91Ó°ÊÓ

Let the diameter of the central beam be D.

The expression to calculate the diameter of the central beam is given by,

$\mathit{θ}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{22}\frac{\mathbf{λ}}{\mathbf{d}}$

Here,$\mathit{λ}$ is the wavelength, d is separation, and$\mathit{θ}$ is angle.

It is known that,

$$\begin{gathered} \mathit{L}\mathit{θ}\mathbf{=}\mathit{D} \\ \mathit{θ}\mathbf{=}\frac{\mathbf{D}}{\mathbf{L}} \end{gathered}$$

From the equation$\mathit{θ}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{22}\frac{\mathbf{λ}}{\mathbf{d}}$ ,

$$\begin{gathered} \mathit{θ}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{22}\frac{\mathbf{λ}}{\mathbf{d}} \\ \frac{\mathbf{D}}{\mathbf{L}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{22}\frac{\mathbf{λ}}{\mathbf{d}} \\ \mathit{D}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{22}\frac{\mathbf{λ}}{\mathbf{d}}\mathit{L} \end{gathered}$$ …….. (1)

Substitute$2000×{10}^3\text{m}$ for L,$0.2×{10}^{-3}\text{m}$ for d , and$1.4×{10}^{-9}\text{m}$ for $\mathrm{λ}$in equation (1).

$$\begin{gathered} D=\frac{1.22\left(1.4×{10}^{-9}\text{m}\right)\left(2000×{10}^3\text{m}\right)}{0.2×{10}^{-3}\text{m}} \\ ≈17.1\text{m} \end{gathered}$$

Therefore, the diameter of the central beam is $17.1m$.

The expression to calculate the beam intensity is given by,

$\frac{I_1}{I_2}={\left(\frac{d}{D}\right)}^2$ …… (2)

Substitute $0.2×{10}^{-3}\text{m}$for d and $17.1\text{m}$for D in equation (2).

$$\begin{gathered} \frac{I_1}{I_2}={\left(\frac{0.2×{10}^{-3}\text{m}}{17.1\text{m}}\right)}^2 \\ =1.37×{10}^{-10} \end{gathered}$$

Therefore, the ratio of the beam intensity is $1.37×{10}^{-10}$.