Question

Scientists use laser range-finding to measure the distance to the moon with great accuracy. A brief laser pulse is fired at the moon, then the time interval is measured until the "echo" is seen by a telescope. A laser beam spreads out as it travels because it diffracts through a circular exit as it leaves the laser. In order for the reflected light to be bright enough to detect, the laser spot on the moon must be no more than $1.0\mathrm{km}$in diameter. Staying within this diameter is accomplished by using a special large diameter laser. If $\lambda =532\mathrm{nm}$, what is the minimum diameter of the circular opening from which the laser beam emerges? The earth-moon distance is$384,000\mathrm{km}.$

Short answer

The minimum diameter of the circular opening from which the laser beam emerges$0.50\mathrm{m}$

Step-by-step solution

Maximum Diameter

We should compute cartesian coordinates and for size of the light's "bullet" by using equations for the length of the main maximum:

$w=\frac{2.44\lambda L}{D}\Rightarrow D=\frac{2.44\lambda L}{w}$

localid="1650221494106" $$\begin{gathered} D=\frac{2.44\times 5.32\times {10}^{-7}m\times 3.84\times {10}^{8}m}{1\times {10}^{3}m} \\ =0.499\mathrm{m} \end{gathered}$$(Numerically)

Minimum Diameter

Lets remember and we want is to round right, still not down due, is because we're seeking for shortest length which might help to attain this. As a findings, we just round the result to $D=0.50m$.