Thanks for the question.
I will slightly rephrase the question and answer:
Consider the quarter-wave stack
of the form
g(HL)^mHa.
where g, the substrate, is glass and a is air, the medium. That is,
incident light travels through air, hits a high-refraction-index
layer, passes through m alternating layers of low-refractive-index and
high-refractive-index layers, and finally reaches the substrate,
glass.
(a) Evaluate the characteristic matrix for such a 3-layer stack (m=1)
We are given the stack:
gHLHa
where g is the substrate and a is the medium.
I will use the discussion in the on-line chapter notes to Optics by
James Wyant, at http://www.optics.arizona.edu/jcwyant/Optics505(2000)/ChapterNotes/Chapter08/multilayerfilms.pdf
. (You will need a pdf viewer to access this; a pdf viewer can be
downloaded from the Adobe download site
http://www.adobe.com/prodindex/acrobat/readstep.html)
The characteristic matrix M of a single film with refraction index y
that is quarter-wave thick is given by
0 i/y
y*i 0
Actually the Wyland web page chooses units so that the matrix is the
negative of this; I will follow the convention on p. 4 equation 9 of
the Macleod article cited below for the sign (choice of sign does not
affect units).
Actually the y is the admittance coefficient, for refraction index n,
but we follow the analysis in the Macleod article cited below to call
it n, the refraction index, which is fine for optical wavelengths.
Now, the characteristic matrix of a layer of films is simply the
product of their characteristic matrices.
In this case, we have two films, the high and low ones, which have
characteristic matrices M_H and M_L.
Since the stack, from incident to substrate, is HLH , the
characteristic matrix of the product is:
M_H M_L M_H
Suppose the refraction indices of the two materials are n_H and n_L
respectively.
Then the product M of these three matrices can be evaluated by hand as
follows:
M=M_H*M_L*M_H
=
(M_H*M_L )*M_H
We have M_H * M_L =
0 i/n_H * 0 i/n_L
in_H 0 in_L 0
=
-n_L/n_H 0
0 -n_H/n_L
Thus,
M=M_H*M_L*M_H =
-n_L/n_H 0 * 0 i/n_H
0 -n_H/n_L in_H 0
=
0 -in_L/n_H^2
-in_H^2/n_L 0
This is the desired characteristic matrix.
----------------Part b-----------------------
(b) Determine the reflectance for the stack for nH=2.32(ZnS) and
nL=1.38(MgF_2). Assume normal incidence and n_g=1.5 and n_a=1.0 .
NOTE: I am going to do the computation for m=1, for a three-layer
stack.
---------------Answer------------------------
In this case, the high-index layers are zinc sulfate with a refraction
index of 2.32, and the low-index layer is magnesium fluoride with a
refraction index of 1.38<2.32 . We are assuming the incoming light
beam through the air at index n_a=1.0 is normal (that is, straight on)
and the refractio index of the substrate on which the three films are
deposited is glass with an index of 1.5 .
We will also assume that the wavelength of the incident light is equal
to the wavelength used to compute the distance of the layers, the
"reference wavelength"; when we say "quarter wave stack" we are using
this wavelength implicitly.
Now, the reflection index, which is the percentage of incident light
that is reflected, is computed using Mathematica on page 4 of the
chapter notes given earlier, and turns out to be
R=r^2
where
r= -m_{21} + m_{12}*n_a*n_g
------------------------
m_{21}+m_{12}*n_a*n_g
Here m_{ij} are the elements of the characteristic matrix, n_a is the
refraction index of the medium, n_g is the refraction index of the
substrate, and we have used the fact that m_{11}=m_{22}=0 .
Now
m_{21}=-in_H^2/n_L = -i 2.32^2/1.38 = -i 3.90
m_{12} = -i n_L/n_H^2 = -i 1.38 / (2.32^2) = -i 0.256
We have n_a*n_g= 1.5,
r= 3.9 i - i 0.256*1.5
-------------------
-3.9i -i 0.256*1.5
Now, 0.256*1.5=.384, so this is, after canceling i,
r= (3.9 - .384)/(-3.9-.384) = -.82
To get the reflection index, we square it:
R=r*r = .67 .
Thus, 67% of the incident light will be reflected back.
VERIFICATION.
I verified this using the formula for reflectance by Macleod on page
18 of the article of his cited below (formula 41).
That formula gives the reflectance for an arbitrary m-layer stack
g(HL)^mHa, but it's hard to type in here.
WARNING: Do not use the Macleod formula as written directly: in his
formula, it looks like one is multiplying by y_sub, but actually due
to a typographical glitch he should be dividing by y_sub (this took me
about 2 hours to figure out!).
------------------------------------------------------------
SEARCH METHODOLOGY
In order to solve this problem, I performed an extensive internet and
library search. Search terms I used on google included terms like:
optics
mirrors
optics mirrors
optics thin films
thin films
refraction
reflection coefficient optics
Unfortunately, I did not find anything that I felt clearly answered
the question in this search.
I then visited a local engineering library, and looked through ten or
twelve textbooks on optics, most of which, however, were primarily
concerned with first principles and derivations.
Finally, I found two books on thin film optics that were helpful:
Optical Thin Films: User Handbook, by James D. Rancourt. Optical
Engineering Press, 1996.
Thin Films for Optical Systems. Francois Flory (editor). Marcel
Dekker, Inc. 1995.
"Thin-film optical coating design" by H.A. Macleod. In Thin Films for
Optical Systems, Francois Flory (ed.), Dekker 1995.
I had to read parts of both of these books to understand and answer
the question. The Rancourt book is a little bit more user-friendly.
Chapter 3 has a long discussion on computing various types of thin
film performance; however, this book did not discuss the
characteristic matrix. The Rancourt book, p. 3 equation (3), explains
that we can ignore the difference between the refraction index and the
gamma used in the characteristic matrix in the Wyant web site
computation (of course the multiplicative factors just cancel at the
end when we go to compute the reflectance).
The characteristic matrix was defined and discussed in the first
chapter of the Flory book; the chapter is
It is worth mentioning that Macleod has his own book on optical thin
films:
Thin Film Optical Filters, by H. A. Macleod, Institute of Physics Pub.
3rd edition, 2001.
This book comes well-recommended, but it was not in the library.
Macleod also has a website at http://www.thinfilmcenter.com/ .
Please do not hesitate to ask for clarification if anything here is
unclear. |