Question

What is the thinnest film of MgF₂ (n = 1.39)on glass that produces a strong reflection for light with a wavelength of 490 nm?

Answer 1

Concept and reason

The concept needed to solve this problem is the thin film interference.

Initially, writer the constructive interference condition for a thin film interference for strongly reflected rays. The condition is that the path length of the interfered waves is equal to the integer multiple of wavelength of the light used in the medium of the film. Substitute two multiplied by the thickness of the film for path difference of the two reflected rays. Later, find the wavelength of the light used in the medium of film. Use this wavelength value in the equation of interference condition and solve for minimum value of the thickness of the film.

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Fundamentals

The constructive interference condition for a thin film interference for strongly reflected rays is,

${\rm{path length difference}} = m\lambda ,{\rm{ }}\left( {m = 1,2..} \right)$

Here, m is an integer and $\lambda$ is the wavelength of the wave.

The wavelength $\lambda$ of a wave in a medium of refractive index n is,

$\lambda = \frac{{{\lambda _{{\rm{air}}}}}}{n}$

Here, ${\lambda _{{\rm{air}}}}$ is the wavelength of the light in vacuum.

The path length difference between two waves one is reflected from top surface of the film and the bottom surface of the film is equal to twice the thickness of the film.

${\rm{path length difference}} = 2t$

Here, t is the thickness of the film.

Substitute $2t$ for ${\rm{path length difference}}$ in the equation ${\rm{path length difference}} = m\lambda$ .

$2t = m\lambda$

Substitute $\frac{{{\lambda _{{\rm{air}}}}}}{n}$ for $\lambda$ .

$2t = m\frac{{{\lambda _{{\rm{air}}}}}}{n}$

Rearrange the equation for t.

$t = m\frac{{{\lambda _{{\rm{air}}}}}}{{2n}}$

The final expression for thickness of the film is,

$t = m\frac{{{\lambda _{{\rm{air}}}}}}{{2n}}$

Substitute 1 for m and ${t_{{\rm{min}}}}$ for t.

$\begin{array}{c}\\{t_{{\rm{min}}}} = \left( 1 \right)\frac{{{\lambda _{{\rm{air}}}}}}{{2n}}\\\\ = \frac{{{\lambda _{{\rm{air}}}}}}{{2n}}\\\end{array}$

The final expression for the minimum thickness of the film is,

${t_{{\rm{min}}}} = \frac{{{\lambda _{{\rm{air}}}}}}{{2n}}$

Substitute $490{\rm{ nm}}$ for ${\lambda _{{\rm{air}}}}$ and 1.39 for n.

$\begin{array}{c}\\{t_{{\rm{min}}}} = \frac{{490{\rm{ nm}}}}{{2\left( {1.39} \right)}}\\\\ = {\rm{176 nm}}\\\end{array}$

Ans:

The thinnest thickness of the film is 176 nm.