Chapter 25, Problem 15Q

To determine

If you use blue light instead of red light, by what factor you can improve resolution.

Answer to Problem 15Q

Solution:

By a factor of 1.56

Explanation of Solution

Given:

The wavelength if blue light = 450 nm

The wavelength if red light = 700 nm.

Formula used:

The Rayleigh criterion gives us the resolution limit for two objects distance D apart when using light of wavelength $\lambda$

$\theta =\frac{1.22\lambda }{D}$

Where;

$\theta$= Angular half width.

$\lambda$= Wavelength of light used

D = Distance between the objects.

Calculation:

The Rayleigh criterion gives us the resolution limit for two objects distance D apart when using light of wavelength $\lambda$

$\theta =\frac{1.22\lambda }{D}$

We can interpret that as the angle is less we will havebetter the resolution.

For blue light

$\begin{array}{c}{\theta }_{blue}=\frac{1.22\lambda }{D}\\ {\theta }_{blue}=\frac{1.22(450\text{ nm)}}{D}\end{array}$

For red light

$\begin{array}{c}{\theta }_{red}=\frac{1.22\lambda }{D}\\ {\theta }_{red}=\frac{1.22(700\text{ nm)}}{D}\end{array}$

The ratio of angular half width of blue light to that of red light is

$\begin{array}{c}\frac{{\theta }_{Blue}}{{\theta }_{red}}=\frac{\frac{1.22(450\text{ nm)}}{D}}{\frac{1.22(700\text{ nm)}}{D}}\\ =\frac{450}{700}\\ =0.64\end{array}$

The angular half width of blue light is 0.64 times that of red light.

$\begin{array}{l}\frac{{\theta }_{Blue}}{{\theta }_{red}}=0.64\\ \Rightarrow {\theta }_{Blue}=(0.64){\theta }_{red}\end{array}$

Since smaller the angle better is the resolution, the resolution with blue light is

$\frac{1}{0.64}=1.56$ times than with red light.

We can improve resolution by 1.56 times if we use blue light.