Index of Refraction, reflection diminishment

In summary, the conversation was about a glass lens coated with a thin film of magnesium fluoride (MgF2) to reduce reflection. The index of refraction for MgF2 is 1.38 and for the glass is 1.50. The task was to calculate the percentage by which reflection is diminished at wavelengths of 445 nm and 695 nm. To find this, one would need to use equations for thin layer optics, but it is not a simple task and requires advanced knowledge.
  • #1
Oijl
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Homework Statement


A glass lens is coated on one side with a thin film of magnesium floride (MgF2) to reduce reflection from the lens surface. The index of refraction of MgF2 is 1.38; that of the glass is 1.50. Assume that the light is perpendicular to the lens surface and that the thickness of the coating is the least possible needed to eliminate the reflections of light of wavelength 550 nm at normal incidence.

hrw7_35-20.gif


(a) Calculate the percentage by which reflection is diminished by the coating at 445 nm.
(b) Calculate this percentage for a wavelength of 695 nm.

Homework Equations


maxima: 2L=(my)/n
minima: 2L=((m+0.5)y)/2

[y is lambda)


The Attempt at a Solution


What is meant (mathematically) by a diminished reflection?
This is an extension of a sample problem in my textbook; in the book, the problem is to find the thickness of MgF2 that'll minimize reflection (99.6nm). But the only equations I can think of relating to this system use the quantities: thickness of MgF2, n-air, n-MgF2, n-glass, wavelength, and integer m. It doesn't seem, to me, that any of these are what I should be comparing in the 99.6nm thickness and 445nm thickness situations in order to find the percentage by which reflection is diminished.
 
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  • #2
"Diminished" here must mean reduction of reflection from the usual 7% or so of the light. This is quite a difficult problem! As wavelength varies away from the value for which the coating was optimized, the destructive interference of the two reflected waves will be imperfect because the coating thickness will not be exactly 1/4 wavelength. AND the amount of reflection at the two surfaces will not be equal (if it was equal at the optimum wavelength) due to variation in the index of refraction with wavelength. Interesting reading at
http://www.edmundoptics.com/technical-support/optics/anti-reflection-coatings/ [Broken]
and http://www.mellesgriot.com/products/optics/oc_2_2.htm
including graphs showing the "answer" for real optical coatings.
I don't see how you can calculate an accurate answer unless you assume equal reflection from the two surfaces and just do the destructive interference part. I don't know how to do even that - perhaps an integral over the sum of two sinusoidals, one phase shifted relative to the other by the double trip through the coating.
 
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  • #3
It is not difficult to derive a formula for the reflectance of a thin film coating if you are familiar with the theory of thin layer optics, but it is not undergraduate level! You can find such expression as below under the topics "thin layer optics":

n0 is the refractive index of air, n0=1,
n1 is the refractive index of the layer, n1=1.38,
n2 is the refractive index of the glass substrate, n2=1.50,
[itex]\lambda[/itex] is the wavelength, d is the thickness of the layer.


[tex]R=\frac{(n_0-n_1)^2(n_1+n_2)^2+(n_0+n_1)^2(n_1-n_2)^2+2(n_0-n_1)(n_1-n_2)\cos(4 \pi/\lambda\cdot n_1d)}{(n_0+n_1)^2(n_1+n_2)^2+(n_0-n_1)^2(n_1-n_2)^2+2(n_0-n_1)(n_1-n_2)\cos(4 \pi/\lambda\cdot n_1d)}[/tex]

ehild
 

1. What is the index of refraction?

The index of refraction is a measure of how much a material can bend light as it passes through it. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the material.

2. How does the index of refraction affect the path of light?

The higher the index of refraction, the more the light will be bent as it passes through the material. This can result in changes in the direction and speed of light, causing effects such as refraction and reflection.

3. What factors affect the index of refraction?

The index of refraction of a material is dependent on its physical properties, such as density and molecular structure. It can also be affected by external factors such as temperature and pressure.

4. What is reflection diminishment?

Reflection diminishment refers to the reduction of reflected light from a surface due to changes in the index of refraction. This can occur when light passes from one material to another with a different index of refraction, causing some of the light to be reflected and some to be transmitted.

5. How is the index of refraction measured?

The index of refraction can be measured using a variety of methods, such as using a refractometer or measuring the angle of refraction of light passing through the material. It can also be calculated using the material's known physical properties and the speed of light in a vacuum.

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