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

Problem Overview

We evaluate the central optical intensity limit:

\lim_{\theta \to 0} \frac{\sin(4\theta)}{2\theta}

using special trigonometric limit rules.

Step-by-Step Mathematical Solution

Step 1: Diagnose the indeterminate form

Direct substitution of $\theta = 0$ yields:

I(0) = \frac{\sin(0)}{2(0)} = \frac{0}{0} \quad \text{(Indeterminate Form)}

A raw sensor reading division by zero would trigger an unhandled exception fault in the tracking software.

Step 2: Separate constants and match the trigonometric argument

Factor the constant scalar $\frac{1}{2}$ out of the denominator and balance the argument $4\theta$ :

I(\theta) = \frac{1}{2} \cdot \frac{\sin(4\theta)}{\theta} = \frac{1}{2} \cdot 4 \cdot \frac{\sin(4\theta)}{4\theta} = 2 \cdot \frac{\sin(4\theta)}{4\theta}

Step 3: Apply the fundamental trigonometric limit identity

Let $u = 4\theta$ . As $\theta \to 0$ , $u \to 0$ . Recall the fundamental limit:

\lim_{u \to 0} \frac{\sin(u)}{u} = 1

Therefore:

\lim_{\theta \to 0} I(\theta) = 2 \cdot \lim_{4\theta \to 0} \frac{\sin(4\theta)}{4\theta} = 2 \cdot 1 = 2\text{ Watts/m}^2

Engineering Conclusion

The exact physical peak intensity along the central optic axis ( $\theta \to 0$ ) is $2\text{ W/m}^2$ . In the embedded firmware, the computer engineer should program a continuous removable patch:

I_{\text{firmware}}(\theta) = \begin{cases} \frac{\sin(4\theta)}{2\theta} & : \theta \neq 0 \\ 2 & : \theta = 0 \end{cases}

This continuous definition ensures the motion tracking optical system accurately reads central beam peak power without dividing by zero or dropping tracking frames.