Question

Estimate the distance for which ray optics is a good approximation for an aperture of $4\,mm$ and wavelength $400\,nm$.

Answer 1

Here we have to use the Fresnel formula and concept to find the distance for which ray optics is good approximation.The minimum distance travelled by a ray of light along the linear path before diffraction is known as the Fresnel distance. The distance at which spread due to diffraction becomes comparable or not to the width of the slit is defined by Fresnel distance.

Complete step by step answer:
The Fresnel distance is usually denoted by ${Z_f}$.
Mathematically, ${Z_f}$ is given by,
${Z_f} = \dfrac{{{d^2}}}{\lambda }$ where $d =$ size of source, $\lambda =$wavelength of light used.
Given,
Length of the aperture, $d = 4\,mm = 4 \times {10^{ - 3}}\,m$
Wavelength of the light incident $\lambda = 400\,nm = 400 \times {10^{ - 9}}\,m$
The distance for which ray optics is a good approximation=?
${Z_f} = \dfrac{{{d^2}}}{\lambda } \\\ \Rightarrow{Z_f}= \dfrac{{{{(4 \times {{10}^{ - 3}})}^2}}}{{400 \times {{10}^{ - 9}}}} \\\ \therefore{Z_f}= 40\,m \\\$
Hence, $40\,m$ is the distance for which ray optics is a good approximation.

Additional information:
The Fresnel distance is the minimal distance from the slit where the diffraction of light is proportional to the size of the slit. If the width of the slits exceeds the size of the Fresnel field, ray optics principles are applicable and if the slits width does not exceed the size of the Fresnel field, wave optics principles are applicable.

Note: The Fresnel distance is usually a dimensionless quantity. It does not have any significance over fundamental units. Also here we have to only remember the formula for Fresnel distance. The validity of ray optics is dependent upon this formula itself. Ray optics is good approximations of distances equal to the distance from the Fresnel. After passing the Fresnel distance a ray of light starts diffracting and spreads out. After this comes the wave optics.