Determination of thin film optical parameters fromphotometric measurements: an algebraic solutionfor the (TRf,Rb) method
Valentin Panayotov and Ivan Konstantinov
A new method of analysis for the simultaneous determination of the optical constants and the thickness of
thin films is proposed. It requires measurements under normal incidence of the transmission and of thereflections from both sides of a thin film deposited on a nonabsorbing substrate. An algebraic inversion
technique is developed involving a numerical interpolation procedure in the last step. There are no missing
solutions. The physical solution can be isolated by a comparison with some film thickness estimates or by
measurements in a wavelength range. Keywords: Thin films, optical constants, determination methods,photometric measurements, algebraic technique.
1. Introduction
The determination of the optical constants (i.e., therefractive index n and the absorption coefficient k) ofthin films is a topic of fundamental and technologicalimportance. Many efforts have been devoted to theproblem but in spite of the variety of methods report-ed,1-3 the results are often subjective and ambiguousand there is no universal approach.
The same is also true for the most widely spreadphotometric methods. The determination in this casegenerally proceeds by measurement of a transmission5 and a reflection ]? at a given wavelength X for a thinfilm with known thickness d deposited on a nonabsorb-ing substrate with known refractive index n,. Usingthese photometric quantities an inversion of the corre-sponding nonlinear equations gives the optical con-stants of the thin film.
In this procedure some inherent problems can beoutlined. Most often this is the discrepancy betweenthe real samples and the ideal conditions assumed (i.e.,plane parallel boundaries, homogeneity of the films,etc.). The mathematical problems are connected withthe inversion and multiplicity of the solutions whichhinders the finding of the correct physical solution.This requires additional measurements and thus dete-
The authors are with Bulgarian Academy of Sciences, CentralLaboratory of Photoprocesses, BU-Sofia 1040, Bulgaria.
riorates the accuracy due to the accumulation of ex-perimental errors. If we take into account also thepractical experimental problems of the optical andthickness measurements and the variety of substancesunder investigation, the absence of universal approachand the complexity of the determination become quiteunderstandable.
To overcome the problems connected with the highnonlinearity of the thin film optical equations a varietyof inversion techniques have been proposed. Usuallythe solutions are searched numerically, by subsequentiterations.4 7 As an alternative an inversion method8
has been developed which uses a formalism of splittingthe interface between the film and the substrate withan imaginary ambient gap of zero thickness.9 Thusthe and 5 measurement data are transformed intothe corresponding values for the unsupported film.Further on, with the aid of an algebraic technique, theoptical constants are obtained as a function of thethickness.
We have investigated this last approach and ouranalysis shows that for a single film on a nonabsorbingsubstrate the formalism with the unsupported film isnot necessary since it only introduces complications.We have found that a similar algebraic technique canbe applied directly to the starting nonlinear equations.
Furthermore, having in mind that for the isolation ofthe physical solution usually an additional photomet-ric measurement is required,3 we succeeded to find alsoan algebraic solution for a system of three photometricmeasurements. Thus from 5, Af, and gAb data sets(indexes f and b denote front and back, i.e., incidencefrom the film and the substrate side, respectively) weobtained solutions of n, k and d sets. The choice of the
correct physical solution can now be facilitated if someestimations of the physical thickness of the thin filmare available. In analogy with some other methods,3 10
the determination can be performed in a wavelengthrange followed by selection of the solutions with ad-missible dispersion and/or with the same film thick-ness.
According to the accepted notation8 "'1 such an ap-proach can be classified as an algebraic (,]f,]b)method. In the following we shall outline the princi-ple of this method together with an illustration of itsapplication and with a discussion about the futureprospects for its perfection and development.
'[A1
1
n + Nk
n5
tf
r f 1 b
air
'PrbFig. 1. Basic configuration for the computations.
11. Principle of the Method
The amplitude coefficients of reflection r and trans-mission t of a thin film on an infinite nonabsorbingsubstrate (Fig. 1) are given for normal incidence oflight by2,13:
-f + = T1 + -2 = a X exp(iof), (la)
(1+ Pl1)(l + -2 f (1 + pi)(+ = 2)# n X exp(iot), (lb)
22r prb = - 22 b = X exp(i4), (1c)
1 + rr 2j32
tb = 1 +ii 1 2 ) - n3T X exp(iot) = n.T1 (ld)
where
Y = (1- 0)/( + n), (2a)
P2 = ( - n,)/(n + nS) (2b)
85 = exp ti~(2c)
n + ik. (2d)
Here T, Rf, and Rb are, respectively, the transmissionand the reflections of the thin film on an infinite sub-strate (these quantities are directly deduced'14"5 fromthe experimentally measured 5, Rf and ]?b); 't, Of, andckb are the corresponding phase angles and the dashdenotes a complex quantity.
The problem for the determination of the opticalconstants of the thin film is reduced to the solution ofsystem [Eq. (1)]. Although the system is undeter-mined in this form, it can be shown that other interre-lations exist which already permit a final solution for n,k, and d to be found from the values of T R, and Rb.Among the different possible approaches we have cho-sen the following scheme to reveal these additionaliterrelations.
Expressing p2 from Eq. (la) and (1c) one obtains
t = rf 1r,2(l - Plpf)
02 = _ rb + P2
Pi(' + rsrb)
and by equalizing h2 can be expressed as a function of Ffand Fb, i.e., of of and Okb,
(5)n2 = n 2Vb -1) - n(Ff -1)
(Ib+l)-n(rf+1) 'or in terms of the optical constants,
2Nx+k = N
2n'
where
ht2 =N. +iN,.
(6)
(7)
(8)
Thus n and k are expressed as functions of the phaseangles kf and qb. At this stage (Pf and q5b are undeter-mined in the 0-27r range. For each pair of f and 10b,according to Eqs. (6) and (7) a solution exists for n andk consistent with the values of R and Rb. However,not every pair is consistent also with the value of T,which up to now has not been considered.
It can be shown that kf and 'kb (respectively, jf and rb)are interrelated and therefore they cannot be variedindependently. The additional relation can be found,for example, by multiplying Eqs. (lb) and (d) andexpressing the resulting expression in terms of If and bwhich finally leads to
rStftb = rfPb + ?f - b - (9where
rS = (1- n)/(l + n). (10)
By multiplying Eq. (9) with its complex conjugate, ktcan be eliminated and, after some manipulations, onecomes to the necessary connection between f and Okb:
A X cosOf + B X COSkb + C X COSOf COSt + D X sinkf sin~b = E,
From Eq. (11), of as a function of Ob can be obtainedin the form:
tan of = y ± Vy2 + X 2 - Z (12)2 X +Z
where
x = A + C X cos~b,
y = D X sin~b,
z = E - B X Cos0b.
Thus it is obvious that for each 40b one determinestwo corresponding kf and that now a single varying of40b will be sufficient.
Furthermore, it can be shown that ebb itself is re-stricted to a smaller than 27r interval, which followsdirectly from the fact that in Eq. (12) the quantityunder the square root must not be negative. Thisleads to:
a X cos2 b + 2b X coSb + c > 0, (13)
where
a = C2 - B22
b = AC+ BE,
c = A2 + D2_ E2
and finally:
-b + b 2 -ac -b - b2 -ac-a +• < cos . (14)
So, starting with iterations of ebb in the interval de-termined by Eq. (14) and computing the correspond-ing of from Eq. (12) one obtains an ordered set of n andk as a function of ebb only.
The problem is to select the correct solutions of nand k together with the corresponding value of d. Thiscan be done by including the exponential dependen-cies which up to now have not been used. Expressingthe right side of Eq. (4) [if one chooses Eq. (3), the finalresult will be the same] in the form:
2= B + y, (15)
and taking into consideration Eq. (2c), the followingquantities can be defined:
nd = A [arctg (B) + 27rm] ,m = 0,1,2,... (16)
kd =4 In JB By. (17)
Thus for each ebb one obtains besides the alreadymentioned set of n and k also an ordered set of nd and
d. Now, following the algebraic approach 8 one canuse the function
F=nXkd-kXnd (18)
to find the solution (n,k,d) from the condition F = 0.
The outline listing of the general flow of the compu-tations is as follows:
1. Input of X, n,, T, Rf, and Rb-2. Determination of the admissible interval of e0b
[Eq. (14)].3. Input of ebb (in the interval defined by step 2).4. Computation of of [Eq. (12)].5. Computation of n and k (Eqs. 6 and 7).6. Computation of nd and kd [Eqs. (16) and (17)].7. Computation of F [Eq. (18)].8. Next ebb. (Go to step 3).9. Selection of ebb for which F = 0.10. Output of n, k, and d = nd/n for the qib in step 9.A numerical interpolation technique (steps 9 and 10)
is needed to null the function F(Ob) and subsequentlyto fix the optical constants and the film thickness.
As far as Eq. (16) is concerned care should be takento the periodicity of nd by the amount of X/2 andsolutions should be searched in each range.
For nonabsorbing thin films (k = 0) the above con-siderations lead to a simple modified method, whichwill be discussed in the Appendix.
Ill. Numerical Results
To test the analysis it is convenient to accept someoptical parameters (i.e., optical constants and thick-ness) for a hypothetical thin film and to obtain thenecessary input for the reverse determination of thoseparameters by computing the corresponding photo-metric values.
Let us consider as a numerical example the followingproblem: for X = 600 nm, d = 100 nm, n, = 1.5, n = 2.8,and k = 0.7, according to Eqs. (1) and (2) one obtains T= 0.1763, Rf = 0.1956, and Rb = 0.0671, respectively.Now, starting with these photometric values the opti-cal parameters can be determined according to thealready described method. As an illustration (and tofacilitate a possible reproduction of the computa-tions), some intermediate results will be presented.The dependence of n and k on ebb [according to Eqs. (6)and (7), respectively] is shown in Fig. 2. The validregion for t0b is restricted by k 2 0. The dependence ofnd and kd on ebb [according to Eqs. (16) and (17)] isshown in Figs. 3 and 4, respectively. The individualranges of nd are marked in Fig. 3 according to theparameter m in Eq. (16). The dependence of F on e0b[Eq. (18)] is shown in Fig. 5. Solutions exist for thoseebb where F is zero. The results are summarized inTable I. The physical solution can be isolated fromthe thickness since in this case, the thicknesses for thedifferent solutions are sufficiently distinct.
As another illustration we present an example previ-ously discussed by Case.8 The wavelength is 1150 nm,the refractive index of the substrate is 1.7, and thereported photometric quantities are, respectively, T =0.476, R =0.414, and Rb = 0.374. The results from ourcomputations, together with those from the originalpaper,8 are summarized in Table II. The method ofCase gives solutions that fit the T and Rb values exactlyand the Rf value within 2%. According to his interpre-tation there are two separate regions of n and k solu-
Fig. 2. Dependence of n and k on q5b according to Eqs. (6) and (7),respectively. The valid region for lb is restricted by k ' 0 (A = 600
nm; n, = 1.5, T = 0.1763, Rf = 0.1956, and Rb = 0.0671).
80
78
76
74
72-
70-
68
66120 140 160 180 200 220 240
phase change on back reflection
Fig. 4. Dependence of kd on ekb according to Eq. (17) (A, ns, T Rf,and Rb, the same as in Fig. 2).
1000
800
i 600 H
400 F
200 H
140 160 180 200 220 240
phase change on back reflection
0
-200121
Fig. 3. Dependence of nd on /lb according to Eq. (16). The individ-ual ranges of nd are marked according to the parameter m in Eq.(16). The region with negative nd is meaningless (, n, T R, and
R, the same as in Fig. 2).
tions-one with high n and low k, and another with lown and high k. The method in the present paper, how-ever, gives solutions that fit all three photometric val-ues exactly and it reveals out that there are only threemathematical solutions consistent with the input data.It is obvious now that the results of Case are groupedaround these solutions, the deviations being a result ofhis mathematical treatment. In this particular exam-ple, however, the choice of the physical solution amongthe three mathematical solutions may be ambiguousbecause the thicknesses are not sufficiently distinct.This choice can be made on the basis of some otherphysical reason or, for example, by measurements atother wavelengths and subsequent choice of those so-lutions with the same film thickness.
IV. Conclusion
A new analysis method for simultaneous determina-tion of both the optical constants and the thickness of athin film from (Yf,]b) photometric measurements is
2
240
phase change on back reflection
Fig. 5. Dependence ofFon c/b accordingtoEq. (18). Solutions (seeTable I) exist for those fb where F is zero (A, n, T Rf, and Rb, the
same as in Fig. 2).
presented. The basic measurement configuration is asingle film on a nonabsorbing substrate. The inver-sion technique can be classified as algebraic, althoughthe last step involves a numerical interpolation proce-dure. The method reveals all solutions, the physicalone being isolated by a comparison with some estima-tion of the film thickness. When the determination isperformed in a wavelength range the solutions withadmissible dispersion or with the same film thicknesscan be selected.
The method is also applicable for nonabsorbingfilms. In this case only one photometric measure-ment, for example, the transmission, is required for thedetermination of all (n,d) sets consistent with thatmeasurement. If one of these parameters (either n ord) is known, all the solutions for the other one can bedetermined.
As a possible disadvantage of the (lf,b) methodcan be pointed out the generally expected poor accura-cy.6 Nevertheless, having in mind that the simplest
Note: An, Ak, and Ad are computed assuming AT = +0.005, ARf= +0.01, and ARb = :0.01 to illustrate the accuracy of the method.
Table II. Solutions for the Example Problem of Case8
X = 1150 nm n= 1.7
T = 0.476 Rf 0.4 14 Rb = 0.374
W. Case present paper
n k -d nm] - n k d [nm]
2.9648 0.1793 86.8
2.9356 0.1709 101.7 2.9521 0.1678 107.1
3.0061 0.1634 113.3
3.2215 0.1547 119.53.4158 0.1498 119.4
3.7590 0.1436 115.5 3
0.1519 1.3422 104.50.1481 1.1000 125.9
0.1480 1.0436 131.6
Note: The n, k, and d values are presented here with meaninglesshigh accuracy (see Table I). The aim is only to allow an easycomparison with the results of Case.
photometric measurements are required, that the pro-posed computational procedure is fast and easy tohandle, and that all solutions are revealed, the de-scribed method can serve as a tool for fast decisionsand preliminary evaluations.
Furthermore, the developed analytical approachcan be extended to other sample constructions as well.It has been shown that the accuracy of determinationof the optical constants of thin films can be substan-tially improved when a reflection measurement with ametal sublayer Rm is used.'1 We have found that thedeveloped algebraic approach can be applied to thiscase as well. In another paper we shall present thealgebraic solutions for the (Y,]mYf) and (Ylm,?b)sets where already enhanced accuracy should be ex-pected. 1 In this sense we assume that the presentapproach can stimulate the further development ofcomputational methods for the determination of thinfilm optical constants.
0 50 100 150 200 250 300 350 400 450 500
d (nm)
Fig. 6. Dependence of the refractive index n on the thickness d =
nd/n for a nonabsorbing thin film (k = 0, A = 700 nm, n8 = 1.5, T =
0.8). For d = 100 nm (the dashed line) the first six solutions arepresented on the figure.
This work has been completed with the financialsupport of the Bulgarian Committee for Science andHigher Education.
I. Konstantinov expresses his gratitude to the Alex-ander v. Humbold Foundation and to the Institute ofQuantum Optics of the University of Hanover for thesupport of his initial investigations in this field.
Appendix: Nonabsorbing Thin Films
For a nonabsorbing thin film Rf = Rb = R = 1 - T.Thus, only one photometric measurement remainswhich is insufficient to determine simultaneously thetwo unknown quantities-the refractive index and thethickness. Another measurement should be made, forexample, the thickness or of some photometric quanti-ty at a different angle of incidence or on a substratewith different refractive index.
The described method can be extended to cover thenonabsorbing range also in the case when the thicknessd is known. For k = 0 the system [Eq. (1)] is muchsimplified. By multiplication of Eq. (3) with its com-plex conjugate one finds the dependence of n2 on ebf tobe
2 R(1 - r) cos/f - n5(R + r5)
R(1-r) cos/l1 + (R - r)
Inverting the expression for ,2 [Eq. (3)], one alsofinds the dependence of nd on of [Eq. (16)]. Thus foreach of one obtains an ordered set of n and nd values(Fig. 6). Now for a known thickness, d, the function
F = n X d - nd
can be defined, giving the refractive index from thecondition F = 0. Again care should be taken to theperiodicity of nd by the amount of X/2 and (see p.3 col.2, line 18) solutions should be searched in each range.
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