Aberration-driven tilted emission in degenerate cavities

The compensation of chromatic dispersion opened new avenues and extended the level of control upon pattern formation in the temporal domain. In this paper, we propose the use of a nearly degenerate laser cavity as a general framework allowing for the exploration of higher contributions to diffraction in the spatial domain. Our approach leverages the interplay between optical aberrations and the proximity to the self-imaging condition, which allows us to cancel or reverse paraxial diffraction. As an example, we show how spherical aberrations materialize into a transverse bi-Laplacian operator and, thereby, explain the stabilization of temporal solitons traveling off-axis in an unstable mode-locked broad-area surface-emitting laser. We disclose an analogy between these regimes and the dynamics of a quantum particle in a double-well potential.

Figure 1

Schematic of the fundamental Hermite-Gauss (HG) mode $A_s\left(x\right)$ (blue) and the corresponding potential $V\left(x\right)$ (orange) of Eq. (1) in the real (left) and Fourier space (right) for (a) $b<0, c>0$ and $s=1$ (b) $b>0, c>0$, and $s=1$. The resulting double-well potential in the Fourier space with the minima located at $k_0$ shown in (b) corresponds to a tilted HG mode in the real space.

Figure 2

Numerical solution of Eq. (1) for different values of $b$ and fixed values of $s=1$ and $c=4.97\times {10}^{-4}$. White lines indicate the values of $\pm \sqrt{〈x^2{\left|A\right|}^2〉}$.

Figure 3

Evolution of $|{\mathrm{\Gamma }}_0{(x,0)|}^2$and $|{\mathrm{\Psi }}_0{(x,0)|}^2$ as a function of $s$. Traces are shifted for clarity. From top to bottom, $s=(5,1,0.5,0.1)$. Other parameters are $(b,c)=(0.7873,0.0004973)$.

Figure 4

Evolution of ${\mathrm{\Gamma }}_n(x,0)$and ${\mathrm{\Psi }}_n(x,0)$ as a function of the modal index $n$. We note the even/odd alternating sequence. Traces are shifted for clarity. From top to bottom, $n=(3,2,1,0)$. Other parameters are $(b,c,s)=(0.7873,0.0004973,1)$.

Figure 5

Birth of two symmetrically located minima for the Fourier potential $\widehat{V}$ for a stable, marginal, and unstable cavity with (from left to right) $b_x=(-0.2,0,0.05)$. Other parameters are $(b_y,s)=(-0.8,1)$.

Figure 6

A schematic of the MIXSEL, where both the gain (green) and the saturable absorption (pink) are contained in the same microcavity. It is coupled face to face to a distant external mirror by an imperfect self-imaging system.

Figure 7

(a) Branches of one-dimensional tilted HG modes ${\mathrm{\Gamma }}_0$ (red) and ${\mathrm{\Psi }}_0$ (blue) of Eq. (25) as a function of the normalized gain bias $J_1/J_{\mathrm{th}}$. The mode ${\mathrm{\Gamma }}_0$ is stable (solid line) between the fold $F_2$ (cyan circle) and the AH point $H_{22}$ as well as between AH points $H_{23}$ and $H_{24}$ (red squares), respectively. The mode ${\mathrm{\Psi }}_0$ gains the stability at the fold $F_1$ and remains stable up to the AH bifurcation point $H_{11}$. (b), (c) Exemplary profiles of ${\mathrm{\Gamma }}_0$ and ${\mathrm{\Psi }}_0$ for $J_1=0.65J_{\mathrm{th}}$ (gray dashed line). Real (turquoise, dashed), imaginary (black, dotted), and intensity fields (blue, red, solid), respectively, are presented. Parameters are ${\alpha }_1=1.5,{\alpha }_2=0.5,J_2=-0.06,\widehat{s}=15,\kappa =0.035,d_f={10}^{-4},b=0.78,c=4.97\times {10}^{-4},s=1.0$.

Figure 8

(a)–(c): Intensity profiles of the 2D pattern obtained by the numerical simulations for three values of $b_x=(0.78,1.02,1.25)$, respectively, at the fixed value of the current $J_1=0.65J_{\mathrm{th}}$. (d)–(f) Corresponding power spectra in the $(k_x,k_y)$ plane. Parameters are $b_y=-0.39,c_x=c_y=4.97·{10}^{-4}$. Other parameters as in Fig. 7.