The concept required to solve this problem is related to thin film interference.
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:
[tex]\gamma = 2t[/tex]
Where,
[tex]\gamma =[/tex] Path length difference
t = thickness
At the same time we have that the constructive interference condition for a thin film interference for strongly reflected rays is
[tex]\gamma = m\lambda[/tex]
Where
[tex]\gamma =[/tex] Path length difference
[tex]\lambda =[/tex] wavelength
m = Any integer (order of the equation) which represent the number of repetition of the spectrum
Equation both expression we have
[tex]2t = m\lambda[/tex]
Re-arrange to find the thickness we have
[tex]t = m\frac{\lambda}{2n}[/tex]
Our values are given as
m = 1
[tex]\lambda[/tex]= 597nm
n = 1.38
Replacing,
[tex]t = (1)\frac{597}{2(1.38)}[/tex]
[tex]t = 216.3nm[/tex]
Therefore the thinnest thickness of the film is 216.3nm
The thinnest film of MgF₂ on glass that produces a strong reflection for orange light is : 216.3 nm
Applying the relation below
γ = m * λ ----( 1 )
also γ = 2t ---- ( 2 )
where : γ = path length difference, λ = wavelength, m = number of spectrum repetition
equating equations ( 1 ) and ( 2 )
2t = mλ
Therefore :
t = m * λ/2n ----- ( 3 )
n = 1.38, m = 1, λ = 597 nm, t = thickness
Insert values into equation ( 3 )
Thickness ( t ) = 1 * 597 nm / 1.38
= 216.3 nm
Hence we can conclude that The thinnest film of MgF₂ on glass that produces a strong reflection for orange light is : 216.3 nm.
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