3 denote the projections on the set of orts, for example, ⃗ e$_{x}$ , ⃗ e$_{y}$ , ⃗ e$_{z}$ , for ⃗ m = m$_{0}$⃗ e$_{z}$ or ⃗ e$_{x}$ , − ⃗ e$_{y}$ , − ⃗ e$_{z}$ for ⃗ m = − m$_{0}$⃗ e$_{z}$ . For the FM state of the dot array ( ⃗ m = ⃗ e$_{z}$ for all dots) the same set of orts in Eq. (2), ⃗ e$_{1}$ = ⃗ e$_{x}$ , ⃗ e$_{2}$ = ⃗ e$_{y}$ and ⃗ e$_{3}$ = ⃗ e$_{z}$ should be used. In the quadratic approximation over the operators a † ⃗ l and a$_{⃗}$ l the Hamiltonian reads where ⃗ δ = l$_{x}$⃗ e$_{x}$ + l$_{y}$⃗ e$_{y}$ is a dimensionless lattice vector, H$_{a}$ = κm$_{0}$ is the anisotropy field, M = m$_{0}$/a 3 is the characteristic value defining the dipolar interaction intensity and having the same dimension as usual the 3D magnetization. The collective modes are introduced via states a$_{k}$ and a † k of definite quasi-momentum ⃗ k where N is the total number of dots in an array. The collective modes are defined by the quadratic Hamiltonian over a$_{k}$ , a † k which acquires the standard form When the form of coefficients A$_{k}$ and B$_{k}$ are established, the collective excitation en- ergy ε ( ⃗ k ) = ℏ ω ( ⃗ k ) may be found by means of u − v Bogolyubov transformations (see, for example, $^{35}$) and universally reads where γ = 2 µ$_{B}$/ ℏ is the gyromagnetic ratio. The concrete forms of A$_{k}$ and B$_{k}$ are defined by the distribution of the magnetic moments within the array. For the case of interest (parallel ordering of dot magnetization) one can find A$_{k}$ = h + β − 3 2 σ (0) + 1 2 [ σ (0) − σ ( ⃗ k ) ] , B$_{k}$ = 3 σ$_{c}$ ( ⃗ k ) , (6) ε ( ⃗ k ) = 2 µ$_{B}$M √ A 2 k − | B$_{k}$ | $^{2}$, ω ( ⃗ k ) = γM √ A 2 k − | B$_{k}$ | $^{2}$, ˆ H = 2 µ$_{B}$M ∑ k [ A$_{k}$a † $_{k}$a$_{k}$ + 1 2 ( B$_{k}$a † $_{k}$a † − k + B ∗ $_{k}$a$_{k}$ a$_{−}$$_{k}$ ) ] , (5) a$_{k}$ = 1 √$_{N}$ ∑ ⃗ l ′ a$_{⃗}$ $_{l}$e i ⃗ k $^{l}$, a † k = 1 √$_{N}$ ∑ ⃗ l a † ⃗ $_{l}$e − i ⃗ k ⃗ $^{l}$, (4) ˆ H = 2 µ$_{B}$ ∑ ⃗ l     H$_{0}$ + H$_{a}$ − ∑ ⃗ δ =0 M | ⃗ δ | $_{3}$  a † ⃗ $_{l}$a$_{⃗}$ l − 1 2 ∑ ⃗ δ =0 M | ⃗ δ | $_{3}$a † ⃗ $_{l}$a$_{⃗}$ l + a ⃗ $_{δ}$  − 3 µ$_{B}$M 2 ∑ ⃗ l    ∑ ⃗ δ =0 ( δ$_{x}$ + iδ$_{y}$ ) 2 ∣ ∣$^{⃗}$ δ ∣ ∣$^{5}$ a † ⃗ $_{l}$a † ⃗ l + a ⃗ δ + h . c (3) III. FERROMAGNETIC STATE OF ARRAY. 6