | 229.1 | 180.6 | 101.8 | 181.3 | |
Means | V_{1-2-M} | 15.9 | 47.9 | 75.6 | 52.6 |
V_{2-2-M} | 45.7 | 38.9 | 49.2 | 52.5 | |
V_{3-2-M} | 80.9 | 67.8 | 49.6 | 49.3 | |
V_{4-2-M} | 57.3 | 45.2 | 25.5 | 45.3 | |
Ranges | R_{2} | 65 | 28.9 | 50.1 | 7.3 |
The influence levels of the four design parameters on the vibration sensitivities S_{z} and S_{x} (S_{y}) are A > B > C > D and A > C > B > D, respectively. The two optimal combinations for the lowest vibration sensitivities S_{z} and S_{x} (S_{y}) are A3B1C2D1 and A1B2C4D4, respectively. Compared with S_{x} (S_{y}), the vibration sensitivity S_{z} is more important, so to ensure that it is the lowest first, and then A3 is determined. B and C are the secondary influence parameters for the vibration sensitivities S_{z} and S_{x} (S_{y}), respectively. So we select B1 and C4. D is the minimum influence parameter for the vibration sensitivities S_{z} and S_{x} (S_{y}), and D1 is determined in the optimal combination. Finally, A3B1C4D1 is determined as the optimal combination by the comprehensive balance method.
The results of the ranges analysis of the normalized sum S_{1} of vibration sensitivities of the fiber spool are demonstrated in Table 6 and Fig. 4. V_{1-S}1, V_{2-S}1, V_{3-S}1, and V_{4-S}1 denote the normalized sum S_{1} of vibration sensitivities for the four levels, respectively. V_{1-S}1_{-M}, V_{2-S}1_{-M}, V_{2-S}1_{-M}, and V_{2-S}1_{-M} present the means of V_{1-S}1, V_{2-S}1, V_{3-S}1, and V_{4-S}1, respectively. Rs1 represents the range of the normalized sum S of vibration sensitivities as the comprehensive object. A > B > D > C is the order of significance of influence on the normalized sum S_{1} of vibration sensitivities. L_{1} and L_{2} have the greatest and least influence on the normalized sum S of vibration sensitivities of the fiber spool, respectively. To achieve the ultra-narrow-linewidth fiber-stabilized laser with higher frequency stability, the normalized sum S_{1} of vibration sensitivities of the fiber spool is as low as possible. According to this, A3B1C2D1 is determined as the optimal combination of four parameters, which is the same as the vibration sensitivity along the axial direction.
TABLE 6 Results of the ranges analysis of the normalized sum S_{1} of vibration sensitivities
Ranges Analysis | Factors (mm) | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Sums | V_{1-S}1 | 218.3 | 105.2 | 165.6 | 124.9 |
V_{2-S}1 | 174.4 | 185.8 | 108.8 | 144.7 | |
V_{3-S}1 | 96.2 | 158.7 | 163.7 | 140.4 | |
V_{4-S}1 | 111.2 | 150.4 | 162.0 | 190.1 | |
Means | V_{1-S}1_{-M} | 54.6 | 26.3 | 41.4 | 31.2 |
V_{2-S}1_{-M} | 43.6 | 46.5 | 27.2 | 36.2 | |
V_{3-S}1_{-M} | 24.1 | 39.7 | 40.9 | 35.1 | |
V_{4-S}1_{-M} | 27.8 | 37.6 | 40.5 | 47.5 | |
Ranges | R_{S}1 | 30.5 | 20.2 | 14.2 | 16.3 |
Table 7 and Fig. 4 show the results of the ranges analysis of the normalized sum S_{2} of vibration sensitivities. The normalized sum S_{2} for the four levels is presented as V_{1-S} 2, V_{2-S} 2, V_{3-S} 2, and V_{4-S} 2, respectively. We use V_{1-S} 2_{-M}, V_{2-S} 2_{-M}, V_{2-S} 2_{-M}, and V_{2-S} 2_{-M} to denote the means of V_{1-S} 2, V_{2-S} 2, V_{3-S} 2, and V_{4-S} 2, respectively. R_{S} 2 represents the range of the normalized sum S_{2}. The order of significance of influence on the normalized sum S_{2} is A > B > C > D, which is different from that of the normalized sum S_{1}. The greatest and least influential on the normalized sum S_{2} of vibration sensitivities are L_{1} and H_{1}, respectively. According to the results of the ranges analysis of the normalized sum S_{2} of vibration sensitivities, we determine A3B1C2D1 as the optimal combination of four parameters, which is the same as that of the normalized sum S_{1}.
TABLE 7 Results of the ranges analysis of the normalized sum S_{2} of vibration sensitivities
Ranges Analysis | Factors (mm) | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Sums | V_{1-S}2 | 196.3 | 108.5 | 170.8 | 127.4 |
V_{2-S}2 | 168.3 | 175.8 | 112.1 | 144.5 | |
V_{3-S}2 | 112.1 | 162.1 | 160.2 | 139.8 | |
V_{4-S}2 | 116.9 | 147.2 | 150.5 | 181.9 | |
Means | V_{1-S}2_{-M} | 49.1 | 27.1 | 42.7 | 31.9 |
V_{2-S}2_{-M} | 42.1 | 44.0 | 28.0 | 36.1 | |
V_{3-S}2_{-M} | 28.0 | 40.5 | 40.1 | 35.0 | |
V_{4-S}2_{-M} | 29.2 | 36.8 | 37.6 | 45.5 | |
Ranges | R_{S}2 | 21.1 | 16.9 | 14.7 | 13.6 |
The fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser is adopted to obtain the final optimal scheme. The frequency relative variation induced by the vibration sensitivity of a fiber spool can be presented as [39, 42].
where v and Δv are the laser frequency and the laser frequency variation. S_{x}, S_{y}, and S_{z} denote the vibration sensitivities of the fiber spool along the x-axis, y-axis, and z-axis, respectively. a_{x}, a_{y}, and a_{z} present the vibration acceleration along the x-axis, y-axis, and z-axis, respectively.
In general, the vibration acceleration in laboratory environments is approximately 1 μg (or even lower) [41–45]. In this work, we assume that the vibration acceleration along the z-axis direction is 1 μg and the vibration acceleration along the horizontal direction (x-axis and y-axis) is 0.5 μg. When the optimal schemes are selected, the fractional frequency stabilities of the ultra-narrow-linewidth fiber-stabilized laser based on Eq. (1) are presented in Table 8.
TABLE 8 Fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser based on the optimal schemes
No. | A | B | C | D | S_{z} | S_{x} (S_{y}) | Frequency Stability |
---|---|---|---|---|---|---|---|
L_{1}/mm | H_{1}/mm | L_{2}/mm | L_{3}/mm | 10−11/g | 10−12/g | ||
A3B1C2D1 | 30.8 | 8.0 | 14.8 | 18 | 0.6 | 4.3 | 0.7 × 10−17 |
A3B1C4D1 | 30.8 | 8.0 | 15.2 | 18 | 1.4 | 9.8 | 1.7 × 10−^{17} |
According to Table 8, it is easy to get that the A3B1C2D1 is the optimal scheme, and the corresponding fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser is 0.7 × 10^{−17}. When the A3B1C2D1 is selected as the optimal combination, the fiber spool’s displacement in the optical axis is shown in Figs. 5 and 6. In general, the displacement of the fiber spool in the z-axis direction [Fig. 5(a)] is an order of magnitude of 10^{−6} mm, which is about ten times bigger than the displacement of the fiber spool in the x-axis direction [Fig. 5(b)]. The displacement of the bottom fixed face is the minimum, which is an order of magnitude of 10^{−9} mm. When 1 g acceleration is loaded along the x-axis direction, the displacements of the fiber spool in the z-axis direction [Fig. 6(a)] and the x-axis direction [Fig. 6(b)] are an order of magnitude of 10^{−6} mm.
The composition of the material affects vibration sensitivities. Jiang’s research compared the vibration sensitivities of bare fibers coated with polyimide and bare fibers under the same conditions, finding that the latter has better sensitivity [17]. In the case of ultra-narrow-linewidth fiber-stabilized lasers, a general method for securing the fiber is tightly winding it on the surface of a fiber spool with a certain winding tension. This tension also affects vibration sensitivities. Jiang [17], Huang et al. [18], and Hu et al. [22] have conducted studies on this. Jiang’s research, for example, shows that the vibration sensitivity deteriorates twice as much for a winding tension of 0.1 N compared to 0.75 N. However, increasing the winding tension further will also lead to a deterioration in vibration sensitivity [17].
Finally, we also simulate and calculate the vibration modes of the fiber spool, and estimate its first six orders eigenfrequencies are about in the range of 3,500 Hz to 6,800 Hz. The vibration shapes of the first six orders are shown in Fig. 7. The first-order vibration shape shows that the fiber spool has a stretching vibration with the z-axis as the center. For the second-order vibration shape and the third-order vibration shape, the fiber spool swings left and right along the yz plane and xz plane, respectively. The fourth-order vibration shape and the fifth-order vibration shape show that the fiber spool undergoes rotational vibration with the y-axis as the center and the x-axis as the center, respectively. For the sixth-order vibration shapes, the fiber spool undergoes torsional motion along the xz plane.
In this work, taking a self-designed fiber spool as an example, we optimize its vibration sensitivity by using a multi-object orthogonal experimental design approach. According to a test of 4 factors and 4 levels, we select an L_{16} (4^{4}) orthogonal table to design an orthogonal experiment. The vibration sensitivities are simulated and analyzed by the finite element approach for four geometric parameters. The vibration sensitivities along the axial and radial directions are determined as single objects. The normalized sums of the vibration sensitivities are selected as the comprehensive object. Based on the range analysis of the single objects and the comprehensive objects, we obtain the influence levels of the four geometric parameters on the object. Through the comprehensive balance method and the comprehensive evaluation method, the optimal parameter combination is determined. According to comparing the fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser, A3B1C2D1 is determined as the final optimal parameter combination, which can achieve the fiber spool with 10^{−12}/g magnitude vibration sensitivity. To our knowledge, the orthogonal experimental design method is being used for the first time to optimize the vibration sensitivity of the fiber spool and to confirm its efficacy. This work aims to provide an approach for vibration sensitivities optimization of a fiber spool.
We would like to acknowledge the contribution to this paper from X. Zhang of Northwestern Polytechnical University.
Partially supported by the Youth Innovation Promotion Association of the Chinese Academy of Sciences (Grant No. 1188000XGJ); The Chinese National Natural Science Foundation (Grant No. 11903041); The Young Innovative Talents of the National Time Service Center of the Chinese Academy of Sciences (Grant No. Y917SC1).
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
All data generated or analyzed during this study are included in this published article.
Curr. Opt. Photon. 2024; 8(1): 45-55
Published online February 25, 2024 https://doi.org/10.3807/COPP.2024.8.1.45
Copyright © Optical Society of Korea.
Jing Gao^{1}, Linbo Zhang^{1}, Dongdong Jiao^{1}, Guanjun Xu^{1,2} , Xue Deng^{1}, Qi Zang^{1}, Honglei Yang^{3}, Ruifang Dong^{1,2} , Tao Liu^{1,2} , Shougang Zhang^{1,2}
^{1}National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China
^{2}School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China
^{3}Science and Technology on Metrology and Calibration Laboratory, Beijing Institute of Radio Metrology and Measurement, Beijing 100854, China
Correspondence to:^{*}xuguanjun@ntsc.ac.cn, ORCID 0000-0003-1108-0459
^{**}taoliu@ntsc.ac.cn, ORCID 0000-0001-6706-8980
^{***}dongruifang@ntsc.ac.cn, ORCID 0000-0003-2353-1819
^{†}These authors contributed equally to this paper.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
A fiber spool with ultra-low vibration sensitivity has been demonstrated for the ultra-narrow-linewidth fiber-stabilized laser by the multi-object orthogonal experimental design method, which can achieve the optimization object and analysis of influence levels without extensive computation. According to a test of 4 levels and 4 factors, an L_{16} (4^{4}) orthogonal table is established to design orthogonal experiments. The vibration sensitivities along the axial and radial directions and the normalized sums of the vibration sensitivities are determined as single objects and comprehensive objects, respectively. We adopt the range analysis of object values to obtain the influence levels of the four design parameters on the single objects and the comprehensive object. The optimal parameter combinations are determined by both methods of comprehensive balance and evaluation. Based on the corresponding fractional frequency stability of ultra-narrow-linewidth fiber-stabilized lasers, we obtain the final optimal parameter combination A3B1C2D1, which can achieve the fiber spool with vibration sensitivities of 10^{−12}/g magnitude. This work is the first time to use an orthogonal experimental design method to optimize the vibration sensitivities of fiber spools, providing an approach to design the fiber spool with ultra-low vibration sensitivity.
Keywords: Fiber spool, Optimization, Orthogonal experimental design, Ultra-narrow-linewidth fiber-stabilized laser, Vibration sensitivity
The fiber spool with ultra-low vibration sensitivity provides important applications including optical clocks, high-precision spectroscopy, gravitational redshift measurement, ultra-narrow-linewidth lasers, relativistic tests, gravitational wave detection, and coherent communication [1–6]. For example, ultra-narrow-linewidth lasers are often achieved by using the fiber method based on a fiber spool with ultra-low vibration sensitivity [7–20], where a fiber serves as a reference for the laser frequency. So the stability of the optical length of the fiber primarily determines the fractional frequency stability of ultra-narrow-linewidth fiber-stabilized lasers [7–20]. However, the stability of the fiber length is mainly defined by the geometric stability of the fiber winding surface of the fiber spool [15–20].
The main factors affecting the geometric stability of the fiber winding surface of fiber spools [15–20] include vibration, temperature, and thermal noise. In general, the index to assess the impact of vibration on the geometric stability of the fiber winding surface is the vibration sensitivity of fiber spools [15–19]. To reduce this impact, the finite element method plays an important role [15, 16, 18, 19]. Benefiting from this, the vibration sensitivity of fiber spools is reduced from 10^{−9}/g to 10^{−11}/g orders of magnitude by the passive method [15–19]. Furthermore, the fractional frequency stability of state-of-the-art laboratory ultra-narrow-linewidth fiber-stabilized laser approaches an order of magnitude of 10^{−15} at 1 s [15, 16, 18, 19].
To design the fiber spool with ultra-low vibration sensitivity, a large number of numerical simulations are often required to obtain a better combination of parameters, and it is difficult to compare the influence levels of parameters on vibration sensitivity under the same conditions. In addition, with the improvement of ultra-narrow-linewidth fiber-stabilized laser performance requirements, it has changed from mainly focusing on vibration sensitivity in the vertical direction (primary vibration direction) in the past to paying attention to the three directions (x-axis, y-axis, and z-axis). In this context, it is apparent that it is hard to obtain the optimal parameters combination with ultra-low vibration sensitivity for the three directions only by the usual single-factor and single-object analysis computation, as well as the common observation method. The orthogonal experimental design method [21] can significantly reduce the number of tests with no reduction in test feasibility. The range analysis can be used to obtain the level of influence of each factor on the object and to determine the optimal combination of parameters required for multi-object. The orthogonal experimental design method can be utilized to optimize the design parameters of the fiber spool to achieve ultra-low vibration sensitivities in the three directions, but to our knowledge, no correlated investigations have been reported.
In this work, on the basis of a multi-object orthogonal experimental design approach, we optimize the geometrical parameters of a self-designed fiber spool with a diameter of 120 mm to achieve the minimum effect induced by vibration. Quasi-static mechanical finite element model is developed to simulate and calculate the vibration sensitivity in accordance with the vibration characteristics of the fiber spool. The L_{16} (4^{4}) orthogonal table is used to design the orthogonal experiment based on a test of 4 levels and 4 factors. The vibration sensitivities in axial and radial directions are determined as single-object values, and the normalized sums of the vibration sensitivities are the comprehensive object values. According to the range analysis, the influence levels of the four geometrical parameters on the object values are analyzed and discussed. The two methods including comprehensive balance and comprehensive evaluation are adopted to achieve the optimal parameter combination. We determine the final optimal parameter combination A3B1C2D1 based on the corresponding fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser. This work provides a method of vibration sensitivities optimization for the fiber spools.
This work is structured as follows: The design of a self-designed fiber spool with a 120 mm diameter is presented in Section 2. An orthogonal experiment is designed in Section 3 including test indexes, design variables, orthogonal experimental table, and test results. According to a comprehensive balance method and a comprehensive evaluation method, we provide optimization of vibration sensitivity in Section 4. The final optimization results are determined in Section 4.3. The conclusions are summarized in Section 5.
A fiber spool with a 120 mm diameter is designed in this work. We present the geometric model of the fiber spool in Fig. 1. In general, the fiber spool is a cylindrical rotating body. The fiber spool mainly includes the fixed surface, the fiber winding surface, and the central hole for the winding of optical fibers. The z-axis is the axial direction, which is the central rotation axis of the fiber spool. The radial direction is the x-axis and y-axis. The fiber winding surface’s height of the fiber spool is 40 mm. The thickness of the bottom fixing surface is designed to be 8 mm. The distance between the fixed surface and the fiber winding surface is 15 mm. To facilitate the winding of optical fibers, the central hole with a diameter of 15 mm is drilled along the z-axis. The fiber spool is made of titanium, which is relatively easy to machine. The specific stiffness of titanium is relatively high, which is beneficial for reducing the vibration sensitivity of the fiber spool [17]. In addition, the fiber spool made of titanium is not sensitive to temperature fluctuation [17]. L_{1}, H_{1}, L_{2}, and L_{3} are the design parameters used to achieve the ultra-low vibration sensitivity of the fiber spool.
Generally, vibrations from the environment can cause changes in the optical properties of the optical fiber by deforming the fiber spool, such as frequency noise, optical power fluctuation, and polarization changes [15–19, 22]. The vibration acceleration in the environment causes mechanical deformation of the fiber spool and thereby leads to the variation in fiber length. Taking ultra-narrow-linewidth fiber-stabilized lasers as an example, this variation of fiber length is then converted into laser frequency noise [15–19, 22]. Vibration-induced deformation of the fiber spool alters the fiber length and stress, which subsequently affects the fiber density and bending. These changes ultimately lead to modifications in the refractive index and total reflection, resulting in variations in optical power [23, 24]. Ideally, the polarization state of a laser beam in an optical fiber remains unchanged. However, vibrations from the environment can be transmitted through the fiber spool to the optical fiber, resulting in mechanical stress, bending, and non-circularity variations. When a laser beam enters the fiber, it leads to fluctuations in the polarization state of the beam [25, 26].
For ultra-narrow-linewidth fiber-stabilized lasers, the optical fibers on the fiber spool are mainly bare fibers, including fiber core (diameter about 9 μm), cladding (diameter about 125 μm), soft coating (diameter about 190 μm), hard coating (diameter about 250 μm), but not the fiber jacket [12–20]. In this work, the optical fibers we study are bare fibers.
A quasi-static mechanical model is adopted to analyze the vibration sensitivities of the cylindrical fiber spool. This analytical approach is widely used in the vibration sensitivities analysis of the optical reference cavity [27–40] and fiber spool [15–19, 22]. When analyzing the vibration sensitivity of the fiber spool, low Fourier frequencies are considered crucial, as higher-frequency components can be effectively attenuated using traditional vibration isolation methods [15–18, 22, 28]. Therefore, this work mainly investigates the effect of vibrations with Fourier frequencies below 100 Hz. Based on the definition of the vibration sensitivity of the fiber spool [15–19, 22], the vibration sensitivity calculation method is developed. In this work, we consider the fiber spool as a single elastic body in the finite element analysis model. Simulation deformations of the fiber spool are within the elastic limit of the material. When the fiber spool is loaded by vibration acceleration, elastic deformations of the fiber spool cause the fiber winding surface to change, and then vary the fiber length. The fiber spool is vibrated at a 1 g acceleration to determine the vibration sensitivity. In this work, we take Titanium as an example to study the vibration sensitivities of the fiber spool. Theoretically, the deformation of the fiber spool is proportional to the specific stiffness (the ratio of elastic modulus to density) of the material [17, 18, 22]. For ultra-narrow-linewidth fiber-stabilized lasers, Titanium is widely used for the fiber spool. The main reasons include Titanium being relatively easy to machine, specific stiffness (about 2.3 × 10^{7} m^{2}/s^{2}) being relatively large, and not being sensitive to temperature fluctuation, which is more than 10 K during the measurement [17, 18, 22]. In the simulation model, the mechanical characteristics of titanium [17] are used, which are shown in Table 1. A half model is adopted in the simulation of vibration sensitivities along the axial and radial direction to save simulation analysis time. Tetrahedral elements are used to mesh the fiber spool’s geometry. Each tetrahedral element has four nodes, and its side length is approximately 1.5 mm.
TABLE 1. Material properties of the fiber spool.
Material Properties | Elastic Modulus (GPa) | Poisson Ratio (%) | Density (kg/m^{3}) |
---|---|---|---|
Titanium | 105 | 0.37 | 4,500 |
For the fiber spool, vibration sensitivities along axial and radial directions are the important indexes. For the identical material and shape of the fiber spool, various geometric parameters can have different effects on the vibration sensitivity of the fiber spool. Vibration sensitivity ultimately affects the fractional frequency stability of ultra-narrow-linewidth fiber-stabilized lasers. In this work, we select the vibration sensitivities along axial (S_{z}) and radial directions (S_{x} and S_{y}) and the normalized sums (S_{1} and S_{2}) of the vibration sensitivities as single-object values and comprehensive object values, respectively.
In this work, the main geometric parameters that affect the vibration sensitivities of the fiber spool include L_{1}, H_{1}, L_{2}, and L_{3}, which at different levels influence the vibration sensitivity. The choice of the range of design variables is very important in the optimization process of these parameters, which not only affects whether the selected range has optimal or second-optimal solutions, but also influences the search efficiency of the optimization process. To achieve the minimum vibration sensitivity, we adopt the range of design variables in Table 2. There are four design variables, including L_{1}, H_{1}, L_{2}, and L_{3}, and each variable is divided into four levels. In Table 2, A, B, C, and D represent L_{1}, H_{1}, L_{2}, and L_{3}, respectively.
TABLE 2. Factors and levels of orthogonal test.
Level | Factors (mm) | |||
---|---|---|---|---|
A (L_{1}) | B (H_{1}) | C (L_{2}) | D (L_{3}) | |
1 | 30.4 | 8.0 | 14.6 | 18 |
2 | 30.6 | 8.2 | 14.8 | 20 |
3 | 30.8 | 8.4 | 15.0 | 22 |
4 | 31.0 | 8.6 | 15.2 | 24 |
Generally, the combination of all the different test conditions is known as a comprehensive test [21]. This work is a test of 4 factors and 4 levels. There are 256 different test combinations to be performed individually in a comprehensive test, which is apparently time-consuming and unnecessary. The L_{16} (4^{4}) orthogonal table is chosen to design an orthogonal experiment requiring only 16 tests, approximately 94% less than the comprehensive test. In the test, we get the vibration sensitivity values for various parameter combinations. Table 3 provides the test scheme and test results. The absolute values of vibration sensitivities along axial and radial directions are represented by S_{z}, S_{x} (S_{y}). S_{1} and S_{2} denote the normalized sums of vibration sensitivities, which are also the comprehensive object values. In general, the vibration in the axial direction is primary in the laboratory environment [41–45]. Consequently, we assume that S_{z} accounts for 80% and 70%, and S_{x} and S_{y} account for 20% and 30% in the normalization process, respectively.
TABLE 3. Orthogonal table of experiment L_{16} (4^{4}) and test data.
Level | A | B | C | D | S_{z} | S_{x} (S_{y}) | S_{1} | S_{2} |
---|---|---|---|---|---|---|---|---|
L_{1}/mm | H_{1}/mm | L_{2}/mm | L_{3}/mm | 10−11/g | 10−12/g | - | - | |
1 | 30.4 | 8.0 | 14.6 | 18 | 6.8 | 13 | 30.2 | 27.5 |
2 | 30.4 | 8.2 | 14.8 | 20 | 10.6 | 50.3 | 51.7 | 49.5 |
3 | 30.4 | 8.4 | 15.0 | 22 | 13.3 | 0.1 | 56.37 | 49.3 |
4 | 30.4 | 8.6 | 15.2 | 24 | 18.8 | 0.1 | 80.0 | 70.0 |
5 | 30.6 | 8.0 | 14.8 | 22 | 5.7 | 60.4 | 32.0 | 33.2 |
6 | 30.6 | 8.2 | 14.6 | 24 | 14.6 | 23 | 65.1 | 58.9 |
7 | 30.6 | 8.4 | 15.2 | 18 | 7.3 | 85.4 | 42.3 | 44.4 |
8 | 30.6 | 8.6 | 15.0 | 20 | 7.9 | 13.9 | 35.0 | 31.8 |
9 | 30.8 | 8.0 | 15.0 | 24 | 2.5 | 117.6 | 26.1 | 33.0 |
10 | 30.8 | 8.2 | 15.2 | 22 | 5 | 15.6 | 22.8 | 21.3 |
11 | 30.8 | 8.4 | 14.6 | 20 | 5.1 | 145.1 | 41.1 | 48.4 |
12 | 30.8 | 8.6 | 14.8 | 18 | 0.2 | 45.4 | 6.2 | 9.4 |
13 | 31.0 | 8.0 | 15.2 | 20 | 4.1 | 0.7 | 16.9 | 14.8 |
14 | 31.0 | 8.2 | 15.0 | 18 | 8.8 | 66.6 | 46.2 | 46.1 |
15 | 31.0 | 8.4 | 14.8 | 24 | 3.3 | 40.6 | 18.9 | 20.0 |
16 | 31.0 | 8.6 | 14.6 | 22 | 3.1 | 121.2 | 29.2 | 36.0 |
We use the two methods including a comprehensive balance method and a comprehensive evaluation method to achieve the optimal parameter combination of ultra-low vibration sensitivities. As shown in Fig. 2, the optimization process is as follows: Firstly, according to range analysis, we can get the optimal combinations of two single objects, which are both vibration sensitivities along axial and radial directions; And then a comprehensive balance method is adopted to determine the global optimal combination. Secondly, the optimal parameters’ combinations are achieved based on range analysis of the normalization sums S_{1} and S_{2}. When the optimal parameter combination schemes determined by the comprehensive balanced method and comprehensive evaluation method are selected, the final optimal parameter combination scheme is determined by the comparison of the corresponding fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser.
According to a single index, the vibration sensitivities along the axial and radial directions are simulated and analyzed. The optimal combination of four design parameters including L_{1}, H_{1}, L_{2}, and L_{3} is determined for each single index. Based on the importance of the vibration sensitivity indexes, as well as the major and minor factors in each index, we finally identify the overall optimal combination of parameters. In this work, the range approach is adopted to analyze and obtain the relationship between four factors and indexes.
According to the results of the range analysis of the vibration sensitivity along the axial direction, we can obtain the influence of four parameters on the vibration sensitivity S_{z} of the fiber spool. The results of the ranges analysis of the vibration sensitivity S_{z} are shown in Table 4 and Fig. 3. L_{1} and L_{3} have the greatest and least influence on vibration sensitivity S_{z}, respectively. The order of the degree of influence on vibration sensitivity S_{z} is A > B > C > D. In Table 4, V_{1-1}, V_{2-1}, V_{3-1}, and V_{4-1} represent the sums of the vibration sensitivity along the axial direction for four different levels, respectively. V_{1-1-M}, V_{2-1-M}, V_{3-1-M}, and V_{4-1-M} denote the means of V_{1-1}, V_{2-1}, V_{3-1}, and V_{4-1}, respectively. R_{1} denotes the range of the vibration sensitivity along the axial direction. Theoretically, the smaller the vibration sensitivity S_{z} of the fiber spool, the better the fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser. The optimized combination of four geometrical parameters is determined as A3B1C2D1, that is, L_{1}, H_{1}, L_{2}, and L_{3} are 30.8 mm, 8.0 mm, 14.8 mm, and 18 mm, respectively.
TABLE 4. Results of the ranges analysis of S_{z}.
Ranges Analysis | Factors (mm) | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Sums | V_{1-1} | 49.5 | 19.1 | 29.6 | 23.1 |
V_{2-1} | 35.5 | 39.0 | 19.8 | 27.7 | |
V_{3-1} | 12.8 | 29.0 | 32.5 | 27.1 | |
V_{4-1} | 19.3 | 30.0 | 35.2 | 39.2 | |
Means | V_{1-1-M} | 12.4 | 4.8 | 7.4 | 5.8 |
V_{2-1-M} | 8.9 | 9.8 | 5.0 | 6.9 | |
V_{3-1-M} | 3.2 | 7.3 | 8.1 | 6.8 | |
V_{4-1-M} | 4.8 | 7.5 | 8.8 | 9.8 | |
Ranges | R_{1} | 9.2 | 5.0 | 3.8 | 3 |
The influence of four design parameters on the vibration sensitivity S_{x} (S_{y}) along the radial direction and the vibration sensitivity range’s analysis are shown in Table 5 and Fig. 3. In Table 5, we use V_{1-2}, V_{2-2}, V_{3-2}, and V_{4-2} to denote the sums of the vibration sensitivity S_{x} (S_{y}) for four levels, respectively. V_{1-2-M}, V_{2-2-M}, V_{3-2-M}, and V_{4-2-M} are adopted to express the means of V_{1-2}, V_{2-2}, V_{3-2}, and V_{4-2}. The range of the vibration sensitivity S_{x} (S_{y}) is denoted as R_{2}. The order of the degree of influence on vibration sensitivity S_{x} (S_{y}) is A > C > B > D. which is different from the vibration sensitivity S_{z} along the axial direction. Similar to the vibration sensitivity S_{z}, L_{1}, and L_{3} have the greatest and least influence on vibration sensitivity S_{x} (S_{y}) along the radial direction, respectively. In theory, the ultra-narrow-linewidth fiber-stabilized laser requires that the vibration sensitivity S_{x} (S_{y}) of the fiber spool is as low as possible. Based on this, we determine A1B2C4D4 as the optimal combination of four design parameters, which means that L_{1}, H_{1}, L_{2}, and L_{3} are 30.4 mm, 8.2 mm, 15.2 mm, and 24 mm, respectively.
TABLE 5. Results of the ranges analysis of S_{x} (S_{y}).
Ranges Analysis | Factors (mm) | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Sums | V_{1-2} | 63.5 | 191.7 | 302.3 | 210.4 |
V_{2-2} | 182.7 | 155.5 | 196.7 | 210 | |
V_{3-2} | 323.7 | 271.2 | 198.2 | 197.3 | |
V_{4-2} | 229.1 | 180.6 | 101.8 | 181.3 | |
Means | V_{1-2-M} | 15.9 | 47.9 | 75.6 | 52.6 |
V_{2-2-M} | 45.7 | 38.9 | 49.2 | 52.5 | |
V_{3-2-M} | 80.9 | 67.8 | 49.6 | 49.3 | |
V_{4-2-M} | 57.3 | 45.2 | 25.5 | 45.3 | |
Ranges | R_{2} | 65 | 28.9 | 50.1 | 7.3 |
The influence levels of the four design parameters on the vibration sensitivities S_{z} and S_{x} (S_{y}) are A > B > C > D and A > C > B > D, respectively. The two optimal combinations for the lowest vibration sensitivities S_{z} and S_{x} (S_{y}) are A3B1C2D1 and A1B2C4D4, respectively. Compared with S_{x} (S_{y}), the vibration sensitivity S_{z} is more important, so to ensure that it is the lowest first, and then A3 is determined. B and C are the secondary influence parameters for the vibration sensitivities S_{z} and S_{x} (S_{y}), respectively. So we select B1 and C4. D is the minimum influence parameter for the vibration sensitivities S_{z} and S_{x} (S_{y}), and D1 is determined in the optimal combination. Finally, A3B1C4D1 is determined as the optimal combination by the comprehensive balance method.
The results of the ranges analysis of the normalized sum S_{1} of vibration sensitivities of the fiber spool are demonstrated in Table 6 and Fig. 4. V_{1-S}1, V_{2-S}1, V_{3-S}1, and V_{4-S}1 denote the normalized sum S_{1} of vibration sensitivities for the four levels, respectively. V_{1-S}1_{-M}, V_{2-S}1_{-M}, V_{2-S}1_{-M}, and V_{2-S}1_{-M} present the means of V_{1-S}1, V_{2-S}1, V_{3-S}1, and V_{4-S}1, respectively. Rs1 represents the range of the normalized sum S of vibration sensitivities as the comprehensive object. A > B > D > C is the order of significance of influence on the normalized sum S_{1} of vibration sensitivities. L_{1} and L_{2} have the greatest and least influence on the normalized sum S of vibration sensitivities of the fiber spool, respectively. To achieve the ultra-narrow-linewidth fiber-stabilized laser with higher frequency stability, the normalized sum S_{1} of vibration sensitivities of the fiber spool is as low as possible. According to this, A3B1C2D1 is determined as the optimal combination of four parameters, which is the same as the vibration sensitivity along the axial direction.
TABLE 6. Results of the ranges analysis of the normalized sum S_{1} of vibration sensitivities.
Ranges Analysis | Factors (mm) | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Sums | V_{1-S}1 | 218.3 | 105.2 | 165.6 | 124.9 |
V_{2-S}1 | 174.4 | 185.8 | 108.8 | 144.7 | |
V_{3-S}1 | 96.2 | 158.7 | 163.7 | 140.4 | |
V_{4-S}1 | 111.2 | 150.4 | 162.0 | 190.1 | |
Means | V_{1-S}1_{-M} | 54.6 | 26.3 | 41.4 | 31.2 |
V_{2-S}1_{-M} | 43.6 | 46.5 | 27.2 | 36.2 | |
V_{3-S}1_{-M} | 24.1 | 39.7 | 40.9 | 35.1 | |
V_{4-S}1_{-M} | 27.8 | 37.6 | 40.5 | 47.5 | |
Ranges | R_{S}1 | 30.5 | 20.2 | 14.2 | 16.3 |
Table 7 and Fig. 4 show the results of the ranges analysis of the normalized sum S_{2} of vibration sensitivities. The normalized sum S_{2} for the four levels is presented as V_{1-S} 2, V_{2-S} 2, V_{3-S} 2, and V_{4-S} 2, respectively. We use V_{1-S} 2_{-M}, V_{2-S} 2_{-M}, V_{2-S} 2_{-M}, and V_{2-S} 2_{-M} to denote the means of V_{1-S} 2, V_{2-S} 2, V_{3-S} 2, and V_{4-S} 2, respectively. R_{S} 2 represents the range of the normalized sum S_{2}. The order of significance of influence on the normalized sum S_{2} is A > B > C > D, which is different from that of the normalized sum S_{1}. The greatest and least influential on the normalized sum S_{2} of vibration sensitivities are L_{1} and H_{1}, respectively. According to the results of the ranges analysis of the normalized sum S_{2} of vibration sensitivities, we determine A3B1C2D1 as the optimal combination of four parameters, which is the same as that of the normalized sum S_{1}.
TABLE 7. Results of the ranges analysis of the normalized sum S_{2} of vibration sensitivities.
Ranges Analysis | Factors (mm) | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Sums | V_{1-S}2 | 196.3 | 108.5 | 170.8 | 127.4 |
V_{2-S}2 | 168.3 | 175.8 | 112.1 | 144.5 | |
V_{3-S}2 | 112.1 | 162.1 | 160.2 | 139.8 | |
V_{4-S}2 | 116.9 | 147.2 | 150.5 | 181.9 | |
Means | V_{1-S}2_{-M} | 49.1 | 27.1 | 42.7 | 31.9 |
V_{2-S}2_{-M} | 42.1 | 44.0 | 28.0 | 36.1 | |
V_{3-S}2_{-M} | 28.0 | 40.5 | 40.1 | 35.0 | |
V_{4-S}2_{-M} | 29.2 | 36.8 | 37.6 | 45.5 | |
Ranges | R_{S}2 | 21.1 | 16.9 | 14.7 | 13.6 |
The fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser is adopted to obtain the final optimal scheme. The frequency relative variation induced by the vibration sensitivity of a fiber spool can be presented as [39, 42].
where v and Δv are the laser frequency and the laser frequency variation. S_{x}, S_{y}, and S_{z} denote the vibration sensitivities of the fiber spool along the x-axis, y-axis, and z-axis, respectively. a_{x}, a_{y}, and a_{z} present the vibration acceleration along the x-axis, y-axis, and z-axis, respectively.
In general, the vibration acceleration in laboratory environments is approximately 1 μg (or even lower) [41–45]. In this work, we assume that the vibration acceleration along the z-axis direction is 1 μg and the vibration acceleration along the horizontal direction (x-axis and y-axis) is 0.5 μg. When the optimal schemes are selected, the fractional frequency stabilities of the ultra-narrow-linewidth fiber-stabilized laser based on Eq. (1) are presented in Table 8.
TABLE 8. Fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser based on the optimal schemes.
No. | A | B | C | D | S_{z} | S_{x} (S_{y}) | Frequency Stability |
---|---|---|---|---|---|---|---|
L_{1}/mm | H_{1}/mm | L_{2}/mm | L_{3}/mm | 10−11/g | 10−12/g | ||
A3B1C2D1 | 30.8 | 8.0 | 14.8 | 18 | 0.6 | 4.3 | 0.7 × 10−17 |
A3B1C4D1 | 30.8 | 8.0 | 15.2 | 18 | 1.4 | 9.8 | 1.7 × 10−^{17} |
According to Table 8, it is easy to get that the A3B1C2D1 is the optimal scheme, and the corresponding fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser is 0.7 × 10^{−17}. When the A3B1C2D1 is selected as the optimal combination, the fiber spool’s displacement in the optical axis is shown in Figs. 5 and 6. In general, the displacement of the fiber spool in the z-axis direction [Fig. 5(a)] is an order of magnitude of 10^{−6} mm, which is about ten times bigger than the displacement of the fiber spool in the x-axis direction [Fig. 5(b)]. The displacement of the bottom fixed face is the minimum, which is an order of magnitude of 10^{−9} mm. When 1 g acceleration is loaded along the x-axis direction, the displacements of the fiber spool in the z-axis direction [Fig. 6(a)] and the x-axis direction [Fig. 6(b)] are an order of magnitude of 10^{−6} mm.
The composition of the material affects vibration sensitivities. Jiang’s research compared the vibration sensitivities of bare fibers coated with polyimide and bare fibers under the same conditions, finding that the latter has better sensitivity [17]. In the case of ultra-narrow-linewidth fiber-stabilized lasers, a general method for securing the fiber is tightly winding it on the surface of a fiber spool with a certain winding tension. This tension also affects vibration sensitivities. Jiang [17], Huang et al. [18], and Hu et al. [22] have conducted studies on this. Jiang’s research, for example, shows that the vibration sensitivity deteriorates twice as much for a winding tension of 0.1 N compared to 0.75 N. However, increasing the winding tension further will also lead to a deterioration in vibration sensitivity [17].
Finally, we also simulate and calculate the vibration modes of the fiber spool, and estimate its first six orders eigenfrequencies are about in the range of 3,500 Hz to 6,800 Hz. The vibration shapes of the first six orders are shown in Fig. 7. The first-order vibration shape shows that the fiber spool has a stretching vibration with the z-axis as the center. For the second-order vibration shape and the third-order vibration shape, the fiber spool swings left and right along the yz plane and xz plane, respectively. The fourth-order vibration shape and the fifth-order vibration shape show that the fiber spool undergoes rotational vibration with the y-axis as the center and the x-axis as the center, respectively. For the sixth-order vibration shapes, the fiber spool undergoes torsional motion along the xz plane.
In this work, taking a self-designed fiber spool as an example, we optimize its vibration sensitivity by using a multi-object orthogonal experimental design approach. According to a test of 4 factors and 4 levels, we select an L_{16} (4^{4}) orthogonal table to design an orthogonal experiment. The vibration sensitivities are simulated and analyzed by the finite element approach for four geometric parameters. The vibration sensitivities along the axial and radial directions are determined as single objects. The normalized sums of the vibration sensitivities are selected as the comprehensive object. Based on the range analysis of the single objects and the comprehensive objects, we obtain the influence levels of the four geometric parameters on the object. Through the comprehensive balance method and the comprehensive evaluation method, the optimal parameter combination is determined. According to comparing the fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser, A3B1C2D1 is determined as the final optimal parameter combination, which can achieve the fiber spool with 10^{−12}/g magnitude vibration sensitivity. To our knowledge, the orthogonal experimental design method is being used for the first time to optimize the vibration sensitivity of the fiber spool and to confirm its efficacy. This work aims to provide an approach for vibration sensitivities optimization of a fiber spool.
We would like to acknowledge the contribution to this paper from X. Zhang of Northwestern Polytechnical University.
Partially supported by the Youth Innovation Promotion Association of the Chinese Academy of Sciences (Grant No. 1188000XGJ); The Chinese National Natural Science Foundation (Grant No. 11903041); The Young Innovative Talents of the National Time Service Center of the Chinese Academy of Sciences (Grant No. Y917SC1).
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
All data generated or analyzed during this study are included in this published article.
TABLE 1 Material properties of the fiber spool
Material Properties | Elastic Modulus (GPa) | Poisson Ratio (%) | Density (kg/m^{3}) |
---|---|---|---|
Titanium | 105 | 0.37 | 4,500 |
TABLE 2 Factors and levels of orthogonal test
Level | Factors (mm) | |||
---|---|---|---|---|
A (L_{1}) | B (H_{1}) | C (L_{2}) | D (L_{3}) | |
1 | 30.4 | 8.0 | 14.6 | 18 |
2 | 30.6 | 8.2 | 14.8 | 20 |
3 | 30.8 | 8.4 | 15.0 | 22 |
4 | 31.0 | 8.6 | 15.2 | 24 |
TABLE 3 Orthogonal table of experiment L_{16} (4^{4}) and test data
Level | A | B | C | D | S_{z} | S_{x} (S_{y}) | S_{1} | S_{2} |
---|---|---|---|---|---|---|---|---|
L_{1}/mm | H_{1}/mm | L_{2}/mm | L_{3}/mm | 10−11/g | 10−12/g | - | - | |
1 | 30.4 | 8.0 | 14.6 | 18 | 6.8 | 13 | 30.2 | 27.5 |
2 | 30.4 | 8.2 | 14.8 | 20 | 10.6 | 50.3 | 51.7 | 49.5 |
3 | 30.4 | 8.4 | 15.0 | 22 | 13.3 | 0.1 | 56.37 | 49.3 |
4 | 30.4 | 8.6 | 15.2 | 24 | 18.8 | 0.1 | 80.0 | 70.0 |
5 | 30.6 | 8.0 | 14.8 | 22 | 5.7 | 60.4 | 32.0 | 33.2 |
6 | 30.6 | 8.2 | 14.6 | 24 | 14.6 | 23 | 65.1 | 58.9 |
7 | 30.6 | 8.4 | 15.2 | 18 | 7.3 | 85.4 | 42.3 | 44.4 |
8 | 30.6 | 8.6 | 15.0 | 20 | 7.9 | 13.9 | 35.0 | 31.8 |
9 | 30.8 | 8.0 | 15.0 | 24 | 2.5 | 117.6 | 26.1 | 33.0 |
10 | 30.8 | 8.2 | 15.2 | 22 | 5 | 15.6 | 22.8 | 21.3 |
11 | 30.8 | 8.4 | 14.6 | 20 | 5.1 | 145.1 | 41.1 | 48.4 |
12 | 30.8 | 8.6 | 14.8 | 18 | 0.2 | 45.4 | 6.2 | 9.4 |
13 | 31.0 | 8.0 | 15.2 | 20 | 4.1 | 0.7 | 16.9 | 14.8 |
14 | 31.0 | 8.2 | 15.0 | 18 | 8.8 | 66.6 | 46.2 | 46.1 |
15 | 31.0 | 8.4 | 14.8 | 24 | 3.3 | 40.6 | 18.9 | 20.0 |
16 | 31.0 | 8.6 | 14.6 | 22 | 3.1 | 121.2 | 29.2 | 36.0 |
TABLE 4 Results of the ranges analysis of S_{z}
Ranges Analysis | Factors (mm) | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Sums | V_{1-1} | 49.5 | 19.1 | 29.6 | 23.1 |
V_{2-1} | 35.5 | 39.0 | 19.8 | 27.7 | |
V_{3-1} | 12.8 | 29.0 | 32.5 | 27.1 | |
V_{4-1} | 19.3 | 30.0 | 35.2 | 39.2 | |
Means | V_{1-1-M} | 12.4 | 4.8 | 7.4 | 5.8 |
V_{2-1-M} | 8.9 | 9.8 | 5.0 | 6.9 | |
V_{3-1-M} | 3.2 | 7.3 | 8.1 | 6.8 | |
V_{4-1-M} | 4.8 | 7.5 | 8.8 | 9.8 | |
Ranges | R_{1} | 9.2 | 5.0 | 3.8 | 3 |
TABLE 5 Results of the ranges analysis of S_{x} (S_{y})
Ranges Analysis | Factors (mm) | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Sums | V_{1-2} | 63.5 | 191.7 | 302.3 | 210.4 |
V_{2-2} | 182.7 | 155.5 | 196.7 | 210 | |
V_{3-2} | 323.7 | 271.2 | 198.2 | 197.3 | |
V_{4-2} | 229.1 | 180.6 | 101.8 | 181.3 | |
Means | V_{1-2-M} | 15.9 | 47.9 | 75.6 | 52.6 |
V_{2-2-M} | 45.7 | 38.9 | 49.2 | 52.5 | |
V_{3-2-M} | 80.9 | 67.8 | 49.6 | 49.3 | |
V_{4-2-M} | 57.3 | 45.2 | 25.5 | 45.3 | |
Ranges | R_{2} | 65 | 28.9 | 50.1 | 7.3 |
TABLE 6 Results of the ranges analysis of the normalized sum S_{1} of vibration sensitivities
Ranges Analysis | Factors (mm) | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Sums | V_{1-S}1 | 218.3 | 105.2 | 165.6 | 124.9 |
V_{2-S}1 | 174.4 | 185.8 | 108.8 | 144.7 | |
V_{3-S}1 | 96.2 | 158.7 | 163.7 | 140.4 | |
V_{4-S}1 | 111.2 | 150.4 | 162.0 | 190.1 | |
Means | V_{1-S}1_{-M} | 54.6 | 26.3 | 41.4 | 31.2 |
V_{2-S}1_{-M} | 43.6 | 46.5 | 27.2 | 36.2 | |
V_{3-S}1_{-M} | 24.1 | 39.7 | 40.9 | 35.1 | |
V_{4-S}1_{-M} | 27.8 | 37.6 | 40.5 | 47.5 | |
Ranges | R_{S}1 | 30.5 | 20.2 | 14.2 | 16.3 |
TABLE 7 Results of the ranges analysis of the normalized sum S_{2} of vibration sensitivities
Ranges Analysis | Factors (mm) | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Sums | V_{1-S}2 | 196.3 | 108.5 | 170.8 | 127.4 |
V_{2-S}2 | 168.3 | 175.8 | 112.1 | 144.5 | |
V_{3-S}2 | 112.1 | 162.1 | 160.2 | 139.8 | |
V_{4-S}2 | 116.9 | 147.2 | 150.5 | 181.9 | |
Means | V_{1-S}2_{-M} | 49.1 | 27.1 | 42.7 | 31.9 |
V_{2-S}2_{-M} | 42.1 | 44.0 | 28.0 | 36.1 | |
V_{3-S}2_{-M} | 28.0 | 40.5 | 40.1 | 35.0 | |
V_{4-S}2_{-M} | 29.2 | 36.8 | 37.6 | 45.5 | |
Ranges | R_{S}2 | 21.1 | 16.9 | 14.7 | 13.6 |
TABLE 8 Fractional frequency stability of the ultra-narrow-linewidth fiber-stabilized laser based on the optimal schemes
No. | A | B | C | D | S_{z} | S_{x} (S_{y}) | Frequency Stability |
---|---|---|---|---|---|---|---|
L_{1}/mm | H_{1}/mm | L_{2}/mm | L_{3}/mm | 10−11/g | 10−12/g | ||
A3B1C2D1 | 30.8 | 8.0 | 14.8 | 18 | 0.6 | 4.3 | 0.7 × 10−17 |
A3B1C4D1 | 30.8 | 8.0 | 15.2 | 18 | 1.4 | 9.8 | 1.7 × 10−^{17} |