What Equation Was Used for the Reflectance Plot from Thin Films?

[roam]

Homework Statement

Here is a plot of Reflectance vs Wavelength from a multilayer thin film stack of alternating high and low refractive indices:

I want plot this in Matlab, but what equation was used for this plot?

The diagram is from my textbook, there are many like it online but there is no indication as to what equation was used for the plot.

Homework Equations

Reflectance from and N layers of film:

$\left( \frac{(n_0/n_s)(n_L/n_H)^{2N} -1}{(n_0/n_s)(n_L/n_H)^{2N} +1} \right)^2$


Spectral width of the high reflectance zone:

$\Delta \lambda = \frac{4}{\pi} \lambda_0 sin^{-1} \left( \frac{n_H-n_L}{n_H+n_L} \right)$

The Attempt at a Solution

Well the first equation is not a function of wavelength. So I tried using the second one, with setting $\lambda_0=514 \ nm$ as the center wavelength:

n0=1; ns=1.52; nh=2.35; nl=1.38;
l=[300e-9 : 1 : 900e-9];
dl=((4/pi)*l.*asin((nh-nl)/(nh+nl)));
plot(l,dl);

However Matlab does not return any plots (it's blank). I also think this is not the correct equation since it is independent of the number of layers (N). I need an equation that takes both width and height of the curve into account...

What equation do I need to use here?

 

[ehild]

Your formulae are not functions of the wavelength. The plots show reflectance in terms of wavelength. To model R(λ) you can use the method based on the optical admittance. You find it in the book Optics of thin films by Vasicek, for example, or at http://www.edmundoptics.com/technical-resources-center/optics/an-introduction-to-optical-coatings .

ehild

 

[roam]

Thank you for the reply. The only parameters that I have are the refractive indices, N, and λ. So how do I plug these into the equation in your link?

Here is the equation:

$R= \left( \frac{\eta_0-Y}{\eta_0+Y} \right) \left( \frac{\eta_0-Y}{\eta_0+Y} \right)$

How does wavelength factor into this equation?

The link doesn't really explain much. Unfortunately I don't have the textbook, and we haven't covered optical admittances.

 

[ehild]

Thank you for the reply. The only parameters that I have are the refractive indices, N, and λ. So how do I plug these into the equation in your link?

Here is the equation:

$R= \left( \frac{\eta_0-Y}{\eta_0+Y} \right) \left( \frac{\eta_0-Y}{\eta_0+Y} \right)$

How does wavelength factor into this equation?

Through Y.
By the way, the formula is not quite correct. The second factor has to be complex conjugated.

Y depends on A and B. A and B are obtained by the product of matrices.
The elements of the matrices depend on lambda and the refractive indices and thickness of the layers.

The link doesn't really explain much. Unfortunately I don't have the textbook, and we haven't covered optical admittances.

Every necessary formula is given.
Try search for a book about Optics of Thin Layers .

ehild

 

[paranormal]

As a scientist, it is important to be able to accurately plot and analyze data in order to draw meaningful conclusions. In this case, it is crucial to know the equation used for the plot in order to properly interpret the data and make accurate predictions.

Based on your provided equations, it seems like the first equation is the correct one to use for this plot. However, it is not a function of wavelength, so it may be necessary to manipulate the equation to solve for reflectance at different wavelengths.

Another option is to reach out to the author of your textbook or the source of the online plots and ask for the specific equation used for the plot. This will ensure that you have the correct equation and can confidently analyze the data. Additionally, you may be able to find more information or resources on the specific thin film stack being studied, which may include the equation used for the plot.

In conclusion, it is important to accurately determine the equation used for a plot in order to properly analyze and interpret the data. This may require further research or reaching out to the source of the plot for more information.