Laser Telemetry Diffraction Small-Angle Approximation (Entertainment & Computer Engineering)

In automated live entertainment stage rigging and motion capture tracking, a laser scanner tracks performers across the stage using high-speed optical galvo mirrors.

When the scanning laser passes through an optical diffraction aperture, the measured light intensity $I(\theta)$ (in Watts/$\text{m}^2$) detected by the photodiode sensor array as a function of the deflection angle $\theta$ (in radians) is given by:

1. Attempt direct substitution of $\theta = 0$ into $I(\theta)$ and explain why this creates an indeterminate form for the optical sensor micro-controller.
2. Apply the special trigonometric limit identity $\lim_{u \to 0} \frac{\sin(u)}{u} = 1$ to calculate the exact peak centerline intensity $\lim_{\theta \to 0} I(\theta)$.
3. State the calibrated intensity value the embedded software should assign to the central on-axis point $\theta = 0$ to prevent computational division-by-zero crashes.