Open Access
Add to BibSonomy
Abstract
Physical parameters of a spectral beam combining (SBC) system for multiple Yb-doped fiber lasers (YDFLs) were identified and numerically analyzed to obtain the optimal beam quality and the combining efficiency. We proposed an optimal range of the parameters that can be utilized in SBC systems. For a practical SBC system composed of a multi-layer dielectric grating and a transform mirror, we systematically varied input laser parameters such as the incident angle, beam diameter, laser linewidth, spectral spacing, number of beams, and their spatial separation. Characteristics of diffracted beams by the SBC system were numerically analyzed using a Fourier modal method (FMM). The beam quality M2 and the combining efficiency, η, were optimized by varying the laser beam parameters. We found that M2 and η were most affected by the angle of incidence and the laser linewidth, respectively. We presented the optimal parameters for three, five, and seven linear beam array SBCs along with a range of allowed parameters that could be used in the laser power scaling.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Beam combining is one of the efficient methods to achieve laser power scaling-up by spatially merging multiple input laser beams into a single beam using either diffractive or refractive optical elements [1,2]. Compared with the master oscillator power amplifier (MOPA) method, the beam combining has clear advantages of avoiding thermal issues and obviating nonlinear optical effects within amplifier chains to provide a consistent beam quality [3–6]. Beam combining methods can be divided into coherent beam combining (CBC) [7–10] and incoherent beam combining techniques [11,12]. Among the incoherent techniques, spectral beam combining (SBC) has been the most actively studied in recent years and widely implemented in the fields. In SBC systems, lasers oscillating at neighboring wavelengths are spatially combined with optical elements such as a prism [13,14], spectral filter [15,16], and diffraction grating [17–19]. The SBC technique can substantially reduce the heavy burdens of elaborated electro-optic control of phases of individual input lasers in CBC [20–22]. In principle, SBC is known to provide the output beam quality, M2, close to the diffraction limit and a high beam combining efficiency, η, as well as high-power handling capability [23].
In experiments, multi-kilowatt (kW) level attempts have been reported in SBC systems combining high power Yb-doped fiber lasers (YDFLs) [24–26]. The SBC system based on an edge filter for two YDFLs at λ=1070 and 1090 nm with a spectral linewidth of Δλ∼4 nm showed an output power of 10.25 kW and η of ∼97% [24]. Yet, its output beam quality M2 was 12. Adopting a diffraction grating with a groove density of 960 lines/mm, M2 significantly reduced to 4.3, and the output power of 8.2 kW has been obtained with η=97% for four YDFLs near λ=1050 nm with a spectral linewidth of Δλ∼0.9 nm [25]. SBC technology was further developed to combine 96 input YDFLs producing an output power over 30 kW with an η of 94.4% and M2 of 1.6 [26]. The diffraction grating has been a critical component in recent SBC systems, and continuous improvements in η and M2 have been reported in experimental studies [27–30], which revealed the importance of optimization of input laser parameters as well as geometrical alignment parameters related to the grating.
In contrast to these notable experimental achievements in recent years, only a few systematic numerical analyses have been reported, especially updating optimal laser and optical parameters in SBC technologies. In a prior report, the lens aberration has been numerically analyzed to understand its impacts on M2, the diffraction direction, and the number of lasers that can be combined [31]. A statistical study has been reported to find the relationship between M2 and the pointing deviation [32,33]. An analytic relation between the beam quality M2 and the spectral linewidth Δλ has been proposed for an ideal case where only an infinite diffraction grating was assumed without using other optical components [34]. However, these numerical analyses have not sufficiently included the elements and experimental conditions in recent SBC systems. Therefore, systematic numerical studies of SBC parameters became indispensable for further power scale-up.
This paper thoroughly analyzed how to optimize the beam combining efficiency η and the beam quality M2 by identifying essential physical parameters in a practical SBC system. The parameters are the number of input laser beams (N), the input laser spectral linewidth (Δλ), the spectral separation of the input beams (dλ), the spatial separation of the input beams (D), the spectral spacing (λS), the incident angles of input lasers about the grating (the vertical angle θv, and the horizontal angle θh), and the laser beam diameter incident on the grating (dB). We varied them systematically to find an optimal condition for simultaneously maximizing η and minimizing M2. Our analyses were based on the current SBC systems composed of laser collimators, a transform mirror, and a dielectric grating [26,35–37], which can be readily applied in the power scaling of YDFLs.
2. Defining spectral beam combining (SBC) parameters for numerical analysis method
We schematically showed the SBC system used in our numerical study in Fig. 1(a). Multiple laser beams of number N have an identical spectral linewidth Δλ. Here we set the center laser wavelength λ0 = 1064.0 nm, the common wavelength in conventional YDFLs. The input lasers were fed through a fiber optic collimator linear array with a spatial separation D. The laser beam was assumed to be an ideal Gaussian beam with M2 = 1.0. Its waist was 23 µm, the mode field diameter of a commonly used YDF (for instance, LMA-YDF-25/400 from Nufern). These values represent recent high-power fiber laser applications [38–40]. Lenses have been used to collimate and collect the beams in early SBC systems [34,35]. Still, they have suffered from thermal distortion in high-power lasers [41] and in current SBC systems, transform mirror [42] has been widely adopted, obviating thermal issues. The beam array is then collimated using a transform mirror and spatially combined at the grating. This study assumed an off-axis parabolic mirror (such as #35–609, Edmund Optics), which has an aspherical shape to reduce aberrations [43]. We assumed the transform mirror had a reflectivity of 100% at the laser wavelengths, producing a collimated beam with a diameter, dB, at the grating surface. The schematic diagram of the off-axis parabolic transform mirror is shown in Fig. 1(b), along with the collimated beams. The off-axis and focal lengths were 138 mm and 30 mm, respectively. A multi-layer dielectric grating was assumed to have a groove density of 1740 lines/mm, showing a high diffraction efficiency near λ=1060 nm [44]. The grating is schematically shown in Fig. 1(c), which has a Littrow configuration [45], and the diffraction order of −1 is used as the output. Since the transform mirror produces a collimated beam at the grating surface, the beam characteristics do not depend significantly on the distance after the grating. Therefore, we estimated the output beam combining efficiency η and the beam quality M2 immediately after the diffraction grating in the analyses.
Fig. 1. (a) The schematic diagram of an SBC system in this study. Each laser beam has a spectral linewidth Δλ. The center wavelength is λ0 = 1064.0 nm, the adjacent beam has the spectral separation of ± dλ, and the lasers have an equal spectral spacing λS. The lasers form a 1-dimensional array with a D+ and D- spatial separation for longer and shorter wavelengths. A transform mirror merged the input beams with a beam diameter of dB on a dielectric grating. The incident direction is characterized by the vertical angle θv and the horizontal angle θh. The output has the beam combining efficiency ηC and the beam quality M2. (b) The transform mirror has an off-axis length and a focal length of 130 mm and 30 mm, respectively. (c)The grating has a groove density d = 1740 lines/mm, duty cycle = 0.3, depth = 630 nm, and matched layer thickness = 450 nm. It has 15 alternating layers with nH = 1.87, nL = 1.44. The incident beam propagates along the z-axis and makes a horizontal angle θh with the grating. The diffracted beam has the order of −1. in the x-z plane, both the incident and diffracted beams have the same θh. The incident beam and the diffracted beam make a vertical angle θv with the z-direction.
The parameters used in our analyses and their ranges are summarized in Table 1, which covers most experimental SBC systems in Fig. 1. The dimensions of the optical elements such as the mirror and the grating were designed sufficiently large to adopt the multiple laser beams fully.
Table 1. SBC parameters and their corresponding ranges
This study used a commercial diffraction analysis program, VirtualLab fusion [46], to numerically analyze the light propagation in free space. The light diffraction by the grating and free space laser propagation were calculated using a Fourier modal method (FMM), which is based on the rigorous coupled-wave analysis (RCWA) [47]. After expanding the electric field and magnetic field into the Floquet-Fourier series and expanding the permittivity and magnetic permeability in the medium into the Fourier series, the FMM solves the matrix eigenvalue problem to calculate the output field [48] of the light in the given optical conditions.
3. Parametric analyses for a single laser beam diffraction
3.1 Optimization of incident angles θv, and θh
We started the analyses with a single laser beam with the center wavelength λ0 = 1064.0 nm, input M2 = 1, and the beam diameter at the gating dB = 1 mm. We assumed the laser was incident on the grating as shown in Fig. 1(c) and tried to understand the impacts of the incident angle (θv and θh) on the diffraction efficiency (ηD) and the beam quality (M2) of the diffracted beam.
In conventional diffraction gratings, the vertical angle is usually θv = 0°, and the horizontal angle θh or the Littrow angle is found approximately by the grating equation for a given incident angle θi
where, m is the diffraction order, λ is the light wavelength, d is the grating groove density 1740 lines/mm, and θ is an incident beam divergence angle. [49]. However, in SBC applications, the vertical angle θv of the input beam is kept at a non-zero value to separate the diffracted output beam at the diffraction order m=−1 from the input beam [34]. See the right of Fig. 1(c). SBC systems also require the Littrow configuration such that θh for the incident beam and m=−1 order diffracted beam is the same. See the left of Fig. 1(c). Therefore, we started with optimization of θv and θh for a single laser beam as the first step in our numerical analyses. The diffraction efficiency ηD is defined as where, Δφ is the optical phase thickness determined by the groove depth and the refractive index of grating [50]. We used a numerical tool, a diffraction order analyzer of the VirtualLab fusion, and varied θv from 0 to 90°.For the grating parameters identified in Fig. 1(c), we varied the vertical angle, θv, to find the corresponding diffraction efficiency and the Littrow angle. And the results are summarized in Fig. 2(a). For high diffraction efficiency ηD > 99%, θv should be less than 20°, and the corresponding Littrow angle should be in a range 67°<θh < 69°. See the red vertical line in Fig. 2(a). For a moderate requirement of ηD > 95%, the upper limits of θv and θh are extended to ∼38° and ∼73°, respectively. See the green vertical line in Fig. 2(a).
Fig. 2. (a) The Littrow angle (θh) and diffraction efficiency (ηD) as a function of the incident vertical angle θv of a single input beam, (b) beam quality M2 as a function of the incident vertical angle θv of a single input beam. The input beam has a wavelength of λ0 = 1064.0 nm and beam diameter dB =1 mm at the grating.
The beam quality M2 of the diffracted beam with the order of m=−1 was then calculated as a function of θv. The results are plotted in Fig. 2(b). We assumed a beam diameter dB = 1mm with M2 = 1 in both x and y directions for the incident laser beam. We focused on the case of ηD > 99% and found that M2 rapidly increased for θv > 60°. In a range of 0<θv < 60°, M2 reached its maximum of 1.44 at θv ∼42°. From Fig. 2, we concluded that θv < 20° is required to obtain both the high ηD > 99% and low M2<1.2, which is consistent with experimental SBC systems previously reported [27,28,51].
3.2 Optimization of the input beam diameter (dB) at the grating
In previous experimental SBC systems, dB at the grating has been in a range of a few mm, yet its value varied among research groups [28,51–54]. In this study, we implemented a transform mirror shown in Fig. 1(b) to systematically change dB up to 100 mm by adjusting its focal length and the distance between the mirror and the grating. We assumed an ideal linear-polarized Gaussian beam at λ0 = 1064.0 nm with M2 = 1 was reflected by the transform mirror and then incident on the grating. See Fig. 1(a).
Here we set the incident angles as θv = 12.4° and θh = 68.2° to provide both a high ηD and a low M2 in the diffraction process at the grating, as discussed in the above section.
In Fig. 3, we plotted the diffraction efficiency ηD as a function of dB. As dB increased from 1 to 100 mm, no significant variation was found in ηD, but it slightly decreased with increasing dB, as shown in Fig. 3(a). In contrast, M2 abruptly increased in 8 mm < dB < 30 mm. As dB further increased, M2 was maintained at ∼1.35 in the x-direction and ∼1.25 in the y-direction. See Fig. 3(b). In Fig. 3, we found dB < 15 mm is an optimal range to keep both ηD high and M2 low, which is consistent with prior experimental reports [51,52,55].
Fig. 3. (a) Diffraction efficiency ηD and (b) beam quality M2 versus the input beam diameter dB at the grating. We assumed an ideal Gaussian beam at λ0 = 1064.0 nm with M2 = 1 and the incident angles as θh = 68.2° and θv =12.4° at the grating.
3.3 Optimization of the input laser linewidth (Δλ)
In the above sections 3.1 and 3.2, we assumed an ideal case such that the input laser has a delta function-like distribution in the spectral domain. However, this assumption is not applicable in practical SBC systems due to a finite Δλ of input fiber lasers. Since the grating is a dispersive element, the finite linewidth Δλ can directly affect the SBC performances. We investigated the impacts of Δλ on ηD and M2 assuming an ideal Gaussian input beam with λ0 = 1064.0 nm and M2 = 1. We assumed dB = 8 mm, θv = 12.4° and θh = 68.2° to meet the requirements in Fig. 2 and Fig. 3.
We varied Δλ and ηD was almost independent, as shown in Fig. 4(a). However, M2 monotonically increased with Δλ in contrast in Fig. 4(b). An analytic relation between the M2 and Δλ has been suggested [34] as Eq. (3). But only a single infinite diffraction grating was assumed without other optical components.
Fig. 4. (a) Diffraction efficiency ηD and (b) beam quality M2 plotted as a function of the input laser spectral linewidth (Δλ). We assumed θv = 12.4°, θh = 68.2° dB = 8 mm, and for the incident laser λ0 = 1064.0 nm, M2 = 1.
4. Parametric analyses for multiple laser beam SBC system
4.1 Spatial and spectral separation (D and dλ) in a linear beam array
When multiple beams are incident on a grating in an SBC system, one of the fundamental questions is the geometric and spectral configuration of the incident beams to optimize the output efficiency and the beam quality. In most prior experiments, incident beams have been arranged in a linear array [62,63]. In our study, we also assumed a linear array of Gaussian beams, and we investigated three cases of beam-combining: (1) three-beam N = 3, (2) five-beam N = 5, and (3) seven-beam N = 7, as shown in the top of Fig. 5(a). The grating combined these incident beams and the transforming mirror as in Fig. 1(a), satisfying the requirements for θv, θh, Δλ, and dB discussed in Section 3.
Fig. 5. (a) Configuration of multiple beams in the spatial domain (upper row), and in the spectral domain (lower row). (b) Optimization of beam quality M2 by varying the spatial separation (D) and spectral separation (dλ).
In addition to these, we found that the linear beam array should make an optimal tilt angle, θt, with respect to the x-axis, to satisfy the primary requirement at the grating, θv = 12.4° for each beam. After a series of trials, we obtained optimal θt ∼9° such that θt is greater or smaller than the optimal value M2 increases abruptly. This might be attributed to the finite size of the grating. We also found that the spatial separation among beams should be optimized individually to minimize M2 for the same attribute. Therefore, we defined D- and D+ as the spatial separation of beams in the shorter wavelengths and the longer wavelengths, respectively. The central beam (λ0) position was set to be the origin. See the top of Fig. 5(a).
We assumed a uniform λS such that all the beams are separated equally in the wavelength domain, as shown at the bottom of Fig. 5(a). To designate a particular beam, we further defined the spectral separation, dλ, between the central beam (λ0) and the particular beam. We have either dλ>0 for a beam at a longer wavelength or dλ<0 for a beam at a shorter wavelength.
In the analyses, assumed dB = 8 mm, for all the combining beams to minimize its impacts as shown in Fig. 3. We also assumed Δλ=0, ideal monochromatic light for each beam. This assumption can be justified since Fig. 4 implies a single laser beam with Δλ<10 GHz would not significantly affect η and M2, which is consistent with experimental reports [59]. Further investigation of how the spectral linewidth would impact SBC is being pursued by the authors.
We numerically investigated the impacts of dλ and D on M2 and ηOut. And the results are summarized in Fig. 5(b) and Table 2. Here we defined ηOut = POut/Pin. We optimized those parameters to achieve the maximum ηOut and the minimum M2. It is noted that the behavior of beams in the shorter wavelengths, dλ<0, differed from those in the longer wavelengths λ0 + dλ as increasing dλ.
Table 2. Three cases of beam combining
In Table 2, we summarized the beam quality M2 and output efficiency ηOut for three cases of beam combining.
4.2 Multiple beam analysis for the spectral spacing (λS) and spectral linewidth (Δλ)
This section investigated the impact of the spectral spacing λS on SBC performances. Here we defined the combining efficiency ηC as the average output efficiency ηOut for each input laser. The location of side beams was arranged as shown in Fig. 5 and Table 2. In addition, we investigated the impacts of the linewidth Δλ=10 GHz. The results are summarized in Fig. 6. ηC decreased for λS > 5 nm with increasing N, irrespective of Δλ. Significant degradation in M2 was observed for λS > 3 nm. For Δλ=10 GHz, the degradation was higher than 0.4, and the authors found that the spectral linewidth should be further reduced SBC systems with N > 10.
Fig. 6. Combining efficiency ηC and beam quality M2 as a function of the spectral spacing (λS). The open symbols represent the delta function-like spectral distribution and solid symbols for Δλ =10 GHz. The number of beams is N, the center beam wavelength λ0 = 1064.0 nm, and the input M2 = 1. θh =68.2° and θv = 12.4°. The input beam diameter dB = 8 mm.
In Fig. 7, we summarized the parameter design diagram for an SBC system (N≤7) that satisfy ηC >99% and M2 <1.5.
Fig. 7. The suitable range of essential parameters in SBC for the combining efficiency ηC > 99% and beam quality M2<1.5.
In practical SBC systems, the oblique angle of incidence of the beams at the grating distorts the symmetric Gaussian beam to make it elliptic. This asymmetric beam shape could be one of the attributes of beam quality degradation. As the spectral and spatial separation of each beam increase from the central beam, the diffraction angle difference gets larger for each wavelength according to the grating equation [49], which also contributes to beam quality degradation [49]. A significant angular spread is caused by the lights spectrally separated from the central wavelength [34], which can serve as an attribute to beam quality degradation.
5. Conclusion
In a spectral beam combining (SBC) system consisting of a transform mirror and a Littrow grating, the efficiency η and beam quality M2 were systematically analyzed by varying the key parameters. We investigated linear arrays of three, five, and seven Yb-doped fiber lasers numerically to assess the impacts of laser parameters and geometrical parameters and find the most critical factors that affect η and M2. We found that individual laser beams should keep the vertical angle θv <20° to keep a high η and a low M2. To meet the requirement, the linear array of the input laser beams was aligned with a tilted angle of ∼9° in the incident plane. Spectral linewidth (Δλ) of the input laser was the main contributor to deteriorate M2, and Δλ<10 GHz was required. Spectral spacing (λS) between adjacent lasers was found to be less critical, but λS <7 nm was optimal. The optimal range of the beam diameter (dB) incident on the grating was dB < 15 mm. Our numerical analyses can guide the power scaling of lasers using SBC and can be updated for an increased number of lasers.
Funding
National Research Foundation of Korea (2019R1A2C2011293).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. S. J. Augst, J. K. Ranka, T. Y. Fan, and A. Sanchez, “Beam combining of ytterbium fiber amplifiers (Invited),” J. Opt. Soc. Am. B 24(8), 1707–1715 (2007). [CrossRef]
2. T. Y. Fan, “Laser Beam Combining for High-Power High-Radiance Sources,” IEEE J. Quantum Electron. 11(3), 567–577 (2005). [CrossRef]
3. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001). [CrossRef]
4. A. Flores, C. Robin, A. Lanari, and I. Dajani, “Pseudo-random binary sequence phase modulation for narrow linewidth, kilowatt, monolithic fiber amplifiers,” Opt. Express 22(15), 17735–17744 (2014). [CrossRef]
5. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011). [CrossRef]
6. C. Jauregui, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Passive mitigation strategies for mode instabilities in high-power fiber laser systems,” Opt. Express 21(16), 19375–19386 (2013). [CrossRef]
7. H. Tünnermann, J. H. Pöld, J. Neumann, D. Kracht, B. Willke, and P. Weßels, “Beam quality and noise properties of coherently combined ytterbium doped single frequency fiber amplifiers,” Opt. Express 19(20), 19600–19606 (2011). [CrossRef]
8. C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett. 36(14), 2686–2688 (2011). [CrossRef]
9. P. A. Thielen, J. G. Ho, D. A. Burchman, G. D. Goodno, J. E. Rothenberg, M. G. Wickham, A. Flores, C. A. Lu, B. Pulford, C. Robin, A. D. Sanchez, D. Hult, and K. B. Rowland, “Two-dimensional diffractive coherent combining of 15 fiber amplifiers into a 600 W beam,” Opt. Lett. 37(18), 3741–3743 (2012). [CrossRef]
10. S. M. Redmond, D. J. Ripin, C. X. Yu, S. J. Augst, T. Y. Fan, P. A. Thielen, J. E. Rothenberg, and G. D. Goodno, “Diffractive coherent combining of a 2.5 kW fiber laser array into a 1.9 kW Gaussian beam,” Opt. Lett. 37(14), 2832–2834 (2012). [CrossRef]
11. G. Yang, L. Liu, Z. Jiang, J. Guo, and T. Wang, “Incoherent beam combining based on the momentum SPGD algorithm,” Opt. Laser Technol. 101, 372–378 (2018). [CrossRef]
12. C. Wirth, O. Schmidt, I. Tsybin, T. Schreiber, T. Peschel, F. Brückner, T. Clausnitzer, J. Limpert, R. Eberhardt, and A. Tünnermann, “2 kW incoherent beam combining of four narrow-linewidth photonic crystal fiber amplifiers,” Opt. Express 17(3), 1178–1183 (2009). [CrossRef]
13. Lu C. Andrew, “Beam combining of fiber lasers and vertical external cavity surface emitting lasers using volume Bragg gratings,” The University of New Mexico, (2014).
14. Tobias Koenning, K. Alegria, Z. Wang, A. Segref, D. Stapleton, W. Faßbender, M. Flament, K. Rotter, A. Noeske, and J. Biesenbach, “Macro-channel cooled high power fiber coupled diode lasers exceeding 1.2 kW of output power,” Proc. SPIE 7918, 79180E (2011). [CrossRef]
15. C. Fan, J. Ma, R. Zhu, Q. Yuan, W. Zhou, J. Su, J. Xu, and S. Pan, “Coupling efficiency model for spectral beam combining of high-power fiber lasers calculated from spectrum,” Appl. Opt. 56(10), 2574–2579 (2017). [CrossRef]
16. O. Schmidt, C. Wirth, D. Nodop, J. Limpert, T. Schreiber, T. Peschel, R. Eberhardt, and A. Tünnermann, “Spectral beam combination of fiber amplified ns-pulses by means of interference filters,” Opt. Express 17(25), 22974–22982 (2009). [CrossRef]
17. C. E. Hamilton, S. C. Tidwell, D. Meekhof, J. Seamans, N. Gitkind, and D. D. Lowenthal, “High-power laser source with spectrally beam-combined diode laser bars,” Proc. SPIE 5336, 1–10 (2004). [CrossRef]
18. Z. Zhu, M. Jiang, S. Cheng, Y. Hui, H. Lei, and Q. Li, “Narrow linewidth operation of a spectral beam combined diode laser bar,” Appl. Opt. 55(12), 3294–3296 (2016). [CrossRef]
19. H. Yan, Y. Ma, Y. Sun, L. Zhao, F. Tian, K. Zhang, C. Tang, X. Wang, W. Peng, T. Li, J. Li, S. Wang, and W. Zhang, “Scalable hybrid beam combining of kilowatt fiber amplifiers into a 5-kW beam,” Opt. Commun. 397, 95–99 (2017). [CrossRef]
20. S. J. Augst, T. Y. Fan, and A. Sanchez, “Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers,” Opt. Lett. 29(5), 474–476 (2004). [CrossRef]
21. E. C. Cheung, J. G. Ho, G.D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. 33(4), 354–356 (2008). [CrossRef]
22. J. Song, Y. Li, D. Che, J. Guo, and T. Wang, “Coherent beam combining based on the SPGD algorithm with a momentum term,” Optik 202, 163650 (2020). [CrossRef]
23. E. J. Bochove, “Theory of spectral beam combining of fiber lasers,” IEEE J. Quantum Electron. 38(5), 432–445 (2002). [CrossRef]
24. F. Chen, J. Zhang, J. Ma, C. Wei, R. Zhu, B. Han, and Q. Wang, “Beam quality analysis and optimization for 10 kW-level spectral beam combination system,” Opt. Commun. 444, 45–55 (2019). [CrossRef]
25. C. Wirth, O. Schmidt, I. Tsybin, T. Schreiber, R. Eberhardt, J. Limpert, A. Tünnermann, K. Ludewigt, M. Gowin, and E. T. Have, “High average power spectral beam combining of four fiber amplifiers to 8.2 kW,” Opt. Lett. 36(16), 3118–3120 (2011). [CrossRef]
26. E. Honea, R. S. Afzal, M. Savage-Leuchs, J. Henrie, K. Brar, N. Kurz, D. Jander, N. Gitkind, D. Hu, C. Robin, A. M. Jones, R. Kasinadhuni, and R. Humphreys, “Advances in fiber laser spectral beam combining for power scaling,” Proc SPIE 9730, 97300Y (2016). [CrossRef]
27. M. Strecker, M. Plötner, F. Stutzki, T. Walbaum, S. Ehrhardt, T. Benkenstein, U. Zeitner, T. Schreiber, R. Eberhardt, A. Tünnermann, U. Stuhr, M. Jung, and K. Ludewigt, “Highly efficient dual-grating 3-channel spectral beam combining of narrow-linewidth monolithic cw Yb-doped fiber amplifiers up to 5.5 kW,” Proc. SPIE 10897, 108970E (2019). [CrossRef]
28. Y. Zheng, Y. Yang, J. Wang, M. Hu, G. Liu, X. Zhao, X. Chen, K. Liu, C. Zhao, B. He, and J. Zhao, “10.8 kW spectral beam combination of eight all-fiber superfluorescent sources and their dispersion compensation,” Opt. Express 24(11), 12063–12071 (2016). [CrossRef]
29. A. Sevian, O. Andrusyak, I. Ciapurin, V. Smirnov, G. Venus, and L. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33(4), 384–386 (2008). [CrossRef]
30. M. Haas, S. Rauch, S. Nagel, L. Irmler, T. Dekorsy, and H. Zimer, “Thin-film filter wavelength-stabilized, grating combined, high-brightness kW-class direct diode laser,” Opt. Express 25(15), 17657–17670 (2017). [CrossRef]
31. Y. Zhang and B. Zhang, “Analysis of beam quality for the laser beams after spectral beam combining,” Optik 121(13), 1236–1242 (2010). [CrossRef]
32. G. Bai, H. Shen, Y. Yang, X. Zhao, J. Zhang, H. Zhang, Y. Qi, B. He, and J. Zhou, “Theoretical analysis of beam quality degradation in spectral beam combining of fiber laser array with beam deviation,” Opt. Laser Technol. 105, 281–287 (2018). [CrossRef]
33. G. Bai, H. Shen, J. Zhang, X. Niu, M. Liu, Y. Yang, B. He, and J. Zhou, “Analysis of beam characteristic in the single and dual grating spectral beam combining of fiber laser array with pointing deviation,” J. Opt. Soc. Am. B 36(8), 2154–2159 (2019). [CrossRef]
34. T. H Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Quantum Electron. 13(3), 487–497 (2007). [CrossRef]
35. H. Meng, X. Ruan, W. Du, Z. Wang, F. Lei, J. Yu, and H. Tan, “Scaling the spectral beam combining channel by multiple diode laser stacks in an external cavity,” Laser Phys Lett. 14(4), 045811 (2017). [CrossRef]
36. E. Honea, R. S. Afzal, M. Savage-Leuchs, N. Gitkind, R. Humphreys, J. Henrie, K. Brar, and D. Jander, “Spectrally beam combined fiber lasers for high power, efficiency, and brightness,” Proc. SPIE 8601, 860115 (2013). [CrossRef]
37. Y. H. Cha, G. Lim, Y. H. Kim, J. S. Shin, and D. Y. Jeong, “300-W spectral beam combination of narrow-linewidth pulsed rod-type fiber lasers,” in Proc. 2015 Conf. on Lasers and Electro-Optics Pacific Rim (CLEO-PR) (2015).
38. Y. Ye, X. Xi, C. Shi, H. Zhang, B. Yang, X. Wang, P. Zhou, and X. Xu, “Experimental study of 5-kW high-stability monolithic fiber laser oscillator with or without external feedback,” IEEE Photonics J. 11(4), 1–8 (2019). [CrossRef]
39. B. Yang, C. Shi, H. Zhang, Q. Ye, H. Pi, R. Tao, X. Wang, P. Ma, J. Leng, and Z. Chen, “Monolithic fiber laser oscillator with record high power,” Laser Phys. Lett. 15(7), 075106 (2018). [CrossRef]
40. T. Liu, S. Chen, X. Qi, and J. Hou, “High-power transverse-mode-switchable all-fiber picosecond MOPA,” Opt. Express 24(24), 27821–27827 (2016). [CrossRef]
41. Z. Wu, S. You, M. Zhang, J. Pu, Y. Zhang, H. Xu, Q. Du, and Y. Huang, “Effect of thermal errors of a transform lens on beam properties in spectral beam combining systems,” Optik 219, 164994 (2020). [CrossRef]
42. Anping Liu and Kefeng Li, “Thermal lensing mitigation for high power laser processing,” Proc. SPIE 11679, 116790L (2021). [CrossRef]
43. J. Lv, M. Wang, H. Huan, and B. Zhang, “Fringe analysis with Hilbert transform and its application to the measurement of aspheric mirror,” Proc. SPIE 6723, 67231D (2007). [CrossRef]
44. I. Kim, S. So, J. Mun, K. H. Lee, J. H. Lee, T. Lee, and J. Rho, “Optical characterizations and thermal analyses of HfO2/SiO2 multilayered diffraction gratings for high-power continuous wave laser,” J. Phys. Photonics 2(2), 025004 (2020). [CrossRef]
45. T. Clausnitzer, T. Kämpfe, E. B. Kley, A. Tünnermann, A. Tishchenko, and O. Parriaux, “Investigation of the polarization-dependent diffraction of deep dielectric rectangular transmission gratings illuminated in Littrow mounting,” Appl. Opt. 46(6), 819–826 (2007). [CrossRef]
46. Wyrowski Photonics GmbH, “Physical optics simulation and design software,” Wyrowski VirtualLab Fusion, developed by Wyrowski Photonics GmbH, distributed by LightTrans International UG, Jena, Germany.
47. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71(7), 811–818 (1981). [CrossRef]
48. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14(10), 2758–2767 (1997). [CrossRef]
49. J. E. Harvey and R. N. Pfisterer, “Understanding diffraction grating behavior: including conical diffraction and Rayleigh anomalies from transmission gratings,” Opt. Eng. 58(08), 1 (2019). [CrossRef]
50. H.J. Cho, K.H. Lee, S.I. Kim, J.H. Lee, H.T. Kim, W.S. Kim, D.H. Kim, Y.S. Lee, S. Kim, T.Y. Kim, and C.K Hwangbo, “Analysis on Design and Fabrication of High-diffraction-efficiency Multilayer Dielectric Gratings,” Curr. Opt. Photonics 2(2), 125–133 (2018).
51. T. H. Loftus, A. Liu, P. R. Hoffman, A. M. Thomas, M. Norsen, R. Royse, and E. Honea, “522 W average power, spectrally beam-combined fiber laser with near-diffraction-limited beam quality,” Opt. Lett. 32(4), 349–351 (2007). [CrossRef]
52. S. Klingebiel, F. Röser, B. Ortaç, J. Limpert, and A. Tünnermann, “Spectral beam combining of Yb-doped fiber lasers with high efficiency,” J. Opt. Soc. Am. B 24(8), 1716–1720 (2007). [CrossRef]
53. Z. Liu, P. Ma, R. Su, R. Tao, Y. Ma, X. Wang, and P. Zhou, “High-power coherent beam polarization combination of fiber lasers: progress and prospect [Invited],” J. Opt. Soc. Am. B 34(3), A7–A14 (2017). [CrossRef]
54. M. Zwilich and B. Ewers, “Coherent beam combining of multipass thin-disk lasers with active phase control,” OSA Continuum 3(11), 3176–3186 (2020). [CrossRef]
55. S. J. Augst, A. K. Goyal, R. L. Aggarwal, T. Y. Fan, and A. Sanchez, “Wavelength beam combining of ytterbium fiber lasers,” Opt. Lett. 28(5), 331–333 (2003). [CrossRef]
56. I. Gursoy and S. Tunc, “The effects of M2 factor on steel materials,” Proc. SPIE. 11539, 14 (2020). [CrossRef]
57. R. T. Sean,Laser beam quality metrics (SPIE Press, 2013).
58. Guoqing Yang, Lisheng Liu, Zhenhua Jiang, Jin Guo, and TingFeng Wang, “The effect of beam quality factor for the laser beam propagation through turbulence,” Optik 156, 148–154 (2018). [CrossRef]
59. Z. Chang, Y. Wang, Y. Sun, W. Peng, W. Ke, Y. Ma, R. Zhu, and C. Tang, “1.5 kW polarization-maintained Yb-doped amplifier with 13 GHz linewidth by suppressing the self-pulsing and stimulated Brillouin scattering,” Appl. Opt. 58(23), 6419–6425 (2019). [CrossRef]
60. J. Zhang, H. Peng, X. Fu, Y. Liu, L. Qin, G. Miao, and L. Wang, “CW 50W/M2 = 10.9 diode laser source by spectral beam combining based on a transmission grating,” Opt. Express 21(3), 3627–3632 (2013). [CrossRef]
61. M. Haas, S. Rauch, S. Nagel, R. Beißwanger, T. Dekorsy, and H. Zimer, “Beam Quality Deterioration in Dense Wavelength Beam-Combined Broad-Area Diode Lasers,” IEEE J. Quantum Electron. 53(3), 1–11 (2017). [CrossRef]
62. V. Daneu, A. Sanchez, T. Y. Fan, H. K. Choi, G. W. Turner, and C. C. Cook, “Spectral beam combining of a broad-stripe diode laser array in an external cavity,” Opt. Lett. 25(6), 405–407 (2000). [CrossRef]
63. Q. Beilun, Y. Xiao, X. Tang, H. Peng, W. Yang, and Z. Chen, “Spectral interval compressing structure in spectral beam combination system for diode laser arrays,” Opt. Eng. 58(06), 1 (2019). [CrossRef]
