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Q: optics: Characteristic matrix and reflectance ( Answered ,   0 Comments )
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 Subject: optics: Characteristic matrix and reflectance Category: Science > Physics Asked by: monga-ga List Price: \$100.00 Posted: 27 Oct 2002 01:54 PDT Expires: 26 Nov 2002 00:54 PST Question ID: 90432
 ```Very high reflectance mirrors can be fabricated using multi-layer quarter-wave stacks of form g(HL)mHa. {here g and a represent glass and air}. (a) Evaluate the characteristic matrix for such a 3-layer stack (m=1) (b) Determine the reflectance for the stack for nH=2.32(ZnS) and nL=1.38(MgF2). Assume normal incidence and ng=1.5 and na=1.0 i need this by 30/10/02``` Request for Question Clarification by rbnn-ga on 27 Oct 2002 14:41 PST ```In the phrase: g(HL)mHa is the m superscripted? That is, is the actual phrase supposed to be m g(HL) Ha ? Similarly, when you speak later of ng, and na, is it n_g and n_a, that is, the g and a are subscripted?``` Clarification of Question by monga-ga on 27 Oct 2002 20:27 PST ```hi rbnn-ga yes m is superscript and g and a are subscript thanks monga```
 ```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.``` Clarification of Answer by rbnn-ga on 27 Oct 2002 22:30 PST ```For some strange reason the URL for the Wyant paper did not show up correctly. It is probably clear from the context what the URL. I will try to copy/paste the URL again here: http://www.optics.arizona.edu/jcwyant/Optics505(2000)/ChapterNotes/Chapter08/multilayerfilms.pdf but if that fails, I am afraid you will have to type the URL in yourself.``` Clarification of Answer by rbnn-ga on 28 Oct 2002 09:58 PST ```I contacted Dr. Macleod, who confirmed that the misleading or erroneous formula for reflectance he'd used was only due to a typesetting glitch.``` Request for Answer Clarification by monga-ga on 28 Oct 2002 18:36 PST ```hi rbnn-ga thanks for answering my question thanks monga``` Clarification of Answer by rbnn-ga on 28 Oct 2002 20:11 PST `OK, thank you.`