Question #43148

A continuous multi-lens optical medium is designed such that its effective optical power per unit length depends non-linearly on the transverse displacement $y$ from the optical axis ($z$-axis). Under the paraxial slope approximation ($|dy/dz| \ll 1$), the ray trajectory $y(z)$ satisfies the differential equation: $\frac{d^2y}{dz^2} = \alpha y - \beta y^3$ where $\alpha = 4 \text{ mm}^{-2}$ and $\beta = 1 \text{ mm}^{-4}$. A ray injected near the optical axis ($y \approx 0$) experiences a diverging effect. However, due to the non-linear spherical aberration term ($\beta y^3$), there exists a critical off-axis injection height $y_c > 0$ where a ray injected with zero initial slope ($y'(0) = 0$) propagates perfectly parallel to the optical axis, analogous to a particle at an equilibrium point in an electrostatic potential. If a ray is injected at $z=0$ with a slight offset $y(0) = y_c + \delta y$ (where $0 < \delta y \ll 1 \text{ mm}$) and $y'(0) = 0$, what is the spatial period $L$ of its trajectory along the $z$-axis?