At present, the step-index fiber is taken as the gain medium for most high-power fiber laser systems. Without loss of generality, the HOMs excited from a step-index fiber is investigated in this study. It is considered that the coordinate z-axis is parallel to the geometrical axis of the fiber with the core radius a. In the weakly guiding approximation, the field distribution of the LP _nm -mode can be expressed as [52]. E n m ( r , ϕ ) = R n m ( r ) cos ( n ϕ ) , (1) where ( r , ϕ ) are the polar coordinates. The radial dependence of the approximately transverse field amplitude in Eq. 1 is defined as follows R n m ( r ) = { k 1 J n ( U m r a )         0 ≤ r ≤ a k 2 K n ( W m r a )                 r > a , (2) where k 1 J n ( U m ) = k 2 K n ( W m ) and J _n and K _n are the n-order Bessel function of the first kind and the modified Bessel function of the first kind, respectively. U _m and W _m are the solutions of the characteristic equations (53). J n ( U m ) U J n + 1 ( U m ) = K n ( W m ) W K n + 1 ( W m ) (3) and U 2 + W 2 = V (4)

The normalized frequency V is defined as V = 2 π a λ n core 2 − n clad 2 , (5) where n _core and n _clad are the core refractive index and cladding refractive index, respectively. The numerical aperture (NA) can be written as N A = n c o r e 2 − n c l a d 2 . The exemplary fiber that will be considered here has an ideal step-index profile with a core/inner-cladding diameter of 20/400 μm and an NA of 0.06.

The HOMs excited in the fiber are magnified 200 times by a large diameter collimator and then combined in the beam combiner system. It is assumed that a HOM beam array consists of seven beamlets located as z = 0, which are arranged in a tiled hexagonal architecture by coherent beam combining, as shown in Figure 1. The distance between the centers of neighboring sub-aperture is r ₀ and the diameter of the whole beam array is D. The optical field of each beamlet is E n m l . The electric field distribution of the HOM beam array with a hard aperture is expressed as E = A coe ∑ l = 1 N E n m l [ r 2 + r 0 2 + 2 r r 0 ⁡ cos ( ϕ − ϕ l ) ] × c i r c [ r 2 + R 2 + 2 r R ⁡ cos ( ϕ − ϕ l ) ] , (6) where ϕ l = π l / 3 . The circ(•) denotes the hard aperture truncated function with a diameter of R. The coefficient A _coe can be obtained according to the well-known relationship between power P and the electric field E n m l [54]. P = ∫ 0 2 π d ϕ ∫ 0 ∞ | E | 2 r d r , (7)

In the parabolic approximation, the electric field E satisfies the Maxwell wave equation (43). 2 i k ∂ E ∂ z = ∇ ⊥ 2 E + k 2 ( n 2 n 0 2 − 1 ) E , (8) where ∇ ⊥ 2 = ∂ 2 / ∂ x 2 + ∂ 2 / ∂ y 2 and n, and n ₀ are the refractive indices of the atmosphere with and without disturbance, respectively. k = 2 π / λ is the wave number related to the wavelength λ. According to the hydrodynamic equation, the atmospheric density ρ ₁ with disturbance caused by thermal blooming can be obtained [43]. ∂ ρ 1 ∂ t + v ⋅ ∇ ρ 1 = − γ − 1 c s 2 α I , (9) where v, γ, c _s, and α are the wind speed, specific heat, sound speed capacity ratio, and absorption coefficient in the atmosphere, respectively. The intensity I is given by I = | E | 2 ⁡ exp ( α z ) .

Based on Eqs 1–9, we designed a 4D computer code to simulate the time-dependent propagation of a HOM beam array propagating through the atmosphere by using the multiphase screen method and finite-difference method [43]. A lens with focus z _f = 5 km located at z = 0 is considered in this study. In the following calculations, the parameters are taken as a = 50 μm, R = 4.5 cm, λ = 1 . 064 μm , n ₀ = 1.00031, v = 2 m/s along x-axis, ρ ₀ = 1.30246 kg/m³, c _s = 340 m/s, α = 0.07/km, P = 1 kw, and N = 7, z = 5 km.

In this section, the propagation properties of the LP ₁₁ mode beam array propagating in free space are demonstrated. As we all know, the intensity distribution of the LP ₀₁ mode is circular symmetry, and the far field intensity distribution of the LP ₀₁ mode coherent beam array is comprises a central lobe with a number of side lobes. But for the LP ₁₁ mode, the intensity distribution is axial symmetry, and therefore, the arrangement of the LP ₁₁ mode has a significant impact on the focal intensity distributions.

The intensity distributions of the LP ₁₁ mode beam array with centrosymmetric arrangement are shown in Figure 2. It is assumed that the angle of the LP ₁₁ mode around the central beamlet in Figure 2A is set as θ = 0, and the different rotation angles for the initial beamlet arrangement are shown in Figures 2B–D. It can be seen that the beam shapes of the LP ₁₁ mode beam array at the receiver plane are quite different from those of the LP ₀₁ mode beam array. The beam shapes of the LP ₁₁ mode beam array are a radial spot beam array without a central lobe. As θ changes from 0 to 90°, the number of side lobes gradually changes from 6 to 12. By comparing the beam shapes at the initial plane, it is clearly seen that the focal intensity distributions are consistent with the first ring of the hexagonal mesh of the fiber laser array (see the red circle highlight in Figures 2A–D). These observations indicate that the desired beam shape of the focusing spots can be obtained by simply rotating the surrounding beamlets.

It can be clearly seen that different focal spots can be obtained by changing the initial arrangement of the LP ₁₁ mode beam array as mentioned in the Linear propagation of HOMs beam array section. Therefore, the impact of thermal blooming on the LP ₁₁ mode beam array can be quite different for different arrangements. In this section, based on the results in the Linear propagation of HOMs beam array section, the effects of thermal blooming on the special arrangements of the LP ₁₁ mode beam array are investigated in detail.

The intensity distributions of the LP ₁₁ mode beam array for centrosymmetric arrangement under the conditions of thermal blooming are shown in Figures 3, 4. From Figures 4I–L, it can be observed that the influence of thermal blooming on the LP ₁₁ mode beam array can be quite different for different rotation angles. In addition, the focal beam shapes are not symmetrical except for the arrangement of Figure 3A. The difference between Figure 3 and Figure 4 is that the initial central beamlets in Figure 4 are rotated by 90 degrees. It can be seen that the focal beam shapes under thermal blooming are quite different, although the focal beam shapes in free space are the same. The phenomena illustrates that the arrangement of the central beamlet has little influence on the focal beam shapes in free space but has a significant effect on the focal beam shapes under thermal blooming.

It is assumed that the directions of the central beamlet in Figures 3A–D are parallel to that of the wind and that in Figures 4A–D are vertical to that of the wind. Generally, the power of the bucket-based beam width is used to describe beam spreading and energy focusability, which is expressed as [55] ∫ 0 w η I r d r = η ∫ 0 + ∞ I r d r , where w η is the bucket half-width chosen. The beam width w 86.5 % is adopted in this study. On the other hand, the beam centroid position is changed due to the effect of thermal blooming, which is defined as [55] j ¯ = ∬ j I d x d y / ∬ I d x d y , where j = x and y. The center of the bucket is taken as ( x ¯ , y ¯ ) in the following calculations. The changes of the beam width at the target for different values of rotation angles are shown in Figure 5. It can be seen that the value of beam width w 86.5 % of the parallel direction is lower than that of the vertical direction. Thus, the beam focusability of the parallel direction is higher than that of the vertical direction. That means the thermal blooming becomes more severe for the vertical central beamlet arrangement, especially when θ = 20°. As the θ increases, the difference of the beam width w 86.5 % between parallel and vertical directions decreases. Thus, the laser energy focusability can be controlled simply by rotating the central beam.

Here, we choose four arrangement types of initial beamlets (see Figures 6A–D) to investigate the influence of transverse wind speed on the energy focusability. As can be seen from Figure 6E, the beam width decreases and becomes closer as the wind speed increases. The physical reason is that the absorbed energy in the propagation path is carried away more quickly as the wind speed increases. That is to say, increasing the transverse wind speed can help increase the energy focusability. In addition, the beam width of Figure 6A is the largest for different values of wind speed. Thus, the arrangement of Figure 6A should be avoided in order to improve the energy focusability.

In practical applications, it is difficult to obtain the pure LP ₀₁ mode even at relatively high conversion efficiency. Therefore, the case of the mixture of LP ₀₁ and LP ₁₁ modes is worth studying. Considering that the model superposition states comprise different admixtures of the LP ₀₁ and LP ₁₁ modes, the initial field can be expressed as E mix = ∑ l = 1 N { A L P 11 E 11 l [ r 2 + r 0 2 + 2 r r 0 ⁡ cos ( ϕ − ϕ l ) ] + 1 − A L P 11 E 01 l [ r 2 + r 0 2 + 2 r r 0 ⁡ cos ( ϕ − ϕ l ) ] } , (10) where A L P 11 is the power fraction of the LP ₁₁ mode and the value of A L P 11 is 0 ≤ A L P 11 ≤ 1 .

The intensity distributions of the mixed-mode beam array are shown in Figures 7–9. It can be seen from Figures 7–9 that in free space, as the content of the LP ₀₁ mode increases, the energy is gradually concentrated from the side lobes to the center lobe. That is to say, the energy distribution between the central lobe and side lobes can be controlled by changing the content of the LP ₀₁ mode. The difference in Figures 7–9 is that the initial arrangement of the outer-ring beamlets is different. As can be seen from Figure 7, the focal beam shape of the pure LP ₁₁ mode beam array comprises six radial spots, and the energy is concentrated in the central lobe for the pure LP ₀₁ mode beam array.

As we rotate the outer-ring beamlets 180 degrees on the basis of Figure 7, the intensity distributions of the mixed mode beam array under different conditions are shown in Figure 8. It can be clearly seen that when the mixed-mode beam array propagates in free space, the energy of the central lobe in Figure 8 is more concentrated than that in Figure 7. The physical reason is that the beam distribution at the initial plane is more compact in Figure 8. However, the intensity distributions under thermal blooming in Figure 8 are more dispersive than those in Figure 7. Thus, the arrangements of beamlets at the initial plane in Figure 7 are more resistant to the degrading effect of thermal blooming.

As we rotate the outer-ring beamlets 90 degrees on the basis of Figure 8, the intensity distributions of the mixed-mode beam array under different conditions are shown in Figure 9. As mentioned previously, the focal intensity distribution comprises 12 radial spots when the initial intensity distribution is shown in Figure 9A. However, the number of side lobes decreases as the content of the LP ₀₁ mode increases, that is, the side lobes are six when the content of the LP ₀₁ mode is 0.6. In order to compare the energy focusability under the three conditions more intuitively, the beam width w 86.5 % versus the LP ₀₁ fraction for different initial beamlet arrangements is investigated in Figure 10. It can be seen from Figures 10A–C that the beam width decreases as the wind speed increases. Thus, increasing the value of wind speed can be helpful in increasing energy focusability. On the other hand, the beam width in Figure 8 is the smallest under the same wind speed. Thus, the initial beamlet arrangements in Figure 8 can also be helpful in increasing energy focusability. In addition, the beam width in Figure 9 is smaller than that in Figure 10 when the wind speed is small. However, when the wind speed increases, the beam width in Figure 9 is larger than that in Figure 10. It indicates that compared with Figure 9, the effect of wind speed has a greater impact on the focusability of Figure 10.