reflection of light in a long-wavelength approximation from an n-layer system of inhomogeneous...

Peep Adamson Vol. 21, No. 3 /March 2004/J. Opt. Soc. Am. B 645

Reflection of light in a long-wavelengthapproximation from an N-layer system

of inhomogeneousdielectric films and

optical diagnostics of ultrathinlayers. II. Transparent substrate

Peep Adamson

Institute of Physics, University of Tartu, Riia 142, Tartu 51014, Estonia

Received June 12, 2003; revised manuscript received September 23, 2003; accepted October 21, 2003

The reflection of linearly polarized electromagnetic plane waves from an N-layer system of inhomogeneous,dielectric films on a transparent, homogeneous substrate is investigated in the long-wavelength limit. Ap-proximate formulas are obtained for changes in the reflectance of s- or p-polarized light and in the ellipsometricangles that are caused by a multilayer, thin-film system. An analysis of the influence of a multilayer, ultra-thin surface film on the reflectance of p-polarized light at the Brewster angle is carried out. All approximateanalytical results are correlated with the exact computer solution of the reflection problem for a multilayersystem of inhomogeneous films. The possibilities are discussed of using the obtained approximate expressionsfor resolving the inverse problem of ultrathin dielectric films on transparent substrates. Novel options aredeveloped for determining the parameters of ultrathin films by integrating differential reflectance and ellip-sometry. © 2004 Optical Society of America

OCIS codes: 260.2110, 240.0310, 310.6860, 260.2130, 120.5700, 120.4290.

1. INTRODUCTIONThe issue regarding the role of transition layers on trans-parent materials has a very important place in the theoryof specular reflection of light because the actual physicalinterfaces are not perfectly smooth (geometrical) planeswhere the characteristics of the medium change in step-wise manner, and, therefore, the famous Fresnel’s formu-las are only the first approximation to reality. But thecorresponding corrections to Fresnel’s formulas describ-ing the contribution of the interfacial domain to the re-flection coefficient of light were obtained only within theframework of a model1 in which this domain is regardedas a single, (in)homogeneous layer, although in practicethe near-surface region can often consist of several differ-ent layers.

On the other hand, optical diagnostics of ultrathin lay-ers has grown in importance not only in the physics of thesurface region of solids2 and liquid systems,3 but in mod-ern materials technology.4 It has been revealed that,generally, the contribution of an ultrathin dielectric layerto the reflection characteristics of a fully transparent sys-tem is very low.3,5 Because of this, common differentialmethods,2–9 in which the direct contribution of an ultra-thin layer to the reflection characteristics is measured,are a subject of particular interest. In addition, the non-absorbing substrate has a useful specific property.Namely, for this substrate, by the use of p-polarized lightincident at the Brewster angle, one can obtain the essen-tial enhancement of relative change of reflectance that iscaused by the ultrathin layer. This is due to the fact thatthe specular reflectance of bare (without ultrathin surface

0740-3224/2004/030645-10$15.00 ©

layer), transparent, massive substrate tends to zero at theBrewster angle. Thus, the special case in whichp-polarized light is incident in the neighborhood of theBrewster angle has been of main interest for transpar-ent systems from the standpoint of measuring tech-nique.5,10–12

A purpose of this paper is first, to study the reflectioncharacteristics in the long-wavelength limit for anN-layer system of nonuniform dielectric films on a trans-parent, homogeneous substrate. A second aim is to in-vestigate further the possibilities for determining the pa-rameters of ultrathin films on the basis of second-orderapproximate expressions for reflection characteristics.

The present paper is an extension of our earlier work9

(where we considered the same problem in the case ofstrongly absorbing substrates), and therefore it is called‘‘II. Transparent substrate.’’ In comparison with an ab-sorbing substrate the fully transparent system is morecomplicated: in the absence of absorption, the contribu-tion of the ultrathin layer to the reflectance in the first or-der at small values of d/l (thickness d is much less thanan optical wavelength l) is equal to zero; therefore, thecontribution of the ultrathin layer must be expressed atleast in the second order with respect to d/l.1,13–15 Theultrathin films are considered phenomenologically withinthe framework of macroscopic electrodynamics by usingthe concept of local dielectric constant. It then followsthat we restrict ourselves to films whose thickness ismuch greater than the spatial dimension on which the re-fractive index is formed (d > a few nanometers).

The paper is organized as follows. In Section 2 the

2004 Optical Society of America

646 J. Opt. Soc. Am. B/Vol. 21, No. 3 /March 2004 Peep Adamson

new general, second-order approximate expressions arederived for the amplitude reflection coefficient and reflec-tance of s- and p-polarized light for an N-layer system ofinhomogeneous, dielectric films on homogeneous, trans-parent substrate. The differential reflectance andchanges of ellipsometric parameters caused by an ultra-thin multilayer are also considered. In Section 3 a closerexamination of peculiar features of the reflectance ofp-polarized light at the Brewster angle is carried out.Next, Section 4 is concerned with the possibility of deter-mining the optical parameters of ultrathin dielectric filmson nonabsorbing substrates by differential reflection char-acteristics. A number of novel, straightforward expres-sions for the determination of the dielectric constant ofuniform layers are established. The obtained approxi-mate expressions can be applied not only for ultrathinfilms; in the infrared and microwave regions these formu-las work also for thick films (the only condition beingd/l ! 1). Finally, it is worth pointing out that all ap-proximate analytical results obtained in this paper arecorrelated with the exact computer solution of the reflec-tion problem for a multilayer system of nonuniform films.

2. BASIC FORMULASAssuming that all the media are nonmagnetic, we con-sider the reflection of s- and p-polarized, time-harmonic,electromagnetic plane waves with a vacuum wavelength lin the medium with dielectric constant ea [ na

2 from amultilayer system consisting of N inhomogeneous dielec-tric layers of thickness di ! l and dielectric constante i(z) [ ni

2(z) (i 5 1,..., N so that eN11 5 es [ ns2 is the

dielectric constant of the semi-infinite, nonabsorbing, iso-tropic, and homogeneous substrate) varying only in thedirection perpendicular to the layers (along the z axis).

We use the standard matrix method16,17 for calculatingthe contribution of ultrathin layers to the reflection char-acteristics. For the system of N inhomogeneous layersone may obtain the characteristic matrix

MN~ s! 5 Fm11

~ s! m12~ s!

m21~ s! m22

~ s!G , (1)

where s [ s, p is the index of polarization, in the long-wavelength approximation in the second order for smallvalues of kdi 5 2p(di /l), and where

m11~ s! 5 1 2 k2(

i51

N

ai~ s!di

2 2 k2 (i51

N21

bi~ s!di (

j5i11

N

cj~ s!dj ,

m12~ s! 5 2ik(

i51

N

bi~ s!di , m21

~ s! 5 2ik(i51

N

ci~ s!di ,

m22~ s! 5 1 2 k2(

i51

N

gi~ s!di

2 2 k2 (i51

N21

ci~ s!di (

j5i11

N

bj~ s!dj ,

ai~s ! 5 di

22E0

di

dzEz

di

@e i~z! 2 ea sin2 fa#dz,

bi~s ! 5 1, ci

~s ! 5 di21E

0

di

@e i~z ! 2 ea sin2 fa#dz,

gi~s ! 5 di

21E0

di

@e i~z ! 2 ea sin2 fa#dz

2 di22E

0

di

@e i~z ! 2 ea sin2 fa#zdz,

ai~ p ! 5 di

22E0

di

@1 2 e i21~z !ea sin2 fa#dzE

z

di

e i~z!dz,

bi~ p ! 5 1 2 di

21ea sin2 faE0

di

e i21~z !dz,

ci~ p ! 5 di

21E0

di

e i~z !dz,

gi~ p ! 5 di

21E0

di

e i~z !dz 2 di22E

0

di

e i~z !zdz

2 di22ea sin2 faE

0

di

e i~z !dzEz

di

e i21~z!dz,

where fa is the angle of incidence. If N 5 1, then all theterms containing summation over ( i51

N21 vanish in all for-mulas here and below.

On the basis of characteristic matrix (1) one can calcu-late the reflection coefficient for complex amplitudes bythe formula

rN~ s! 5 $@m11

~ s! 1 m12~ s!Ps

~ s!#Pa~ s! 2 @m21

~ s! 1 m22~ s!Ps

~ s!#%

3 $@m11~ s! 1 m12

~ s!Ps~ s!#Pa

~ s!

1 @m21~ s! 1 m22

~ s!Ps~ s!#%21, (2)

in which

Pa~s ! 5 na cos fa , Ps

~s ! 5 ns cos fs ,

Pa~ p ! 5 na /cos fa , Ps

~ p ! 5 ns /cos fs ,

cos fs 5 ~1 2 eaes21 sin2 fa!1/2,

and which in the second order in kdi leads to the follow-ing result:

rN~ s! ' r0

~ s!F1 12k2Pa

~ s!Ps~ s!

@Pa~ s!#2 2 @Ps

~ s!#2 X(i51

N

@ gi~ s! 2 ai

~ s!#di2

1 (i51

N21

(j5i11

N

@ci~ s!bj

~ s! 2 cj~ s!bi

~ s!#didj

2 $Ps~ s!@Pa

~ s! 1 Ps~ s!#%21(

j51

N

(i51

N

$ci~ s!

2 @Ps~ s!#2bi

~ s!%$cj~ s! 1 Pa

~ s!Ps~ s!bj

~ s!%didjC1 i

2kPa~ s!

@Pa~ s!#2 2 @Ps

~ s!#2 (i51

N

$ci~ s! 2 @Ps

~ s!#2bi~ s!%diG.

(3)

Here r0( s) is the amplitude reflection coefficient from bare

substrate (di [ 0): r0( s) 5 @Pa

( s) 2 Ps( s)#/@Pa

( s) 1 Ps( s)#.


Hence, for the reflectance RN( s) 5 urN

( s)u2 we obtain the fol-lowing expression accurate to the second order in smallvalues of kdi:

RN~ s! ' R0

~ s!F1 14k2Pa

~ s!Ps~ s!

@Pa~ s!#2 2 @Ps

~ s!#2 X(i51

N

@ gi~ s! 2 ai

~ s!#di2

1 (i51

N21

(j5i11

N

@ci~ s!bj

~ s! 2 bi~ s!cj

~ s!#didj

1 $@Pa~ s!#2 2 @Ps

~ s!#2%21(i51

N

(j51

N

$ci~ s!

2 @Ps~ s!#2bi

~ s!%$cj~ s! 2 @Pa

~ s!#2bj~s!%didjCG, (4)

where R0( s) 5 ur0

( s)u2.In what follows, we consider the relative contributions

@rN( s) 2 r0

( s)#/r0( s) [ DrN

( s)/r0( s) and @RN

( s) 2 R0( s)#/R0

( s)

[ DRN( s)/R0

( s) of the multilayer system of dielectric films.From relations (3) and (4) we obtain for s-polarization:

where A [ 8p2nans cos fa cos fs , B [ ea 2 es , C[ ea cos2 fs 2 es cos2 fa , h i [ 1 2 eaeni

21 sin2 fa , andthe quantities e ti , eni , a i , and b i are expressed in termsof e i(z) as

e ti 5 di21E

0

di

e i~z !dz, (9)

DrN~s !

r0~s !

'A

Bl2 F(i51

N

~e ti 2 2a i!di2 1 (

i51

N21

(j5i11

N

~e ti 2 e tj!didj

2 (i51

N

(j51

N~e ti 2 es!~e tj 2 ea sin2 fa 1 nans cos fa

~es cos2 fs 1 nans cos fa cos fs!

DRN~s !

R0~s !

'2A

Bl2 F(i51

N

~e ti 2 2a i!di2 1 (

i51

N21

(j5i11

N

~e ti 2 e tj!didj

and for p-polarization:

DrN~ p !

r0~ p !

'A

Cl2 F(i51

N

~e ti 2 2a i 1 b iea sin2 fa!di2 1 (

i51

N21

(j5i11

N

2 (i51

N

(j51

N~e ti cos2 fs 2 esh i!~e tjenj cos fa cos fs 1

enj cos fs~es cos fa 1 n

1 i4pna cos fa

Cl(i51

N

~e ti cos2 fs 2 esh i!di , (7)

DRN~ p !

R0~ p !

'2A

Cl2 F(i51

N

~e ti 2 2a i 1 b iea sin2 fa!di2 1 (

i51

N21

(j5i11

N

1 C21(i51

N

(j51

N

~esh i 2 e ti cos2 fs!~eah j 2 e tj cos2 f

eni21 5 di

21E0

di

e i21~z !dz, (10)

a i 5 di22E

0

di

e i~z !zdz, (11)

b i 5 di22E

0

diF e i~z !E0

z

e i21~z!dz 2 e i

21~z !E0

z

e i~z!dzGdz.

(12)

In the case of normal incidence (fa 5 0) and single layer(N 5 1), the formula for differential reflectance @R1

(s, p)

3 (fa 5 0) 2 R0(s, p) (fa 5 0)]/R0

(s, p) (fa 5 0) [ D R1(0) /

R0(0) takes significantly simpler form:

DR1~0 !/R0

~0 ! ' 16p2nans~ea 2 es!22@~ea 2 es!~e t1 2 2a1!

1 ~ea 2 e t1!~es 2 e t1!#~d1 /l!2. (13)

Note that the differential reflectance is an immediatelymeasurable quantity because the relative change in theintensity of the reflected signal

DIN~R !/I0

~R ! 5 @IN~R ! 2 I0

~R !#/I0~R !

5 @RN~ s!I ~i ! 2 R0

~ s!I ~i !#/R0~ s!I ~i ! 5 DRN

~ s!/R0~ s! ,

where IN(R) and I0

(R) are the reflected intensities frommultilayer and bare substrate, respectively, and I (i) is theintensity of the incident light.

fs!didjG 1 i

4pna cos fa

Bl(i51

N

~e ti 2 es!di , (5)

21(i51

N

(j51

N

~es 2 e ti!~ea 2 e tj!didjG , (6)

ih j 2 e tjh i!didj

nsenj 2 nansea sin2 fa!

cos fs!didjG

ih j 2 e tjh i!didj

idjG , (8)

cos

1 B

~e t

na

ans

~e t

a!d


Next we consider the ellipsometric quantities in thelong-wavelength limit. If we represent the ratio of theamplitude reflection coefficients for p- and s-polarized ra-diation in the conventional form as rN

(p)/rN(s) [ tan C

3 exp(iD), where C and D are the ellipsometric angles,then to first order with respect to small values of di /l onthe basis of relation (3) for contributions dC5 C 2 C0 and dD 5 D 2 D0 (C0 , D0 are the ellipsomet-ric angles of a bare substrate) of an ultrathin multilayerto the ellipsometric parameters, we obtain

dD ' 4pna cos fa sin2 fa~es cos2 fa

2 ea cos2 fs!21(

i51

N

g i~di /l!, (14)

g i 5 di21E

0

di

@e i~z ! 2 ea#@es 2 e i~z !#e i21~z !dz

5 @ea~eni 2 es! 1 eni~es 2 e ti!#eni21, (15)

if fa Þ fB (for fa 5 fB , we have dD 5 p/2), and

dC ' p~ea 1 es!1/2U~es 2 ea!21(

i51

N

g i~di /l!U , (16)

if fa 5 fB [for fa Þ fB the quantity dC ; (di /l)2].Hence, in the first order approximation in di /l the con-tribution of each layer in an N-layer system to the ellip-sometric angles is expressed in a purely additive form,i.e., it is not sensitive to the presence of the other layers.Note that in the first order in di /l the contributions of aninhomogeneous layer to dD and dC are equal to zero ifes(eni 2 ea) 5 eni(e ti 2 ea).

It follows from the preceding relations (6) and (8) thatin the case of transparent systems, the contribution of aninhomogeneous layer to the reflectance in the long-wavelength limit in the second order in di /l can be char-acterized by two independent integral parameters e ti anda i for s-polarization and four integral quantities e ti , eni ,a i , and b i for p-polarization. In contrast to the reflec-tance the ellipsometric quantities dD and dC can be de-scribed in the first order in di /l and by only two averagedmaterial parameters e ti and eni . It should be mentionedthat if the contribution of an inhomogeneous dielectriclayer to the reflectance also has the first order of small-ness in di /l as in the case of strongly absorbingsubstrate,9 the number of integral parameters needed foreach layer is half as many: only e ti in the case ofs-polarization and e ti , eni , for p-polarization.

The refractive index of inhomogeneous layers is de-scribed by one of the following functions:

ni~z ! 5 n0i 1 ~ndi 2 n0i!~z/di!g, (17)

ni~z ! 5 n0indi@ndig 2 ~ndi

g 2 n0ig!~z/di!#

21/g, (18)

ni~z ! 5 n0i~ndi /n0i!z/di [ n0i exp@ln~ndi /n0i!~z/di!#,

(19)where n0i and ndi are the values of refractive index at z5 0 and di , respectively, and g Þ 0 is a certain realnumber. As an illustration, Fig. 1 shows the depen-dences of functions (17)–(19) on z for different values of g.Note that for a uniform layer we have e ti 5 eni [ e i (e i is

the dielectric constant of the ith uniform layer), b i 5 0,e ti 2 2a i 5 0, and g i 5 (e i 2 ea)(es 2 e i)e i

21. From thelast formula it is apparent that in the homogeneous case,g i has the same values for two different values of e i de-termined by the simple relation e i

(1)e i(2) 5 eaes , and that

g i reaches its maximum value in the interval ea , e i, es if e i equals the geometric mean of the dielectric con-stants of the substrate and the surrounding medium, i.e.,e i 5 Aeaes.

Two questions arise regarding the above-derived for-mulas. First, what is the accuracy of approximate for-mulas? Second, how much faster is this approximationcompared with the standard numerical techniques fornonuniform films9,18: (i) by direct numerical integrationof the second-order differential equation,19 and (ii) by di-viding the nonuniform film into a large number of uni-form layers and using a matrix or iterative method to cal-culate reflection characteristics.20

The relative errors n 5 $@DRN( s)/R0

( s)#ex 2 DRN( s)/

R0( s)%/@DRN

( s)/R0( s)#ex , where @DRN

( s)/R0( s)#ex was obtained

by using the numerical techniques for exact solution ofthe reflection problem and DRN

( s)/R0( s) was calculated by

use of approximate relations (6) and (8), are presented,e.g., for a two-film system in Figs. 2 and 3. The profilesof n1(z) and n2(z) are described by Eqs. (17) and (18), re-spectively, with g 5 1 (Fig. 2) and with g 5 3 (Fig. 3).Note that the quantities l and di are measured in all fig-ures in arbitrary common units. As shown by exact nu-merical calculations, the error of relations (6) and (8) doesnot exceed a few percent if the approximate maximumvalues of ( i51

N di /l amount to a few hundredths. This isin good agreement with the condition ( i51

N di /l ! 1/2pused in the derivation of the basic formulas. We mayalso note that in comparison with s-polarization, the dif-ferential reflectance of p-polarization as a function of fachanges its sign passing through the zero value and theapproximate formula is inapplicable [Fig. 3(b)] in prin-ciple in the neighborhood of the point where DRN

( p) → 0(even for di /l as small as wished) because it is insuffi-cient to restrict oneself to terms of the second order in theexpansion in di /l in the vicinity of this point.

Fig. 1. (a) Profiles according to Eqs. (17) and (19) (dashed curve)and (b) according to Eq. (18) of the refractive index ni(z) for dif-ferent values of g (the numbers on the curves) when n0i 5 1.5and ndi 5 2.


The analogous relative errors of ellipsometric quanti-ties [relations (14) and (16)], e.g., for a three-film systemwhere n1(z) is described by Eq. (17) with g 5 2, n2(z) isdescribed by Eq. (18) with g 5 4, and n3(z) is describedby Eq. (19), are demonstrated in Fig. 4. It is important tobear in mind that the approximate relations (14) or (16)for ellipsometric parameters cease to work if dD → 0 ordC → 0 (or more specifically, if dD or dC < max(di /l);then one should take into account terms of second orderin the expansion in a small value of di /l).

It can be inferred from exact numerical calculationsthat the accuracy of approximate expressions for fixeddi /l depends appreciably on the value of the gradient ofrefractive index: for steep gradients the error is, as arule, between 1 and 2 orders of magnitude larger than thecorresponding error for small gradients (undi 2 n0iu

Fig. 2. Relative errors of approximation formulas for (a)DR2

(s)/R0(s) [relation (6)] and for (b) DR2

( p)/R0( p) [relation (8)] as

functions of l for a two-film system with d1 5 d2 5 4 when fa5 45°; na 5 1; ns 5 1.5 (4, 5), 4 (the other curves); n01 5 1.3(4), 1.4 (2, 3), 4 (1, 5); nd1 5 n02 5 1.5 (1), 2 (2, 5), 2.5 (4), 4 (3);nd2 5 1.3 (5), 1.5 (4), 2 (3), 4 (2), 4.5 (1). Preceding numbers inparentheses are curve labels.

Fig. 3. Relative errors of approximation formulas for (a)DR2

(s)/R0(s) [relation (6)] and for (b) DR2

( p)/R0( p) [relation (8)] as

functions of fa for a two-film system with d1 /l 5 0.01 andd2 /l 5 0.02 when na 5 1; ns 5 2; n01 5 1.3 (1), 2.5 (2); nd15 n02 5 1.5 (2), 4.3 (1); nd2 5 1.3 (1), 2 (2). Preceding numbersin parentheses are curve labels.

, 1). These calculations also indicate notably that theerror is in a lesser degree sensitive to the shape of the in-dex profile; however, generally the error is larger for g. 1 than for g 5 1 in Eqs. (17) or (18). Note that in theinfrared for ultrathin films with nanometric thickness,the obtained approximations of relations (5)–(8), (14), and(16) reach theoretically high exactness, but there could bedifficulties in measuring very small values of differentialreflection quantities DRN

( s)/R0( s) or dD and dC.

The speed of calculation of approximate expressions forinhomogeneous ultrathin films is determined first of allby the speed with which one can calculate the integrals ofrelations (9)–(12). The calculations show that if the rela-tive error allowed is of the order 1022 –1023, we can re-duce the calculation time between 1 and 2 orders of mag-nitude compared with the method of direct numericalintegration of the wave equation, or with the technique ofsubdividing inhomogeneous film into a large number ofhomogeneous layers. The speed of the approximation athand practically does not depend on the gradient of re-fractive index or on the shape of index profiles of Eqs.(17)–(19).

It must be emphasized that the main advantage of thelong-wavelength approximation lies in its reversibility;i.e., on the basis of approximate relations obtained wecan, as shown later, rapidly calculate not only the reflec-tion characteristics but also the thickness and refractiveindex of ultrathin films. The latter possibility is of primeimportance because the use of numerical techniques tosolve the inverse problem based on exact reflection equa-tions, which are transcendental and highly nonlinear, israther complicated. These equations have, as a rule,many physically meaningful solutions, and often may nothave a solution at all, or it is impossible to obtain a solu-tion with the desired accuracy because of the presence ofmeasurement errors. Also, the use of these techniqueswould need profound improvement in computationalspeed.

Fig. 4. Relative errors of approximation formulas for (a) dD [re-lation (14)] when fa 5 45° (solid curves) and 75° (dashed curves)and for (b) dC [relation (16)] as functions of l for a three-film sys-tem with d1 5 d2 5 d3 5 1 at na 5 1; ns 5 1.5; n01 5 1 (6), 1.5(1, 4), 2 (7), 3 (2, 5), 4 (3); nd1 5 n02 5 1.3 (2, 5), 1.5 (3), 2(1, 4, 7), 4 (6); nd2 5 n03 5 1.5 (6), 2.5 (1, 3, 4), 3 (2, 5), 4 (7);nd3 5 1.2 (3), 1.5 (1, 4, 6), 4 (2, 5, 7). Preceding numbers in pa-rentheses are curve labels.


3. BREWSTER ANGLEIn this section we consider the reflectance at the Brewsterangle. From relation (8) we learn that at the angle fa5 fB for a transparent N-layer system

DRN~ p ! [ RN

~ p !~fB! ' p2~ea 1 es!21F(

i51

N

g i~di /l!G 2

.

(20)

As an illustration, Fig. 5 shows the dependence of R3( p)

3 (fB) on ns for a three-film system, where n1(z) andn2(z) are described by Eq. (17) with g 5 4 and by Eq. (18)with g 5 4, respectively, and n3(z) is described by Eq.(19). Note that the exactness of the second-order ap-proximation of Eq. (20) is insufficient when RN

( p)(fB) ap-proaches a minimum (also see Fig. 5). If N 5 1 then aminimum of Eq. (20) is equal to zero @g1 5 0 at es5 en1(e t1 2 ea)(en1 2 ea)21]. Therefore, in this case, inthe long-wavelength approximation, an inhomogeneouslayer has no effect on reflectance at the Brewster angle,i.e., it is physically tantamount to a homogeneous layerwith e1 5 es .

Let a substrate have an initial surface layer with i5 1. Then the deposition of a second ultrathin layerwith i 5 2 onto the initial layer changes the reflectanceso that DR21

( p)(fB)/R1( p)(fB) ' (g2d2 /g1d1)(g2d2 /g1d1

1 2), where DR21( p) 5 R2

( p) 2 R1( p) . Consequently, the

sensitivity of differential reflectance at the Brewsterangle to the properties of an ultrathin layer deposited onthe real substrate (where always exists an initial surfacelayer: The blurriness of the interface is caused by adsor-bates, defects, microroughness, etc.) is found to be highfor ug2d2 /g1d1u . 1 (the only exclusion is the case ofg2d2 /g1d1 → 22). For homogeneous film when ea 5 1,this requires the fulfillment of the condition ue1 2 esud1, ue2 2 esud2 .

Relation (20) shows that the addition of an ultrathinlayer onto the system with N > 1 may also decrease thereflectance at the Brewster angle. From relation (20) fol-lows also another interesting fact: various permutationsof layers give the same result for the quantity RN

( p)(fB).

Fig. 5. Reflectance R3( p)fB as a function of ns for a three-film

system when d1 /l 5 0.005, d2 /l 5 0.003, and d3 /l 5 0.002 atna 5 1; n01 5 3; nd1 5 n02 5 4; nd2 5 n03 5 1.5; nd3 5 4.Dashed curve corresponds to calculation by relation (20).

Therefore, in the long-wavelength approximation the re-flection of p-polarized light incident at the Brewsterangle, with accuracy to the second order in di /l, is foundto be independent of the way in which these films are ar-ranged. Note that for s-polarized light the change in re-flectance at the Brewster angle is not invariant with re-spect to rearrangements of ultrathin layers.

One further comment should be made in connectionwith the Brewster angle. As for transparent film on ab-sorbing substrate, so for absorbing film on transparentsubstrate, generally DR1

( s) ; d1 /l in the long-wavelength approximation except at the angle fa 5 fBfor p-polarization. If the latter is the case, the reflec-tance for p-polarized light incident at angle fB vanishesin the first order in d1 /l, not only for transparent layersbut for absorbing layers as well. For example, for homo-geneous absorbing film with complex dielectric constante1 5 e1 1 ij1 [ (n1 1 ik1)2, one can obtain for the re-flectance in the third order in d1 /l:

R1~ p !~fB! ' R10

~ p !~fB!@1 2 2p~ea 1 es!21/2

3 ~1 1 eaesu e1u22!j1~d1 /l!#, (21)

R10~ p !~fB! ' p2u e1u22[~ea 1 es!@~ u e1u2 2 2ease1 1 eas

2!1/2

1 u e1u2~ea 1 es!21#2 2 2(e1u e1u2

1 ~j12 2 e1

2!eas 1 $@e1u e1u2

1 ~j12 2 e1

2!eas#2 1 ~ u e1u2

2 2ease1!2j12%1/2)]~d1 /l!2, (22)

where R10( p)(fB) is the reflectance at fa 5 fB in the sec-

ond order in d1 /l, u e1u2 5 e12 1 j1

2, and eas 5 eaes /(ea1 es). Hence, all actual physical surfaces, which as arule have an interfacial transition layer (with thicknessd1), form nonzero reflectance ;(d1 /l)2 for p-polarizedlight incident at the Brewster angle. Note that the ex-actness of second-order relation (22) is inadequate. Therelative error of approximate relation (21) is plotted inFig. 6.

Fig. 6. Relative errors of approximate relation (21) as functionsof (a) d1 /l when k1 5 0.5 and of (b) k1 when d1 /l 5 0.01 atna 5 1; ns 5 1.5 (1, 6, 7), 4 (2–5); n1 5 1.3 (7), 1.5 (5), 2 (4, 6),3.5 (3), 4 (1), 4.5 (2). Preceding numbers in parentheses arecurve labels.


4. OPTICAL DIAGNOSTICSA. Homogeneous Ultrathin FilmNext we consider certain aspects of optical diagnostics ofultrathin layers as an application of the approximate for-mulas obtained. First, we examine the case where asingle uniform layer will be deposited on a substrate. Inthis case the pure ellipsometric method is not applicablefor the determination of e1 on the basis of approximate re-lations (14) and (16), whereas the relation dD/dC is inde-pendent of e1 . But ellipsometry in combination with dif-ferential reflectance is successfully employed fordiagnostics of ultrathin films as in the case of absorbingsubstrate.9 For example, if the differential ellipsometricangle dD is measured at fa [ fa

(D) and the differential re-flectance DR1

(s)/R0(s) at fa [ fa

(s) then from relations (6)and (14) we obtain that

~dD!2/@DR1~s !/R0

~s !#

' na@ns cos fa~s ! cos fs

~s !#21@cos2 fa~D! sin4 fa

~D!#~es 2 ea!2

3 @es cos2 fa~D! 2 ea cos2 fs

~D!#22

3 ~e1 2 ea!~e1 2 es!e122. (23)

The solution of the inverse problem, i.e., the determina-tion of e1 from (dD)2/(DR1

(s)/R0(s)), is generally not unique

(two physically meaningful solutions can be found):

e1 ' $es 1 ea 6 @~es 2 ea!2 1 4eaest#1/2%@2~1 2 t !#21,(24)

t [ns cos fa

~s ! cos fs~s !@es cos2 fa

~D! 2 ea cos2 fs~D!#2

na cos2 fa~D! sin4 fa

~D!~es 2 ea!2

3~dD!2

DR1~s !/R0

~s !. (25)

For ea 5 1 (actual in practice) the numerical simulationshows that if physically correct solution e1

(1) > es , then,as a rule, the extraneous (physically incorrect) solutione1

(2) < 1. But if the condition e1(1) , es is fulfilled, then

e1(2) . 1 and, in addition, e1

(2) may come close to e1(1) if

e1(1) ! es .

Subsequently, for the determination of e1 we can useonly the differential reflection of p-polarized light mea-sured at two different incident angles. Indeed, for twospecial cases, first, for fa 5 0 and, second, for fa 5 fB ,relations (13) and (20) yield

DR1~0 ! ' 16p2nans~na 1 ns!

24~e1 2 ea!

3 ~e1 2 es!~d1 /l!2, (26)

DR1~ p !~fB! ' p2~es 1 ea!21~e1 2 ea!2

3 ~es 2 e1!2e122~d1 /l!2. (27)

From relations (26) and (27) for e1 can be obtained pre-cisely the same formula as relation (24) but the quantity twill be in the form

t [ 16nans~na 1 ns!24~ea 1 es!@DR1

~ p !~fB!/DR1~0 !#.

(28)

Thus, for p-polarized light one can measure the quantity

DI1~R !~fa 5 fB!/DI1

~R !~fa 5 0 !

5 DR1~ p !~fa 5 fB!/DR1

~ p !~fa 5 0 !

[ DR1~ p !~fB!/DR1

~0 ! ,

so that the intensity of the incident rays is the same forboth normal incidence (fa 5 0) and incidence at theBrewster angle fB , and then determine e1 from relation(24) where the quantity t is calculated by Eq. (28). Notethat, equally well, we can use the relation DR1

( p)

3 (fB)/DR1(s)(fB). In this case

t [ 16eaes~ea 1 es!22@DR1

~ p !~fB!/DR1~s !~fB!#. (29)

But it must be emphasized that s-polarization alone is ofno interest for the solution of the inverse problem becausethe angular dependence of its differential reflectance istrivial @DR1

(s)(fa)/R0(s)(fa) 5 cos fa cos fsDR1

(0)/R0(0)#.

For reference we have included a computer simulationfor the possible errors of approximate formulas obtainedfor optical diagnostics. For example, the relative errorsof relation (24) [if t is determined by Eq. (25) or by Eq.(29)] as functions of d1 /l and ns are plotted in Fig. 7. Asseen in Fig. 7(a) the error does not exceed a few percent-age points if d1 /l < a few hundredths. The dependenceof the error on ns [Fig. 7(b)] shows that the methods athand do not work if ns → 1 but this is not urgent in prac-tice. One should also remember that the angle of inci-dence for ellipsometric parameter dD must be differentfrom the Brewster angle for the substrate, i.e., fa

(D)

Þ arctan ns [Fig. 7(b)].Next, if the parameters e1 and d1 of the first homoge-

neous ultrathin layer (i 5 1) are known and, in addition,the condition d2 ! d1 is satisfied for the second layer (i5 2), then at fa 5 fB , for the relative change in reflec-tance

Fig. 7. Relative errors of approximate relations (24) and (25)(solid curves) and (24) and (29) (dashed curves) as functions of (a)d1 /l if ns 5 1.5 and of (b) ns if d1 /l 5 0.01 at na 5 1; n1

5 2; fa(D) 5 fa

(s) 5 65°; m1 5 0 (1), 3% (2), 23% (3). The vari-able m1 is the relative error of the quantities (dD)2/(DR1

(s)/R0(s))

and DR1( p)(fB)/DR1

(s)(fB). Preceding numbers in parenthesesare curve labels.


DR21~ s!~fB!/R1

~ s!~fB! 5 @R2~ s!~e1 , e2 , d1 , d2 , fB! 2 R1

~ s!

3 ~e1 , d1 , fB!#/R1~ s!

3 ~e1 , d1 , fB!

that is caused by deposition of the second layer to thefirst, one can obtain the following from relations (20) and(6) [note that in relations (5)–(8), numbering of the filmsis oppositely directed, i.e., for the first film i 5 2 and forthe second film i 5 1]:

DR21~ p !~fB!/R1

~ p !~fB! ' 2e1e221~e2 2 ea!~e2 2 es!~e1

2 ea!21~e1 2 es!21d2d1

21,

(30)

DR21~s !~fB!/R1

~s !~fB! ' 32p2eas~e2 2 ea!~e1 2 es!

3 ~es 2 ea!22~d1d2 /l2!. (31)

From relations (30) and (31), it follows that

e2 5 es@1 2 16p2eas~es 2 ea!22e121~e1 2 ea!

3 ~es 2 e1!2~d1 /l!2V#21, (32)

where V [ @DR21( p)(fB)/R1

( p)(fB)#/@DR21(s)(fB)/R1

(s)(fB)#.Equation (32) can, e.g., be used for optical monitoring ofdeposition of thin films. It is of considerable importance,because it provides an unambiguous determination of e2with reasonable error (Fig. 8).

B. Inhomogeneous Ultrathin FilmFor an inhomogeneous ultrathin film, in principle, onlythe integral parameters of Eqs. (9)–(12) can be deter-mined on the basis of a given theoretical treatment. Theaveraged material parameters e ti and eni , which are suf-ficient to estimate the reflection characteristics in thefirst-order approximation in di /l, have a simple physicalinterpretation.21 Therefore, a practical implementationof such quantities would be worthwhile. However, thedetermination of e ti and eni for dielectric films on trans-parent substrates is not a straightforward problem. This

Fig. 8. Relative error of Eq. (32) as a function of l when na5 1; n1 5 3; n2 5 2; ns 5 1.5 (2, 3), 4 (1, 4–7); d1 5 1 (1–3, 5, 7), 5 (4), 10 (6); d2 5 0.1 (1, 2, 7), 0.2 (3, 5), 0.5 (4), 2 (6);m5 5 0 (solid curves), 1% (2), 3% (7), 23% (1). The variable m5is the relative error of the quantity V. Preceding numbers inparentheses are curve labels.

is because the reflection characteristics of an inhomoge-neous film also depend on the second-order integral pa-rameters a i and b i . If the thickness of the inhomoge-neous film is known, then one possibility for creating thenecessary relations that contain only e ti and eni , but notsecond-order quantities a i and b i , is as follows. If thestudied inhomogeneous layer (i 5 1) with unknown ma-terial parameters e t1 and en1 and with known thicknessd1 is coated with the second homogeneous ultrathin layer(i 5 2) for which material parameter e2 is known (thethickness d2 can be determined from ellipsometric quan-tities dD or dC), then DR21

( p)(fa)/R1( p)(fa), in which

DR21( p) 5 R2

( p)(e t1 , en1 , d1 , e2 , d2) 2 R1( p)(e t1 , en1 , d1),

yields

DR21~ p !~fa!/R1

~ p !~fa! ' 2A~Cl!22@2~eah2 2 e2 cos2 fa!

3 ~esh1 2 e t1 cos2 fs!d1d2

1 ~eah2 2 e2 cos2 fa!~esh2

2 e2 cos2 fs!d22#. (33)

On the basis of Eq. (33) one can obtain the relationship

es 2 esea sin2 faen121 2 cos2 fse t1 ' T~fa!~l/d1!,

(34)

where

T~fa!

[DR21

~ p !~fa!

R1~ p !~fa!

~ea cos2 fs 2 es cos2 fa!2

32p2nans cos fa cos fs~eah2 2 e2 cos2 fa!

3l

d21

~e2 cos2 fs 2 esh2!

2

d2

l. (35)

Hence, if the quantity DR21( p)(fa)/R1

( p)(fa) is measuredfor two different angles of incidence fa 5 fa

(1) and fa

5 fa(2) , then e t1 and en1 can be calculated by the follow-

ing expressions:

e t1 ' es 1 $T@ fa~1 !#sin2 fa

~2 !

2 T@ fa~2 !#sin2 fa

~1 !%A21~l/d1!, (36)

en121 ' es

21 1 ~esea!21$T@ fa~2 !#cos2 fs

~1 !

2 T@ fa~1 !#cos2 fs

~2 !%A21~l/d1!, (37)

where A [ sin2 fa(1) 2 sin2 fa

(2) . The results of computersimulation of the error for approximate relations (36) and(37), if the errors of the quantities DR21

( p)@ fa(1)#/R1

( p)

3 @ fa(1)# and DR21

( p)@ fa(2)#/R1

( p)@ fa(2)# are equal to zero,

are demonstrated in Fig. 9. It can be inferred from theseresults that, except at relatively steep gradients, formulas(36) and (37) remain a valid approximation, the error be-ing of the order of several percent.

The application of the third measurement to the quan-tity DR21

( p)(fa)/R1( p)(fa) [at the angle of incidence fa

5 fa(3)] does not provide any possibility for the determi-

nation of d1 because the following relationship is true:

AT@ fa~3 !# 5 T@ fa

~1 !#@sin2 fa~3 ! 2 sin2 fa

~2 !# 1 T@ fa~2 !#

3 @sin2 fa~1 ! 2 sin2 fa

~3 !#. (38)


In other words, three quantities DR21( p)@ fa

(1)#/R1( p)@ fa

(1)#,DR21

( p)@ fa(2)#/R1

( p)@ fa(2)#, and DR21

( p)@ fa(3)#/R1

( p)@ fa(3)# are

not independent measurements. A similar statement istrue for DR21

( p)@ fa(1)#/R1

( p)@ fa(1)#, DR21

( p)@ fa(2)#/R1

( p)@ fa(2)#,

and dD (or dC). If two of these quantities are known,then the remaining one can be determined withoutknowledge of the parameters of an ultrathin film from thefollowing relation:

A@es cos2 fa~D! 2 ea cos2 fs

~D!#@4pna cos fa~D! sin2 fa

~D!#21dD

5 T@fa~1 !#@cos2 fs

~2 ! 2 sin2 fa~2 !#

2 T@fa~2 !#@cos2 fs

~1 ! 2 sin2 fa~1 !#. (39)

But if the thickness d1 of the layer to be studied is known,then

e t1 ' es 2 $T@fa~1 !# 2 P sin2 fa

~1 !%

3 @cos2 fs~1 ! 2 sin2 fa

~1 !#21~l/d1!, (40)

en121 ' es

21 1 ~eaes!21$T@ fa

~1 !# 2 P cos2 fs~1 !%

3 @cos2 fs~1 ! 2 sin2 fa

~1 !#21~l/d1!, (41)

where

P 5 dD~es cos2 fa~D! 2 ea cos2 fs

~D!!

3 ~4pna cos fa~D! sin2 fa

~D!! 2 1.

Therefore, we cannot, on the basis of approximate expres-sions given in this paper, analytically resolve the problemof simultaneously determining all three parameters e t1 ,en1 , and d1 of inhomogeneous, ultrathin, dielectric filmson transparent substrates.

5. CONCLUSIONSIn the long-wavelength approximation, the dielectric ul-trathin film in a multilayer system on a transparent sub-strate is sensitive to the presence of other ultrathin films,

Fig. 9. Relative errors of approximate relations (36) (solidcurves) and (37) (dashed curves) as functions of (a) l when d15 3; d2 5 2; ns 5 1.5 (1, 2), 4 (3), and of (b) ns when d1 /l5 0.0015; d2 /l 5 0.001 at fa

(1) 5 0°; fa(2) 5 45°; na 5 1; n1

5 n02 5 1.5 (4), 2 (2, 3), 3 (1), 4 (5); nd2 5 1.5 (1), 2 (4), 2.5(2, 3, 5). Profiles n2(z) are described by Eq. (18) with g 5 1.Preceding numbers in parentheses are curve labels.

and an interaction between them emerges. As a conse-quence, the contribution of each ultrathin film to the re-flection coefficients is not expressed in a purely additiveform. Physically, this is the most dramatic difference be-tween transparent and absorbing substrates. The accu-racy of the long-wavelength approximation for fixed di /lremarkably depends on the value of the gradient of a re-fractive index profile: for steep gradients the error isconsiderably larger than for small ones (undi 2 n0iu, 1). The long-wavelength approximation reduces thecalculation time of reflection characteristics for inhomoge-neous films by between 1 and 2 orders of magnitude com-pared with common numerical techniques.

The reflection of p-polarized light incident at the Brew-ster angle, with accuracy to the second order in di /l, isfound to be independent of the way in which these filmsare arranged, but for s-polarized light the change in re-flectance at the Brewster angle is not invariant with re-spect to rearrangements of ultrathin films. At the Brew-ster angle, the absorbing, ultrathin, interfacial transitionlayer (with thickness d1) also forms the reflectance;(d1 /l)2 as a transparent transition layer.

The essential property of the obtained approximationformulas is that they are relatively easily invertible, al-lowing direct calculation of refractive index and thicknessof uniform, ultrathin dielectric film on transparent sub-strate on the basis of differential measurements. For in-stance, the successful method for this purpose is the com-bination of ellipsometry with differential reflectance.But on the basis of second-order approximation formulasfor differential reflectance of linearly polarized light, onecannot solve analytically the inverse problem for an inho-mogeneous ultrathin film. Only when the thickness ofthe inhomogeneous layer is known can the average mate-rial parameters be easily determined.

P. Adamson’s e-mail address is [email protected].

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