22. Theory of Multilayer Films

Transcription

1 . Theory of Multilayer Fils Trasfer Matrix Reflectace at Noral Icidece Ati-reflectig Fils High-Reflectace Fils r a N N N... MMM 3 MN a MT N N N t

2 Trasfer Matrix for a Sigle Fil o a Substrate: T Mode t is order Cosider two waves ad that have the sae frequecy : r t k r t r t k r t, cos, cos r t i r zzˆ (T-ode) x y (T-ode) x y t i

3 r = s r r r r r i t i t i t r i r i r i t t t t t t t i i r r t t zzˆ (T-ode) x y (T-ode) x y r r i i

4 Trasfer Matrix for a Sigle Fil o a Substrate: T Mode Tagetial copoets of ad -fields are cotiuous across the iterface. O (a) : a r t i cos cos cos cos a r t t i t (a) (b) O (b) : b i r t cos cos cos b i t r t t t r t

5 Trasfer Matrix for a Sigle Fil o a Substrate: T Mode Note : v c Defie : cos cos t cos s t ( a ) cos cos cos cos a r t t i t ( b ) cos cos cos b i t r t t t a r t i b i r s t

6 Trasfer Matrix for a Sigle Fil o a Substrate: T Mode Fro the geoetry of the proble we ca see that : exp exp i ad i i t i r t where k tcos k tcos ktcos t t t (7-35) Usig these expressios we fid : exp exp i b i r t i i t exp exp i i b i r t i s t

7 Trasfer Matrix for a Sigle Fil o a Substrate: T Mode We ca solve the last two equatios for ad i ters of ad : t b b exp i t i b b i b b exp i At the first boudary we the obtai : iexpi exp a t i b i si bcos b iexpi exp b i i i i exp exp exp exp a t i b b isi cos b b

8 Trasfer Matrix for a Sigle Fil o a Substrate: T Mode i si a bcos b isi cos a b b This ca be writte i atrix for : i si cos a b a b i si cos cos cos t cos s t a, a M, r The x trasfer atrix is give by : M cos i si cos Geeralizig to a syste with N layers : a N N N... MMM 3 MN a MN N N N i si M, M N N, N for a ultilayer t

9 I ters of,, ad r t : a r b t a r b s t I atrix for we obtai : i si cos r t t r s t s t i si cos The aplitude reflectio ad trasissio coefficiets are give by : r r t t I ters of r ad t we obtai : r t t r t t s s

10 Reflectio coefficiet (r) (r) ad ad trasissio coefficiet (t) (t) Solvig for r ad t we have : t s s r s s s s These equatios allow us to evaluate the properties of ultilayer fils. a N N N... MMM 3 MN a MT N N N

11 At oral icidece the atrix eleets becoe : 9-. Reflectace at Noral Icidece: Sigle-Layer Fil cos i si i si cos At oral icidece the reflectace coefficiet becoes : cos i si i si cos cos cos t cos s t r scos s scos i s i si si The power reflectace coefficiet is give by : t cos t t t k tk : optical path differece R=rr * s cos s s cos s si si (Reflectace)

12 (Reflectace at oral icidece) t 4 t (air) s.5 (glass) 4.3% s.5 /

13 Reflectace at at Noral Icidece: Sigle-Layer Quarter-Wave Fil For the quarter wave thickess t 4 4 kt 4 : cos =, si = isi i cos i i si i cos i At oral icidece the r R=rr * i i s s s s s s reflectace coefficiet becoes : The power reflectace coefficiet is give by : t 4 4 If s R A sigle /4 fil ca produce perfect atireflectio!

14 Reflectace at Noral Icidece: Two-Layer Quarter-Wave Fils The trasfer atrix for a quarter For two quarter wave fils : M=MM i i i i The reflectace coefficiet becoes : r wave thickess fil is give by : M i s s s s s s s s s s i The power reflectace coefficiet is give by : R=rr * s s Zero reflectace occurs whe s s

15 Reflectace at Noral Icidece: Two-Layer Quarter-Wave Fils s Although the reflectace is greater, it reais less values over the broad rage by itroducig a / layer!

16 Reflectace at Noral Icidece: Two-Layer Fils

17 9-4. Reflectace at Noral Icidece: Three-Layer Atireflectig Fils

18 3 Reflectace at Noral Icidece: Three-Layer Atireflectig Fils For the quarter - wave fil, zero reflectace occurs whe : s

19 CVI Atireflectig Fils

20 CVI roadbad Atireflectig Fils

21 9-5. High-Reflectace Layers A high - reflectace fil rather tha a ati - reflectig fil is ade by reversig the order of depositio of the low - ad high refractive idex layers. i The trasfer atrix for a quarter wave thickess fil is give by : M i For two quarter wave fils, with the high refractive idex layer o top : M = M M HL H L L i i H H L H i H i L L The power reflectace coefficiet for the two layer fil at oral icidece is give by : R= rr * L s H L s H

22 High-Reflectace stack of N Double-Layers For a high - reflectace fil with N pairs of high - ad low re fractive idex layers :... N N H L H L HN LN H L HL M M M M M M M M M M M N N L L H H N H H L L The aplitude ad power reflectace coefficiets for the ulti layer fil at oral icidece are give by : N L H L s H L s H r N N N L H L s H L s H N N R=rr * s L H s L H N N

23 High-Reflectace Layers N=6 N= +(high) N= N-pairs of H/L

24 Reflectace of high-low /4 Layers R=rr * s L H s L H N N

25 CVI High-Reflectace Coatigs

26 CVI High-Reflectace Coatigs