A continuous excitation approach to determine time-dependent dispersion diagrams in 2D magnonic crystals

Let us consider a general field H^exc(t) which excites spin waves in a magnetic system. The excited spin waves will depend on the frequency spectrum present in the excitation

$$\begin{eqnarray} \fl H^{{\rm exc}}(t) &= \int C^{{\rm exc}}(f) {\rm e}^{{\rm i} 2\pi f t}\,{\rm d}f \nonumber\\ &= \int \left\{A^{{\rm exc}}(f) + {\rm i} B^{{\rm exc}} (f)\right\}{\rm e}^{{\rm i} 2\pi f t}\,{\rm d}f \nonumber\\ &=\int |C^{{\rm exc}}(f)|\cos\{2\pi f t + \phi^{{\rm exc}}(f)\} \,{\rm d}f \\ &\quad + {\rm i}\int |C^{{\rm exc}}(f)|\sin\{2\pi f t + \phi^{{\rm exc}}(f)\}\,{\rm d}f \nonumber\\ &=\int |C^{{\rm exc}}(f)| {\rm e}^{{\rm i}[2\pi f t + \phi^{{\rm exc}}(f)]}\,{\rm d}f.\nonumber \end{eqnarray} \tag{ 1 }$$

Indeed, a general excitation contains a distribution of frequencies with corresponding amplitudes

$$\begin{equation} |C^{{\rm exc}}(f)| = \sqrt{\left[A^{{\rm exc}}(f)\right]^2 + \left[B^{{\rm exc}}(f)\right]^2} \end{equation} \tag{ 2 }$$

and corresponding phases

$$\begin{equation} \phi^{{\rm exc}}(f)=\arctan\frac{A^{{\rm exc}}(f)}{B^{{\rm exc}}(f)}. \end{equation} \tag{ 3 }$$

In terms of phasors rotating in frequency space, |C(f)| is the amplitude of the phasor, while φ^exc(f) is its initial phase angle.

Due to the excitation, the magnetization M(r, t) varies in time and space. We now study the spin waves propagating along the x-axis by analysing the z-component of the magnetization along this axis. This is done by extracting the spin wave modes defined by a characteristic wave vector k_x and frequency f by means of a spatial and a temporal Fourier transform of M_z(x, t) respectively.

$$\begin{eqnarray} \fl M_z(x,t) &=& \int\int C^{M_z}(k_x,f) {\rm e}^{{\rm i} 2\pi f t} {\rm e}^{{\rm i} 2\pi k_x x}\,{\rm d}f\,{\rm d}k_x \nonumber\\ \fl &=& \int\int \left\{A^{M_z}(k_x,f) + {\rm i} B^{M_z}(k_x,f)\right\}\nonumber\\ \fl && \times {\rm e}^{{\rm i} 2\pi (ft+k_xx)}\,{\rm d}f\,{\rm d}k_x \\ \fl & =& \int\int |C^{M_z}(k_x,f)| \nonumber\\ \fl && \times {\rm e}^{{\rm i} \left[ 2\pi(ft+k_xx) + \phi^{{\rm exc}}(f) + \phi^{M_z}(k_x,f)\right]}\,{\rm d}f{\rm d}k_x. \nonumber \end{eqnarray} \tag{ 4 }$$

Expression (4) describes the dispersion relation of the different spin wave modes with given (k_x, f). The amplitude

$$\begin{equation} |C^{M_z}(k_x,f)| = \sqrt{\left[A^{M_z}(k_x,f)\right]^2 + \left[B^{M_z}(k_x,f)\right]^2}, \end{equation} \tag{ 5 }$$

depends on the amplitude of the excitation field at that frequency |C^exc(f)| as well as on the geometrical and material properties of the sample supporting this mode. Hence, since one is only interested in the latter, one should correct expression (5) for the excitation field amplitude (2).

The phase has different contributions

$$\begin{equation} \phi^{{\rm exc}}(f) + \phi^{M_z}(k_x,f)=\arctan\frac{A^{M_z}(k_x,f)}{B^{M_z}(k_x,f)} . \end{equation} \tag{ 6 }$$

It indeed depends on the phase of the considered frequency component in the excitation φ^exc(f). Moreover it contains an additional phase difference $\phi^{M_z}(k_x, f)$. The latter can be interpreted as a temporal phase difference between the mode and the excitation—a phase $\phi^{M_z}(k_x,f)=0$ then stands for a mode beating in phase with the excitation—or as a spatial phase difference between spin wave modes. Due to the propagating character of the spin waves, both interpretations are equivalent.

In most micromagnetic simulations a pulse excitation is used to compute spin wave dispersions. In particular, the use of a sinc waveform sinc(2πf_ct) is popular since it has a constant spectrum between the cut-off frequencies ±f_c

$$\begin{eqnarray} \fl |f|>f_c &: \quad C^{{\rm exc}}_{{\rm sinc}}(f) = 0 \nonumber\\ \fl |f|=f_c &: \quad C^{{\rm exc}}_{{\rm sinc}}(f) = 0.5 \\ \fl |f|<f_c &: \quad C^{{\rm exc}}_{{\rm sinc}}(f) = 1. \nonumber \end{eqnarray} \tag{ 7 }$$

Note that also φ^exc(f) = 0. While this wave form extends infinitely in time we still consider it a pulse since it applies a finite amount of energy to the system. In most cases, the pulse is applied on a restricted area of the sample. After applying the pulse, the magnetization dynamics is simulated for some time T_sim, sampling the magnetization data at N_t time instants. This leads to a frequency resolution Δf = 1/T_sim

$$\begin{equation} f_n = \frac{n}{T_{{\rm sim}}} \quad n=0\ldots N_t/2-1. \end{equation} \tag{ 8 }$$

Only N_t/2 frequency components are obtained due to the cyclic nature of Fourier transforms. Expression (8) shows that, aiming at a large frequency resolution and a large frequency range, one needs to simulate the magnetization response for a long simulation time T_sim and store the magnetization multiple times N_t. Based on one single simulation one can build the complete dispersion diagram describing the amplitude of the different excited modes $|C^{M_z}(f_n,k_{x,m})|$ with

$$\begin{equation} k_{x,m} = \frac{m}{L_{x,{\rm sim}}} \quad m=0\ldots N_x/2-1 \end{equation} \tag{ 9 }$$

the discretized wave vector. Here, L_x,sim is the sample length in the studied x-direction, discretized using N_x discretization cells. Equivalently, to obtain a large k-space resolution over a large k-range, one needs to simulate a long sample L_x,sim discretized with a large amount of discretization cells.

Hence, if one is aiming for a large frequency and k-space resolution, the magnetization processes in very large samples are to be computed during very long time windows T_sim. However, while initially only the magnetization at the excited area is affected by the field pulse, the propagating spin waves distribute the finite amount of energy introduced by the pulse through the complete sample leading to a decreasing signal amplitude with time, even when a zero Gilbert damping is considered. This puts an upper limit on useful simulation windows T_sim and sample dimensions L_x,sim and thus intrinsically limits the frequency and wave vector resolution. This a fortiori applies for the real experiment, where the propagating signal strength decreases even more because of Gilbert damping.

While most experiments which use coplanar wave guides to generate and to probe spin waves apply a continuous sinusoidal excitation to the magnetic system, the approach is much less used in numerical simulations. Let us now consider such a continuous sinusoidal excitation, applied on some restricted area of sample

$$\begin{equation} H^{{\rm exc}}(f_0,t) = h \sin(2\pi f_0 t). \end{equation} \tag{ 10 }$$

Spin waves with the same frequency f = f₀, but different k_x wave vector, can now propagate from the excited area along the studied propagation direction. To determine these modes one only needs to perform a Fourier analysis in k-space

$$\begin{eqnarray} \fl M_z(x,f_0,t) &=& \int C^{M_z}(k_x,f_0,t){\rm e}^{{\rm i} 2\pi k_x x}\, {\rm d}k_x \nonumber\\ & =&\int \left\{A^{M_z}(k_x,f_0,t) + {\rm i} B^{M_z}(k_x,f_0,t) \right\} \nonumber\\ && \times {\rm e}^{{\rm i} 2\pi k_x x}\, {\rm d}k_x \\ & =& \int |C^{M_z}(k_x,f_0,t)|{\rm e}^{{\rm i} \left[ 2\pi k_x x + \phi^{M_z}(k_x, f_0)\right]}\, {\rm d}k_x \nonumber\\ & =&\int |C^{M_z}(k_x, f_0)|\sin\{2\pi f_0 t + \phi^{M_z}(k_x,f_0)\} \nonumber\\ & & \times {\rm e}^{{\rm i} 2\pi k_x x}\, {\rm d}k_x. \nonumber \end{eqnarray} \tag{ 11 }$$

Indeed, the mode varies sinusoidally in time with the same frequency as the excitation and a possible phase delay $\phi^{M_z}(k_x, f_0)$. In a simulation, the excitation is first applied for a given number of periods T₀ = 1/f₀ to ensure that the excited modes can propagate into the structure and no transient effects are present. After this, the magnetization is sampled at N_t time instants during half an excitation period

$$\begin{equation} t_i=\frac{i}{N_tf_0}\quad i=0\ldots N_t/2-1 . \end{equation} \tag{ 12 }$$

By inspecting at what time instant t_max = i_max/N_tf₀ the quantity $|C^{M_z}(k_x,f_0,t)|$ is maximal, one can extract the mode phase angle with respect to the excitation as

$$\begin{equation} \phi^{M_z}(k_x,f_0) = \pm\frac{\pi}{2} - \frac{i_{{\rm max}}}{N_t}\pi \end{equation} \tag{ 13 }$$

and the mode amplitude as

$$\begin{equation} |C^{M_z}(k_x, f_0)| = |C^{M_z}(k_x,f_0,t_{{\rm max}})|. \end{equation} \tag{ 14 }$$

Note that $\phi^{M_z}(k_x,f_0)$ is determined up to an angle π. This explains why only half a period is sampled: the other half results in identical amplitudes $|C^{M_z}(k_x, f_0)|$ and opposite phases $\phi^{M_z}(k_x,f_0)$. A visual inspection of the magnetization distribution at t = t_max easily defines the sign ±π. Furthermore, we have chosen the excitation to follow a sine and not a cosine function to limit transient effects when starting the simulations. Indeed, starting from t = 0 the excitation increases gradually from zero towards its maximum value.

To build the complete dispersion diagram, one needs to perform multiple simulations with excitation frequencies distributed over a predefined range. This distribution does not need to be uniform: certain frequency ranges can be simulated with an increased frequency resolution, e.g. to study a specific spin wave mode or a band gap in large detail, while other frequency ranges can be sampled with a lower frequency resolution by considering fewer simulations, which makes the method extremely versatile.

Contrary to applying a pulse excitation, it is not necessary to consider a vanishing damping in order to ensure a spin wave propagation far into the sample. The external sinusoidal field feeds the system continuously leading to a large amplitude wave propagation over extended areas of the magnonic crystal. Consequently, in regime, the Gilbert damping balances the energy that is continuously introduced into the system by the excitation as one would find in the real experiment. Moreover, the total power is not dissipated among all the modes over a wide range of frequencies as with a pulse excitation, but concentrated at the frequency of a single mode (or just a few, if more modes with different k occur at that frequency). Hence the use of a continuous excitation is advantageous for the effective implementation of spin wave based technology where large distances should be covered.

Some other straightforward advantages of the method are as follows: (i) by considering a fixed excitation amplitude h in all simulations for different frequencies f₀, see expression (10), all modes are excited with the same power, making—contrary to a general pulse excitation—a direct comparison between all mode amplitudes possible; (ii) the magnetization profile and its time variation at a given frequency is directly obtained from the simulations enabling an immediate visual interpretation of the possible modes existing at that frequency; (iii) For the complete dispersion spectrum, the phase difference $\phi^{M_z}(k_x,f)$ with the excitation can be obtained in a straightforward manner.

In this approach, many small simulations instead of one single large simulation (pulse excitation) have to be performed. This is in line with current computer hardware evolutions towards powerful computer clusters. All simulations for different frequencies f₀ are completely independent from each other and can thus be performed in parallel. For this contribution we use our graphics processing unit (GPU) cluster containing 16 nodes. Each node in the cluster runs the micromagnetic software package MuMax, speeding up each single simulation by two orders of magnitude [52].

We simulate the dot array by considering an elongated rectangular matrix of 32 × 5 dots, see figure 1, discretized using 4096 × 512 finite difference cells. Only spin waves propagating along the x-axis are studied. Hence, the spectra for perpendicular and parallel bias fields are obtained from distinct simulations with the bias field applied along the y-axis and the x-axis, respectively. The long simulation size L_x,sim = 16.64 µm ensures a high k-space resolution, see (9). Magnetization data M_z(x, t) can be collected along horizontal lines in the central row of the array, e.g. line 'A' for edge modes and line 'B' for bulk modes. We treat five rows of dots as a good approximation of the full 2D character of the magnonic crystal. The dots at the boundary of the crystal—in figure 1 depicted in dark grey—have a high damping constant. In this way, spin waves are not reflected, but absorbed without interfering with the spin waves in the central row of dots. This rules out stationary modes due to confinement effects. Elsewhere, the damping constant is put at 10⁻⁴. As in [53], a spatially uniform excitation profile is applied on a central dot, in figure 1 depicted with the chessboard pattern is used. However, the excitation field is not a pulse, but has a continuous sinusoidal wave form (equation (10)). The amplitude h is two orders of magnitude smaller than the bias field. This is larger than in the experiment, but is required to have a spin wave signal which is above the noise level throughout large portions of the sample. For our GPU computations this relative noise level is $\mathcal{O}(10^{-6})$ since we have a floating point accuracy. The amplitude is however small enough in order not to awake non-linear magnetization dynamics. For each direction of the bias field we have run simulations with excitation frequencies f₀ ranging from 5 GHz to 20 GHz with steps Δf₀ = 50 MHz.

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To explain the phase extraction method, we consider now bulk spin waves propagating perpendicular to the bias field with 15.5 GHz < f₀ < 17.5 GHz. A discussion of the complete dispersion diagram is postponed to the next section. Once the system is relaxed to its ground state with the bias field directed along the y-axis, the sinusoidal excitation is applied for 20 periods to enable the spin waves to propagate in the system and reach equilibrium conditions between excitation and damping. During the 21th excitation period, M_z(x) along line 'B' in figure 1 is stored at time points

$$\begin{equation} t_i = \frac{i}{36} \frac{1}{f_0}\quad i=0\ldots17. \end{equation} \tag{ 17 }$$

In figure 2, left column, each panel shows the 1D spin wave profile M_z(x, f₀, t_i) along line 'B' in figure 1 for successive frequencies f₀ at a different time point t_i. While the magnetization profile is only shown in three central dots, the corresponding dispersion diagram $|C^{M_z}(k_x,f_0,t_i)|$, presented in the second column of figure 2, is obtained by means of a spatial Fourier transform on the complete data set M_z(x = 0 → L_x,sim, f₀, t_i). In the third column of figure 2 we plot only (k_x,f₀) points where the intensity in the power spectrum is maximal at time points t_i. From that, the phase difference with the excitation $\phi^{M_z}(k_x, f_0)$ can be found using (13), resulting in phase sensitive dispersion diagrams. Combining all maximum values in one graph, i.e. collecting all mode amplitude data irrespective of the phase, one can visualize the mode amplitude $|C^{M_z}(k_x,f_0)|$ as in figure 3.

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From a closer inspection of the graphs in figure 2-left column and figure 3, we see that—depending on the time phase with respect to the excitation—a 2-DE-like profile is established in the excited, central cell over a 2 GHz frequency range. Indeed, at different time points, the modes in this frequency range give rise to the same 2-DE-like profile. This is illustrated by the maximum power lines which, remarkably, shift from one frequency to another as time flows indicating that the phase difference $\phi^{M_z}(k_x,f)$ varies continuously with frequency, see the phase dependent dispersions in figure 2-right column. Figure 4-left shows dynamic magnetization maps of the central, excited cell at different time points. For f = 15.45 GHz (f = 16.55 GHz) this dot has a maximal (zero) magnetization amplitude when the excitation is maximal and a zero (maximal) magnetization amplitude when the excitation is zero, meaning that the magnetization profile is beating in phase (with a 90° phase difference) with respect to the excitation.

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However, the dynamic magnetization map in the central dot is ambiguous, since it only visualizes how the spin waves are excited by the sinusoidal excitation field, but not how they propagate through the magnonic crystal. In fact, since the considered excitation has a uniform profile covering the entire central cell, the magnetization map in this specific cell should possess an even symmetry. Indeed, it can be either uniform or have an even number of nodes in both the x- and y-directions (as even-node harmonics of higher order). This should be true at all time points. Consequently, the spin waves need to be stationary: their nodal lines can not move. We can understand this by considering that, in the excited cell, the stationary wave results from a superposition of two propagating waves ψ_−k and ψ_+k with exactly the same cell function, but travelling in opposite directions out of the central cell: −k and +k. The superposition can lead to either functions ψ_a = ψ_−k + ψ_+k or ψ_b = ψ_−k − ψ_+k. To have an even profile, the function ψ_a needs to be a superposition of even functions ψ_−k(ψ_+k), with a resulting power that is maximum at t = 0, and zero at t = T/4, i.e. has a 90° phase difference with respect to the excitation (see figure 4-left, the mode at 16.55 GHz). For the same reason, ψ_b needs to be a superposition of odd functions with a resulting amplitude that is exactly zero at t = 0, and maximum at t = T/4, i.e. it beats in phase with respect to the excitation (see figure 4-left, the mode at 15.45 GHz).

Hence, looking at the excited cell only, it is impossible to study the propagating character of the single ψ_±k functions: one needs to consider a unit cell that is not involved in the excitation process. To that purpose, figure 4-right shows the time evolution of the dynamic magnetization in the first cell on the right of the excited one, the ψ_+k wave. Here, the propagating character is observed by following the shift of extrema and nodal lines from the left to the right. For example, referring to the mode at 16.55 GHz, it is clear that the maximum on the left at t = 0 (the red area) gradually shifts to the right part of the dot as time increases. Similar considerations hold for all other modes at different frequencies.

Moreover, in figure 4-left, we can see that both modes, at 15.45 GHz and 16.55 GHz, show an even profile in the excited cell. However, by inspection of the profiles on the right panel, we understand that the propagating mode ψ_+k at 16.55 GHz is actually a 2-DE mode (even) and that the one at 15.45 GHz is actually a 1-DE mode (odd). Following the explanations above, the superposition of left and right propagating waves results, in the excited cell (left panel), to a maximal signal at t = 0 for the even mode at 16.55 GHz and a maximal signal at t = T/4 for the odd mode at 15.45 GHz. The fact that a continuous, spatially uniform excitation can activate modes independently of their symmetry, simply by tuning the excitation frequency, is a remarkable feature of the presented method.

Going through the broad band shown in figure 3, at each frequency f a different combination of propagating modes ψ_f,+k and ψ_f,−k superimposes to the same magnetization profile in the excited cell: the 2-DE profile. In fact, the magnetization profile can result simply from a single mode (odd or even), or, more often, can result from the superposition of modes with different k. The time point at which the superposition gives rise to the maximum signal defines the phase difference with the excitation. This phase difference turns a magnetization profile with any definite symmetry into an even profile in the central excited cell as required by the excitation process.

We have explained why and how the mode power depends on time, and how this dependence is different if we focus on the central dot (the excited one, where spin waves have a stationary behaviour) or in the rest of the array (where the propagating character of waves becomes apparent). In particular, when performing the spatial Fourier analysis, we can, or can not, take into consideration the central (excited) cell and understand its influence on the dispersion graphs.

In an ideal infinite array, in which edge effects are negligible and no direct excitation is considered, only thermal spin waves exist. Even though their profile varies in time and from cell to cell, these variations are undetectable when averaged over the crystal, leading to a calculated/measured power independent of time. On the contrary, when in this infinite array the waves are excited from a given lattice site, the mode power averaged over the array is weighted by the intensity of the wave at different lattice sites. Consequently, now the mode power definitely depends on time as the Gilbert damping makes an infinite array effectively finite. Indeed, modes with k≠0 have different dynamic profiles in adjacent cells of the crystal, see equation (16). As time flows, the wave propagates and damps out, shifting the different magnetization profiles from one cell to another and decreasing their contribution to the corresponding calculated/measured average power.

This actually happens in our simulations, which deal both with a finite size array and with a localized excitation field, especially when we exclude the central, excited cell from the Fourier analysis. In figure 5 (figure 6) we plot the dispersion relations of the spin waves propagating perpendicular (parallel) to the applied field at time points t = 0 and t = T/4, calculated including (panels (a) and (c)) or excluding (panels (b) and (d)) the excited cell in the centre of the array. While the dispersion graphs that exclude the central cell show a small time dependence, only due to the propagation and damping of modes described above, when the central cell is included in the Fourier analysis the time dependence is more critical. Due to the excitation process and superposition effects, modes in the central cell have a power which strongly varies with time: namely, modes with even (odd) profiles are particularly amplified at t = 0 (t = T/4). Recalling that the BLS cross section is, at a first approximation, proportional to the average mode power [56], this leads to a proposal to measure the presence of odd modes.

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Usually, in a BLS experiment, odd modes have a very small cross section, making them hardly measurable, unless considering large incidence angles [56]. However, in an excitation/detection experiment involving the BLS experiment, the laser beam (which, in the microfocused-BLS-technique [57], has a space FWHM of about 300 nm) can be focused to illuminate an area containing or not containing the excited region, and the spectra collected in the two cases can be compared. The peaks corresponding to odd modes will be present only in the spectrum containing the excited cell while those corresponding to even modes will be present in both spectra. In this way, the spin wave dynamics of magnonic systems can be interpreted with much more confidence.

When comparing our results with the experiments, we recall that the usual resolution of BLS measurements is around 0.5 GHz. This suits the experimental points found for k = 0 at about 10.6 GHz and 11.0 GHz in the first panels of both figures 5 and 6, which should indeed correspond to the same mode. To this extent, the simulations show a good agreement with BLS measurements, especially for t = 0, where even (BLS active) modes are amplified.

Referring to figures 5 and 6, we discuss now only the few modes that are helpful to interpret the experimental BLS points. The mode at about 11 GHz (at k = 0) is rather non-dispersive along the direction perpendicular to the bias field (figure 5-panel a, group velocity nearly zero), and has a slightly negative dispersion along the direction parallel to the bias field (figure 6(a); from inspection of the magnetization profile, we find that its cell function is rather uniformly parallel with the bias field, but with 4 nodes along the perpendicular direction. This mode is usually addressed as a 4-BA-like mode, and propagates parallel to the direction of the applied field. Since this is a mode with an even cell function, it is best seen in panel (a) of both figures 5, 6. We skip modes in the range 11–12 GHz, which show a clear backward-like character, but are outside the scope of this paper.

We focus now our attention on the mode that experimentally occurs at about 12.3 GHz (k = 0, mean value between the two BLS points at k∥H and k⊥H), and which we find numerically at about 12.1 GHz. This is the fundamental mode, which has a rather uniform profile, the largest bandwidth (ω_X(Y) − ω_Γ) in both the x- and y-directions and the largest intensity in the first Brillouin zone (BZ, 0–0.5, in units of 2π/U). This mode stems from the Damon–Eshbach (DE) surface mode of the continuous film. However, because of Bragg diffraction at the void spacers between adjacent dots, its dispersion is discontinuous at zone boundaries and band gaps arise. With reference to figure 5, in the second BZ (k ∈ 0.5–1.0 × 2π/U, and ω/2π ∈ 14.75–15.25 GHz), due to the Bloch factor (equation (16)), the DE mode obtains a profile with one nodal line, we call it 1-DE, and in the 2^nd BZ it obtains the largest intensity in the spectrum. As can be expected from an odd mode, the corresponding dispersion, folded in the 1^st BZ (in the reduced scheme), has a vanishing intensity, if the central dot is not considered (figure 5), or an appreciable one if the central dot is considered (at k = 0 it occurs at about 15.2 GHz).

The DE mode in the range (1.0–1.5, in units of 2π/U, in the frequency range 16.25–17 GHz) has 2 nodal lines, and is called the 2-DE mode when folded to the first BZ (at k = 0 it occurs at about 16.3 GHz). In the range (1.5–2.0, first half of the third BZ) the DE mode obtains 3 nodes, and when considering it in the first BZ (where it shows vanishing power) it is usually addressed as a 3-DE mode (and so on for increasing Bloch wave vector). The same mode, considered along the direction parallel to the applied field (figure 6), has a negative dispersion, because it stems from the backward mode of the continuous film, while the 1-DE and 2-DE modes discussed above show in that zone only poor or slightly negative dispersion. The behaviour of the dispersion curves is found as predicted from the effective vector model, introduced in reference [55]: the effective wave vector, which is limited to the reduced zone, corresponds to the wave vector of the dipolar spin waves in an effective continuous film that replaces the magnonic crystal. This representation enables an easy understanding of the dispersion properties for modes with complex profiles appearing in non-trivial lattice symmetries [29].