91Ó°ÊÓ

The wavelength of the laser, $\mathrm{λ}=515\mathrm{nm}$

The diameter of the laser, d = 3 mm

The focal length of the lens, f = 3.50 cm

The central disk has 84% of the incident power.

The radius of the central disk is given by, $\mathrm{R}=\frac{1.22\mathrm{´Úλ}}{\mathrm{d}}$

Intensity is defined to be the power per unit area:

Using the given formula in the problem, get the radius of the central disk. Now the incident intensity or the power flux density in the incident beam is given using the formula of power per area of focus. Similarly, the intensity at the central disk is found using the radius of the disk and also considering the percentage of intensity.

Formulae:

The average intensity of a beam (or the power flux density),

${\mathrm{I}}_{\mathrm{avg}}=\frac{\mathrm{Power}}{\mathrm{Area}}$ ….. (1)

Area of a circular portion,

$\mathrm{A}={\mathrm{Ï€¸é}}^2\mathrm{or}{\mathrm{Ï€»å}}^2/4$ ….. (2)

Substituting the given data in the given equation of radius, the radius of the central disk is as follows:

$$\begin{gathered} \mathrm{R}=\frac{1.22\left(3.50\mathrm{cm}\right)\left(515\mathrm{nm}\right)}{3.00\mathrm{mm}} \\ =7.33\mathrm{μ³¾} \end{gathered}$$

Hence, the value of the radius is $7.33\mathrm{μ³¾}$.

Using the given data and area value of equation (2) in equation (1), the average intensity of the incident beam is given as:

$$\begin{gathered} {\mathrm{I}}_{\mathrm{avg}}=\frac{\mathrm{P}}{{\mathrm{Ï€»å}}^2/4} \\ =\frac{4\left(5\mathrm{W}\right)}{3.14{\left(3\mathrm{mm}\right)}^2} \\ =7.07×{10}^5\mathrm{W}/{\mathrm{m}}^2 \end{gathered}$$

Hence, the value of the average intensity is $7.07×{10}^5\mathrm{W}/{\mathrm{m}}^2$

As 84% of the incident beam is focused on the region, using the given data and area value of equation (2) in equation (1), the average intensity in the central disk is given as:

$$\begin{gathered} {\mathrm{I}}_{\mathrm{avg}}=\frac{0.84\mathrm{P}}{{\mathrm{Ï€¸é}}^2} \\ =\frac{0.84\left(5\mathrm{W}\right)}{3.14{\left(7.33\mathrm{μ³¾}\right)}^2} \\ =2.49×{10}^{10}\mathrm{W}/{\mathrm{m}}^2 \end{gathered}$$

Hence, the value of the average intensity is $2.49×{10}^{10}\mathrm{W}/{\mathrm{m}}^2$.