Question

A scientist needs to focus a helium-neon laser beam $(\lambda =633nm)$to a $10-\mu m-diameter$spot $8.0cm$behind a lens

(a) what focal-length lens should she use?

(b) what minimum diameter must the lens have?

Short answer

(a) Focal-length lens she uses is $S^{\text{'}}=f=8cm.$

(b) The minimum diameter is $D=0.012m\to 1.2cm$.

Step-by-step solution

Part (a) Step 1: Given information 

We need to find the focal length.

Part (a) Step 2: Simplification 

Parallel rays always focus at focal length,

Therefore, $S^{\text{'}}=f=8cm.$

Here $f$is focal length.

part (b) Step 1: Given Information

We need to find the minimum diameter of the lens.

part (b) step 2: Calculation

By using,

$\omega =\frac{2.44\lambda f}{D}$ (By multiplying both sides with D.)

$D\times \omega =\cancel{D}\times \frac{2.44\lambda f}{\cancel{D}}$

$D\times \omega =2.44\lambda f$ (Divide both the sides by $\omega$.)

$\frac{D\times \cancel{\omega }}{\cancel{\omega }}=\frac{2.44\lambda f}{\omega }$

$D=\frac{2.44\lambda f}{\omega }$ (substitute values in equation.)

$D=\frac{2.44\times \left(6.33\times {10}^9\right)\times \left(0.08\right)}{10\times {10}^{-6}}$

$D=0.012m\to 1.2cm.$

Here, $D$is the diameter, $f$is focal length and $\lambda$is the wavelength.