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BRIT. J. APPL. PHYS. (J. PHYS. D , 1968, SER. ) 2, VOL. 1. PRINTED IN GREAT BRITAIN Optical reflection and transmission formulae for thzn films S . G. TOMLIN Department of Physics, University of Adelaide, Australia MS. received 13th May 1968 Abstract. The very complicated formulae which relate the components of the complex refractive index of a thin film to the measurable optical reflection and transmission coefficients have been put into a much simpler form. This greatly simplifies the programming for a computer solution of the equations, and also leads to some useful approximate formulae. 1. Introduction Although, in principle, measurements at normal incidence of the optical reflection and transmission coefficients of a thin film allow the determination of its complex refractive index n - i k , the usual formulae from which n and k must be calculated are extremely complicated. It is the purpose of this paper to derive much more manageable rigorous expressions and some approximate formulae. 2. Theoretical considerations Let us consider light of wavelength h in a non-absorbing medium of refractive index no incident normally on a film of thickness d and complex refractive index nl - ikl which is supported on a substrate of complex refractive index nz -ikz. R is the reflection coefficient for the upper surface of the film and T i s the coefficient for transmission through the film into the substrate. In practice the first medium is usually air or vacuum and no= 1, but we retain no for generality. Also the substrate is often transparent for the spectral region used and then kz = 0, but again we retain kz for situations where absorption in the substrate may be significant. Following Heavens (1955), R and T a r e given by exp R = (g12+h12)exp (2011)+(gz2+hz2) (-2011)+A cos 2y1+B sin 2y1 exp (2011) +(az hl2) (gz2+h2) (- 2011)+ C cos 2y1+ D sin 2y1 + exp + (1) where ((1 +g1)2+h12} ((1 +gz)Z+h22} T=nz no exp ( 2 4 (g12 h12)(gz2 hz2) exp ( -2011) C cos 2y1+ D sin 2y1 + + + (2) A=2(glgz +hhz), g1= gz = B=2(g1hz -gzh), no2 - n12 - k1z C=2(g1gz-h1hz), Ill D=2(gihz +g2h1) (no +t ~ i+ ki2’ ) ~ - nz2 +k12 - kzz n# = 2n0k1 (no + ki2 + 1212 (n1+ + (kl +k2)*’ and = ( n2)2 (ki a + + +kz)2 2(n& - n2kl) 011 = 2.rrkld/h yl=2mld/h. For the purpose of determining n1 and kl from measured R and T, expressions for any convenient functions of R and T could be used. Inspection of equations (1) and (2) suggested consideration of the functions (1 _+ R)/T, and these were found to be relatively simple when expressed directly in terms of nl and kl. 1667 1668 From (1) and (2), S. G. Tomlin Edl*(g12+h~2)}{exp(2a1)f(gz2+hz2) ( - 2 a i ) } + ( C f A ) cos 2 y l + ( D i . B ) sjn2yl exp (3) nz ( ( 1 +gd2+h1'} ((1 +g2)'+hz2) C i A = 4gigz or . - 4hlh2, D i . B=4gihz or 4g2hl Substituting in (3) leads to the required equations: and (4) (5) Given experimental values for (1 i R)/T, d, SO, 122 and kz these equations may be solved . readily using a computer. After multiplying through by the right-hand denominator and rearranging, the equations may be written ki) = 0 and fz(n1, kl) =0. The derivative 8f22/8kl is readily obtained and the second equation may be solved by Newton's method to find k l for a given nl. These values are then substituted infl(n1, kl) for a succession of values of n1 until the equation fi(n1, kl) = 0 is satisfied to any required accuracy. Care must be taken that when multiple roots occur, as may happen over some ranges of A, the correct root is chosen. This necessitates measurements over a fairly wide range of wavelengths. fi(nl, 2.1. Approximate formulae For a highly absorbing film when CUI> 2.0 or 2.5 the terms involving exp (2011)are dominant in (1) and (2), or (4) and ( 5 ) , and the formulae reduce to and Optical reflection and transmissionformulae for thinJilms 1669 For small values of a1 on the other hand, provided also that k12<n12, one can easily expand the right-hand sides of (4) and (5) in powers of kl to give where Ai = (no2+ a2) +nz2 kz2)+ (no2- M I 2 ) (n12- nzz - kz2)cos 2y1+ 2nkz(noZ - n12) sin 2yl (ni2 Bi =2nz{(no2+ n12)2y1- (no2- n12)sin 2y1) Ci = (2/ni2)[(no2 n12)(ni2 nz2 kz2)y12+{m4 - no2(nz2 k$)} sin2yl Di = 2(nz/n12> {4(no2 n12)y l 3- 2n02y1 no2 sin 2yl} and where + + + + + + + + nlkz{(no2+n12)2y1- no2 sin 2y1}] B2 = (n12 nz2 k22) 2yl+ (ni2-n$ - kz2) sin 2yl+ 4nlkz sinzyl Cz =4nz(y12- sinzyl) + + DZ= (l/n12){4(nl2 nz2 kz2)y13 - (nz2 kz2)(2y1- sin 2y1) +4nikz(y12- sin2 yl)). For a transparent substrate with ke=O these formulae are much simpler, and for a transparent film with kl=0 they reduce to forms given by Heavens (1955). From these equations useful first- and second-order solutions may be obtained. The third- and higher-order equations may be written down, but their numerical solution is no simpler than that of the exact equations. + + + 2.2. First-order solution Retaining only first-order terms in kl we have, from (7), 2n2nlZ 1 - R - T k1=T Bz and on substituting this expression into the first-order equation from ( 6 ) we have which may be solved graphically or numerically for nl and kl. 2.3. Second-order solution If second-order terms in kl are retained, equation (7) yields a quadratic equation in kl of which the correct root, such that 1 - R - T= 0 when kl = 0, is given by Values of kl calculated for chosen values of nl may be substituted into the second-order equation from (6) to obtain a graphical solution for nl and kl. For either of these approximate solutions the computations involved are feasible with a desk calculator, especially in the case of a transparent substrate (kz = 0). 1670 S. G. Tomlin 3. Discussion The range of reliability of the above approximate formulae is indicated in figures 1 and 2, which show results calculated from experimental data for selenium and carbon films, using the exact and second-order formulae. For the relatively thick selenium film (d= 724 A) the approximation is satisfactory up to k = 0 3 (01=0.46), when the error in k is 7.4% and that in n is only 1 %. Beyond this point the discrepancies increase rather rapidly. The firstorder calculation has the much more limited validity indicated by the individual points plotted. 3.01 2.0 n 2' 1.01 1.0 I 0.8 0.6 0.4 0.3 lo h (pm) Figure 1. n and k curves for a selenium film of thickness d=724 A. The full lines are from the exact formulae and the broken lines from the second-order approximation. The isolated points result from first-order calculations. In the case of the thinner carbon film ( d = 322 A) the second-order solution is very good for n over the wavelength range shown, the maximum error being 2.2% at the shortest wavelength, where a=0.6. The greatest discrepancy in k is then 12.2%. At a=0.43 k=0.95 and the error is 6.5 % in k and only 0.4% in n. For a 100 A thick carbon film the maximum error over the same range of h was about 5 % in both n and k. >: X X n 2.0- - 1.0 k I I I I .o Io 0.2 0.8 0.6 h (pin) 0.4 Figure 2. n and k curves for a carbon fl of thickness d=322 A. The full lines are from the im exact formulae and the broken curve is the second-order approximation for k. For n the two curves are barely distinguishable. Isolated points result from first-order calculations. Optical reflection and transmissionformulae for thin films 1671 In both the above examples the second-order approximation is good, particularly for n, for 01 up to about 0.45. This leaves a rather large gap to 01-2.5, beyond which the simplified formulae for a heavily absorbing layer may be applied. For this intermediate region it is essential to use the relatively easily programmed solution of the exact equations (4) and (5). Acknowledgments 1 am indebted to Dr. D. G. McCoy and Messrs. R. D. Campbell and R. E. Denton for permission to use some of their experimental results and computations. References HEAVENS, S., 1955, Optical Properties of Thin Solid Films (London: Butterworth). 0.