Ivan Ohlfdal Vol. 5, No. 4/April 1988/J. Opt. Soc. Am. A 459
Immersion spectroscopic reflectometry of multilayersystems. I. Theory
Ivan Ohlidal
Department of Solid State Physics, J. E. Purkyn6 University, Kotlfask6 2, 61137 Brno, Czechoslovakia
Received February 2, 1987; accepted November 13, 1987In this paper general rules for the spectral dependences of Fresnel coefficients and reflectances of multilayersystems in different transparent ambients are presented. These rules imply some limitations on the opticalanalysis of the multilayer systems that are summarized in this paper. The theoretical results are applied to someprocedures that can be used in the analysis of nonabsorbing multilayer systems.
1. INTRODUCTION
Optical analysis of transparent thin films is often performedby means of immersion spectroscopic reflectometry (ISR).Within the framework of this method the spectral reflec-tances measured at normal incidence in various transparentambients (e.g., liquids) can be employed to determine thethickness and the refractive index of a nonabsorbing thinfilm in a simple way.1-3 This method can be used for theoptical analysis of nonabsorbing double layers as well.4'5 Inprinciple, multilayer systems can also be analyzed by usingISR. The spectral reflectances of a multilayer system obeysome general rules that can be utilized for optical analysis ofthese systems by ISR. In this paper these general rules arepresented. Furthermore, an application of these rules to theanalysis of multilayer systems with ISR will be describedfrom the theoretical point of view.
2. THEORY
A. General Formulas for the Fresnel Coefficients andReflectances of Multilayer Systems Immersed in DifferentAmbientsLet us consider a multilayer system consisting of absorbingthin films and an absorbing substrate. Let the number ofthin films be N. We shall assume that all materials formingthe system under consideration are optically isotropic andhomogeneous. Further, we shall assume that the bound-aries of this system are ideally flat and smooth. The opticalproperties of this system are then determined by 3N + 2optical parameters, namely, the refractive indices hj = nj +ikj and thicknesses dj ( = 1, 2, . . ., N) of the films and therefractive index = n + ik of the substrate (see Fig. 1). Weassume that the ambient is always transparent (o = no isreal). A monochromatic plane wave reflected from the sys-tem at normal incidence is completely described by the Fres-nel coefficient expressed as follows 5' 6:
FR exp(i6) = 1 r2 p(i^ 1 ) (1)+ 2 exp(i1 )'
where
r2 + r 3 exp(X 2 )= 1 + 2r3 exp(iS 2)
. rk + rk+l exp(ixk)
1 + rkrk+l exp(ik)
-n rN + N+1 eXP(iXN)row = 1 N)
'a1 + NPN+l ep(i'N)
r s- 1 -nss - +hhs- + ,
(s = O, 1,2,.. .,N+ 1),
j = 4 njdj, h = n, + ik. ( = 1 .. ., N),
ho - no, N+ l- = n + k.
rk is the Fresnel coefficient of the th boundary; R and 6 arethe reflectance and the phase change of the system, respec-tively; and X denotes the wavelength of the incident light.
Equation (1) can be rewritten as
P +(31 = & - (3)
where P = 2 exp(i1 1)
,, I1
; hence
p R-P1R1 - PI] (4)
The Fresnel coefficients and PI are dependent on therefractive index of the ambient no, whereas the quantity Premains constant when no changes. Thus the followingequation must be fulfilled:
Equation (6) shows that the reflection coefficients of themultilayer system in different nonabsorbing ambients aremutually dependent. The main consequences of Eq. (6) canbe summarized as follows:
(1) At a certain wavelength the reflectance and the
phase change corresponding to only one arbitrary transpar-ent ambient can be used for the optical analysis of the multi-
layer system. This means that a set of experimental values
for the reflectance and the phase change that correspond to
several different ambients and a chosen wavelength cannotbe used to determine all optical parameters characterizing
the multilayer system. In general, by means of this set we
are able to evaluate two optical parameters of the system at
most, if we know the other 3N parameters.(2) If we know the spectral reflectance and phase change
of the multilayer system in an ambient, we can determine
the spectral dependences of these quantities correspondingto any another ambient.
From Eq. (6), the reflectance of the multilayer system in
the ambient with refractive index no' can be expressed by
R(no"), the reflectance of the multilayer system immersed inan ambient with refractive index no", is obtained by puttingno" instead of no' in Eq. (7). From this and Eq. (8) we obtainthe expression for the reflectance R(no") as follows:
R(no")=(no0 " + no2)A + (no"2 - no2)B - 2nono"C
(no"2 + no2)A + (no"2 -n 02)B + 2nn0 "C
(9)
where
A = (n02
- n' 2) [1 + R(no)j[1 -R(no'),
B = (no'- n0 ) 2[1- R(no)R(n')]
+ (no' + n0)2[R(no) -R(no')],
C (no2 -n'2)[1 - R(no)]l -R(no'f.
Equation (9) implies the following conclusions:
(1) At a chosen wavelength, the reflectances of the mul-tilayer system corresponding to no more than two differentnonabsorbing ambients are independent. At wavelengthsfor which the Fresnel coefficient of the multilayer system is
real (i.e., the phase change equals mr, m = ±1, +2,. . .), onlythe reflectance corresponding to one nonabsorbing ambientis independent.
(2) If we know the spectral reflectances of the multilayersystem in two different nonabsorbing ambients with knownrefractive indices no and no' we can determine the spectralreflectance of this system immersed into another nonab-sorbing ambient with refractive index no".
Equation (8) shows that by means of the spectral reflec-tance describing the multilayer system in two different non-absorbing ambients with the known refractive indices no and.no', the spectral dependence of cos 6(no) corresponding to
the ambient with the refractive index no is determined.Thus the spectral dependence of the phase change 6(n0) isnot unambiguously given by the spectral dependence of no,no', R(no), and R(no').
Now let us deal with an absorbing multilayer system on a
transparent substrate. The Fresnel coefficient and the re-flectance of the multilayer system from the substrate sidefulfill the same equations as for light incident from theambient side. This means that the following equations arevalid:
In the foregoing equations (no, n) and p(no, n') denote thereflection coefficients of the system from the substrate sidefor substrates with refractive indices n and n' and one ambi-ent with a refractive index no. cos , (no, n) is again expressedby Eq. (8) if n, n', R(no), and R(no') are replaced by n, n',p(no, n), and p(no, n'), respectively.
Further, let us assume that the multilayer system is non-absorbing. At any wavelength the reflectance of this systemfrom the substrate side is identical with the reflectance ofthe same system from the ambient side for a chosen combi-nation of substrate and ambient media (see, e.g., Ref. 6).This fact and Eqs. (9) and (11) imply that the reflectanceB(no", n") of the system corresponding to an arbitrary ambi-ent and an arbitrary substrate with refractive indices n"and n", respectively, is given as follows:
simple procedure for determining the spectral dependenceof the phase change of any multilayer system will be de-scribed in Subsection 2.B.
B. Determination of the Spectral Dependences of thePhase Change of Multilayer SystemsWith this procedure a nonabsorbing thin film with a certainrefractive index is deposited onto a part of the upper bound-ary of the multilayer system to be studied. Let us assumethat we are able to determine with sufficient accuracy therefractive index and the thickness of this supplementarythin film. Further, onto another part of the upper boundaryof this system that is not covered by the first supplementaryfilm, a second supplementary film is deposited that is of thesame refractive index but of a different thickness. Thespectral reflectances of the multilayer system covered byboth supplementary thin films are expressed by means ofthe following equation 6 7:
R r02 + R(no) + 2r[R(nO)] 1
12 cosx[oi + 6(no)]
1 + r02R(no) + 2r0 [R(no)] 1 2 cos[xoi + 6(no)]
where
(13)
) (no"2 + n 2)ce + (n" 2 - n 2)0 - 2nono"'y
(no"2 + n02)a + (n" 2 - no2)f + 2nono"-y
(12)
where
= (no 2- n0'2)[1 + p(no, n")][ -p(no', n")],
= (n' - n0)2[l - p(no, n")p(no, n")]
+ (no' + no)2[p(no, n") - p(no', n")],
-y = (no 2-no'
2)[1 -p(no, n")][ -p(no', n")].
The conclusions implied by Eq. (12) can be summarized asfollows:
(1) At a chosen wavelength, a maximum of four reflec-tances of any nonabsorbing multilayer system sandwichedbetween two transparent media are independent. Thesereflectances correspond to the following pairs of refractiveindices of the ambient and the substrate: no and n, n andn', n' and n, and n' and n'. Thus, within monochromaticreflectometry, a maximum of four optical parameters of anynonabsorbing multilayer system can be evaluated if the re-maining 2N - 4 parameters of this system are determined inan independent way.
(2) If we know four spectral reflectances of the nonab-sorbing multilayer system corresponding to two differentambients and two different substrates, i.e., the reflectancesp(no, n), p(no, n'), p(no', n), and p(no', n'), then we can calcu-late the spectral reflectance characterizing this systemplaced on an arbitrary substrate and immersed into an arbi-trary ambient.
In practice, measurement of the spectral reflectance char-acterizing an arbitrary multilayer system at normal inci-dence is relatively easy. However, the experimental deter-mination of the spectral dependences of the phase change atnormal incidence is more difficult. A new and relatively
n - noi = 1, 2, ro -n+n, + nO
x0i = A n 0 d0 i,
n, is the refractive index of air, and R(no) and 6(no) are thereflectance and the phase change of the multilayer systemimmersed in an ambient with a refractive index no.
Equation (13) implies that
R(nO)(RirO2- 1) + 2r[R(n)]1 2(Ri - 1)
X cos[xoi + 6(no)] + Ri - rO = 0.
Let
X = [R(n,)] 112 COS 6(no)
and
Y = [R(nO)]'1 2 sin 6(no).
Equation (14) can then be rewritten as
X 2 + Y 2+AiX+BiY+ Ci = 0,
where
A = 2r0(Ri - 1)cos x0 IRjr 0
2 -
R - 1o 2
SlnE. Riro12,-w1otSolving Eq. (15), we obtain
(14)
(15)
2ro(Ri - 1)sin x0 i
Rr2 -1IRiro
i i=1, 2.
X y B2 -B C2 - C1x_ +A-A2 A-A2
and
P1Y 2 + P2Y+P 3 = 0,
where
p = (A -A ) + 1,
(16)
(17)
Ivan Ohlidal
Ivan Ohlidal462 J. Opt. Soc. Am. A/Vol. 5, No. 4/April 1988
2(B2 - Bj)(C 2 -C 1) A,(B 2 - B1)2 l (Al - A2)2 A, -A 2
____ C 2 A(C 2 -CO)P3 = C + A )+ AlC -_ C1Al -+ A2 Al A- A2
From Eqs. (16) and (17) we get
-P2 ± (P22 - 4PlP3) 1/2
2Pa
and _B 1 - 2 +( 2
B2 - B P2 (P22 -4PIP3)12X 1,2= ~ -A_ -A1 - A 2 2P1
Equations (18) and (19) imply that
C2 - C1
Al - A2
Ri(n0) = X12 + Y1
2, tan bi(no) =- (20)
Thus, by means of the spectral reflectances RI and R2, wefind two spectral dependences of 6(no) and R(no). Thismeans that we get two values of the complex reflection coef-
ficient characterizing the multilayer system immersed in theambient with a refractive index no at every wavelength,namely, (X 1
2 + Y12)112 exp[6l(no)] and (X2
2+ Y92)1/2 exp[62 (no)]
[61 (no), 62 (no) e (0, 27r)].
The true spectral dependence of the Fresnel coefficientcan be found in the following way. A third supplementaryfilm with a refractive index no and a thickness do3 is deposit-
ed onto a remaining uncoated part of the multilayer systembeing investigated. The spectral reflectance R3 of this sys-tem covered with the third supplementary film is then mea-sured. Further, the reflectances RI and R2 are calculated byusing the two Fresnel coefficients found by means of themeasured reflectances R1 and R2 , i.e.,
ro2 + Ri(no) + 2rO[Ri(no)]"1
2 COS[X03 + 6i(nO]
1 + ro2 Ri(no) + 2ro[Ri(n 0 1/2 COS[X03 + i(nO)]
where i = 1, 2.
The reflectances R1 and R2 must be compared with themeasured reflectance R3 at all wavelengths. This compari-
son can be performed by evaluating 1R3j - Rji and R3 j - R2jl,
where R3 j, Rlj, and R2j are the reflectances R3, R1 , and R2,
respectively, corresponding to the wavelength k. When therelation -
IR3 j - R1j1 < IR3j - R2jI (22)
is fulfilled, we can expect that the Fresnel coefficient
[Rl(no)]1 12 exp[61(no)] is the true one at the wavelength Xj.
This procedure can be performed at all wavelengths, and sowe can determine the true spectral dependence of the Fres-nel coefficient of the multilayer system immersed in theambient with a refractive index no. The spectral depen-dence of the Fresnel coefficient of this system immersed intoany ambient can be determined by using Eq. (6). If ISR
with two spectral reflectances cannot be applied (e.g., be-cause of the interaction of materials forming the films with
liquids serving as the nonabsorbing ambients), the ISR
method utilizing the measured spectral dependence of theFresnel coefficient of the system can usually be used. In
practice, the use of the spectral dependences of the phase
change and the reflectance determined in the described wayis dependent on the accuracy with which the reflectances R1
and R2 are determined and the accuracy with which thethicknesses dj, d2, and d3 and the refractive index no of thesupplementary films are known.
To conclude this section it is worthwhile to emphasize thefollowing results, which are important from the practicalpoint of view.
In the method of ISR based on the measurement of twospectral reflectances in two different nonabsorbing am-bients, the spectral dependences of the following pairs ofquantities can be employed for analyzing the multilayer
system:
(1) R(no) and R(no'),
(2) R(no) and cos 6(no) and/or R(no) and cos 6(no'),
(3) cos 6(no) and cos 6(no').
Henceforth this method of ISR will be called liquid immer-sion spectroscopic reflectometry (LISR), since its applica-tion would involve measurement of the reflectance of thesystem in at least one nonabsorbing liquid (the second ambi-ent in which the reflectance is measured can be air).
ISR based on measuring four spectral reflectances corre-sponding to all combinations of refractive indices of twodifferent transparent ambients and two different nonab-sorbing substrates will be called double-immersion spectro-scopic reflectometry (DISR). Within the framework of
DISR, the reflectances can be replaced by cos 6 or cos p for
optical analysis of the nonabsorbing multilayer systems.ISR based on measuring the spectral dependence of the
Fresnel coefficient of the system in one medium will hence-
forth be called solid immersion spectroscopic reflectometry(SISR). In this method, the spectral dependences of the.following pairs of quantities can be employed for analyzing
the multilayer system:
(1) R(no) and 6(no) and/or R(no) and 6(no'),(2) R(no) and R(no'),(3) (no) and 6(no').
3. APPLICATIONS
In this section some applications of the theoretical resultspresented above are outlined. First let us deal with thesystem consisting of an absorbing substrate and a nonab-sorbing thin film. Equation (9) enables us to derive an
explicit formula for evaluating the refractive index of thisnonabsorbing film if we perform the following theoretical
consideration: If the spectral dependence of the refractiveindex of the film n, is identical with the spectral dependence
of the refractive index no of the ambient in which the systemis immersed, then the spectral reflectance R describing thesystem considered is equal to the spectral reflectance of theabsorbing substrate immersed in the ambient with refractiveindex no = nI. Thus it holds that
The thickness d can be evaluated by means of one of thefollowing formulas:
= __ kXd= 4rn, (-62 + arccosL) + 2-
and
d=1 = n (-62 -arccosL) +47rn, ~~2n,
where k =0, 1,.. ., 1I + k,
L R(n0)(1 + r 2 P212) - (r12 + 12 )
2r11i21[1 - R(no)]
tan 62 = - 2knjn,2 - n2 - k
and
0 S arccosL < 7r,
ery X. n, n2, d2, n3, d3, .. , nN, dN, n, and k are unknownparameters. The unknown parameters can be sought byusing the least-squares method, for example. In this meth-od, we minimize the quantity
N
Q = 1 i2/bi2
i=l(25)(31)
in a space whose vectors have components identical with theoptical parameters sought [Fi denotes the value of F at the
(26) wavelength Xi, bi is the standard deviation of Fi correspond-ing to the standard deviations of R(no) and R(no'), and N isthe number of wavelengths in which the reflectances aremeasured]. It should be noted that the optical constantsmust be expressed by means of some dispersion formulas. 89
(27) di is sought separately by means of the least-squares methodthrough the minimization of the quantity
(28)
7r < 62 < 2r.
Equations (25)-(28) are derived by using Eq. (1) for N = 1.The true formula for calculating d can be found by using asimple criterion: The thicknesses calculated by means ofthe true formula are identical at all wavelengths within smallexperimental error. d evaluated by using the untrue for-mula is dependent on wavelength. From the foregoing it isevident that the formulas presented above can be used fordetermining the thickness and the spectral dependence ofthe refractive index of the nonabsorbing thin film if thespectral dependences of the optical constants of the sub-strate n and k are evaluated by using an independent meth-od. The main advantage of this procedure is that the thick-ness and the refractive index of either relatively thin or thicknonabsorbing film can be evaluated by means of explicitformulas.
The theoretical results described in Section 2 can be usedto analyze a multilayer system as well. Let a multilayersystem consisting of N nonabsorbing thin films be investi-gated with LISR. Let us assume again that the spectraldependence of the refractive index of the nonabsorbing am-bient equals the spectral dependence of the refractive indexof the upper thin film of the system. The reflectance of thissystem at every wavelength is then given by the reflectanceof the system formed by N - 1 lower nonabsorbing thin filmsexisting in the ambient with a refractive index n (in thiscase, the upper film of the system analyzed is removed fromthe point of view of the reflectance). This means that
R(nl) = IT212, (29)
where R(n 1) and r2 are expressed by Eqs. (9) and (2); thenthe function
F = R(nj) - r2 (30)
must be equal to zero at all wavelengths of interest. Thefunction F is dependent on no, n', R(no), R(no'), n, n, k, n 2,d2, n3, d3, ... , nN, dN, and . The wavelength is theindependent variable, and no, no', R(no), and R(no') areknown quantities that are determined experimentally at ev-
where Ri'(no) and Ri'(no') denote the experimental reflec-tances R(no) and R(no') measured at the wavelength Xi. iand bi denote the standard deviations of Ri'(no) and Ri'(no').
With DISR we can separate the two parameters sought.These two parameters are the thicknesses d and dN, sincewithin for the DISR method the function F can be defined as
F= R(nl, nN) - (33)
where R(nj, nN) is given by Eq. (12) and r is expressed as
- r 2 + 3 exp(ix 2 )
1 + r 3 exp(ix 2)
in which
r 3 + r4 exp(ix 3 )
- 1 + r 4 exp(ix3)
- _ rN-l + rN exp(iXN-l)rN-l = 1--
I + rN-,rN ep(iXN-1)
(34)
(rN+l = r = 0).
This means that by minimization of the function expressedby Eq. (31) the following parameters are sought: n, n2, n 3,.. ., nN, d2, d3 -.. ,dNl. d and dN are then determined byusing a minimization of the function Q similar to that givenby Eq. (32).
It should be noted that the same procedure can be chosenfor analyzing the nonabsorbing multilayer systems withSISR as with LISR. Of course, other procedures can beutilized with SISR as well.
The methods described here can also be used, for example,for analyzing the multilayer systems investigated in Ref. 10.
4. CONCLUSION
In this paper it has been shown that at normal incidence thespectral dependence of the Fresnel coefficient characteriz-ing light reflected from a multilayer system immersed in anonabsorbing ambient with a refractive index no, together
Ivan Ohlidal
464 J. Opt. Soc. Am. A/Vol. 5, No. 4/April 1988
with the spectral dependence of no, determines unambigu-ously the spectral dependence of the Fresnel coefficient ofthis system in any nonabsorbing ambient with another re-fractive index. Further, it has been proved that the spectralreflectance of the multilayer system corresponding to a non-
absorbing ambient with any arbitrary refractive index no" isunambiguously given by the spectral dependences of therefractive indices no and no' of two ambients and the spectralreflectances R(no) and R(no') describing this system in theseambients. Moreover, it has been presented that the spectralreflectance of a nonabsorbing multilayer system placed onan arbitrary nonabsorbing substrate and immersed into anarbitrary ambient is unambiguously determined by thespectral dependences of the refractive indices of two am-bients and two substrates and four reflectances correspond-ing to all combinations of these two ambients and two sub-strates. Thus we can use for analyzing the multilayer sys-tems the spectral dependences of the following pairs ofquantities:
(1) The reflectance and the phase change correspondingto an ambient with a refractive index no.
(2) The reflectances corresponding to two different am-bients with refractive indices no and no'.
(3) The phase changes corresponding to two differentambients with refractive indices no and n'.
If the nonabsorbing multilayer systems placed on the non-absorbing substrates are analyzed, four spectral reflectancescorresponding to four combinations of the refractive indicesof two different nonabsorbing substrates and two differentnonabsorbing ambients can also be employed.
It should be noted that the reflectances can be replaced bycos 6 or cos (p to analyze the multilayer systems.
In addition, a method is described in this paper for deter-mining the spectral dependence of the phase change of themultilayer system.
ACKNOWLEDGMENT
I wish to thank V. Holy for valuable discussions.
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