Figure 1

(a) Splitting of the propagation wave numbers in the case of the reactive coupling of two spatial harmonics (due to the refractive index modulation), $\gamma =im$, around the degeneracy point $\Delta k=\left(k_{\parallel ,2}-k_{\parallel ,1}\right)\to 0$. (b) In the case of dissipative coupling (due to gain-loss modulation), $\gamma =m$, the harmonics lock to one another instead of the mutually pushing. In the latter case the propagation wave numbers are complex, and (c) their imaginary parts (the growth exponents) are nonzero. In (a) and (b) the zero of the vertical axis (the reference spatial frequency) corresponds to $\left(k_{\parallel ,2}+k_{\parallel ,1}\right)/2$. In all cases $m=0.2$.Reuse & Permissions

Figure 2

(a) Gain-loss profile for a GLM material with a 2D harmonic modulation. (b) Reciprocal lattice vectors of the GLM crystal. Parabolas represent the spatial dispersion curves of the harmonic components in the paraxial approximation. The harmonics created by the dashed vectors (and depicted by dashed parabolas) are not considered in the analytical approach, as they do not interfere with the triple-cross point at around $\mathbf{k}=\left(k_x,k_z\right)=0$.Reuse & Permissions

Figure 3

(a) and (b) Propagation eigenvalues (real parts of $K_{\parallel }$), and (c) the growth/decay exponents (imaginary parts of $K_{\parallel }$) for three-coupled-mode model (6) in the self-collimation regime for $Q_{\parallel }=1.6$. (a) illustrates the geometry of the dispersion parabolas of harmonic components with vanishing coupling $\left(m\to 0\right)$. (b) shows the development of a self-collimation plateau on the dispersion curves of the Bloch modes for $m=0.1$.Reuse & Permissions

Figure 4

(Color online) (a) Field amplitude for a propagating beam through a GLM material with $m=0.1$ and $Q_{\parallel }=1.6$. The peak amplitude in each transverse section has been normalized to unity. The initial width of the Gaussian beam is $W_0=13.3$ corresponding to a Rayleigh length $Z_R=88.8$. (b) Profiles of the initial Gaussian beam (thin solid curve), and of the beam propagated by a distance $Z=4Z_R$ along the modulated material and space filtered to remove the amplified transverse modes (bold curve). The profile of the beam propagating through a homogeneous material is shown for comparison (dashed curve). (c) Width of the central part of the beam propagating in the modulated and a homogeneous material (dashed curve). The oscillations appear because of the influence of the amplifying sidebands. (d) Imaginary part of the propagation wave number obtained from the propagation of plane waves through the material entering at varying angle.Reuse & Permissions

Figure 5

(a) and (b) The propagation eigenvalues (real parts of $K_{\parallel }$), and (c) the growth exponents (imaginary parts of $K_{\parallel }$), for three-coupled-mode model (6), in the spatial-filtering regime. (a) Geometry of dispersion parabolas of harmonic components. (b) Plateau of mode locking for $m=0.1$. (c) Angular gain profile. $Q_{\parallel }=1.0$.Reuse & Permissions

Figure 6

(Color online) (a) Propagation of a Gaussian beam through gain-loss modulated medium with $m=0.1$ and $Q_{\parallel }=1.0$. The initial beam is as in Fig. 3. (b) The width of the beam propagated through the modulated medium (thick line) and homogeneous material (dashed line). (c) Imaginary part of the propagation wave number obtained from the propagation of plane waves through the material in different directions. (d) Focalization of the beam behind the material for $m=0.1$ and $Q_{\parallel }=0.85$. White vertical line indicates the rear face of the crystal.Reuse & Permissions