Shadow-Angle Method for Anisotropic and Weakly Absorbing Films

i

Shadow-angle method foranisotropic and weakly absorbing films

Gregory Surdutovich, Ritta Vitlina, and Vitor Baranauskas

A method for determining the optical properties of a film on an isotropic substrate is proposed. Themethod is based on the existence of two specific incidence angles in the angular interference pattern ofthe p-polarized light where oscillations of the reflection coefficient cease. The first of these angles, uB1,is the well-known Abeles angle, i.e., the ambient–film Brewster angle, and the second angle uB2 is thefilm–substrate Brewster angle. In the conventional planar geometry and in a vacuum ambient there isa rigorous constraint ε1 1 ε . ε1ε on the film and the substrate dielectric permittivities ε1 and ε,respectively, for the existence of the second angle uB2. The limitation may be removed in an experimentby use of a cylindrical lens as an ambient with ε0 . 1, so that both angles become observable. This,contrary to general belief, allows one to adopt the conventional Abeles method not only for films with ε1

close to the substrate’s value ε but also for any value of ε1. The method, when applied to a wedge-shapedfilm or to any film of unknown variable thickness, permits one to determine ~i! the refractive index of afilm on an unknown substrate, ~ii! the vertical and the horizontal optical anisotropies of a film on anisotropic substrate, ~iii! the weak absorption of a moderately thick film on a transparent or an absorbingisotropic substrate. © 1999 Optical Society of America

OCIS codes: 120.5410, 120.5700, 260.5430.

H

1. Introduction

Previously Abeles showed that the determinationprocesses of the film’s refractive index and its thick-ness may be completely separated.1 When therelative reflectances of p-polarized light at the film-covered and uncovered surfaces are equal, the angleof incidence is equal to the Brewster angle uB1 of theambient–film interface, independently of the filmthickness. But one can look at this effect from adifferent point of view. For a film with nonuniformthickness2,3 ~such as a wedge-shaped film or a grow-ng film! under variation of the angle of incidence the

interference fringes of all interference patterns thatcorrespond to the different places ~thicknesses! of thefilm disappear at this angle, whereas the reflectanceof a sample becomes equal to the reflectance of asubstrate. It looks as if at this angle, uB1, the film

The authors are with the Departamento de Semicondutores,Instrumentos e Fotonica, Faculdade de Engenharia Eletrica e deComputacao, Unicamp, CEP 13083-970, Campinas, Sao Paulo,Brazil. G. Surdutovich’s e-mail address is [email protected].

Received 4 December 1998; revised manuscript received 23March 1999.

0003-6935y99y194164-08$15.00y0© 1999 Optical Society of America

4164 APPLIED OPTICS y Vol. 38, No. 19 y 1 July 1999

effectively ceases to exist, and only the angular scan-ning displaces the film from a shadow region. Thuswe now refer to the ambient–film interface Brewsterangle as the shadow Brewster angle ~SBA!. Here wewould like emphasize that in addition to theambient–film angle uB1 there is still one characteris-tic angle, uB2, that corresponds to the film–substrateinterface Brewster angle. At this Brewster anglethe sample reflectance also ceases to depend on thefilm thickness, the interference fringes disappear,and the reflectance of a sample becomes equal to thereflectance of a clean substrate. Previously Abeles’smethod for determining the refractive index of a thinfilm after finding only angle uB1 was applied solely toisotropic and transparent films on a known substrate.The presence of the angle uB2 means the appearanceof the additional measurable parameter and allowsus to extend Abeles’s method to isotropic transparentfilms on an unknown substrate or on absorbing anduniaxially anisotropic films. In addition, by use ofthis angle it is possible to determine the permittivityε1 of the film, not only when it is near the substratevalue ε, as is implied under the conventional Abelesapproach, but also for arbitrary values of ε1 and ε.

owever, there is a rigorous constraint ε1 1 ε . ε1ε onthe film and the substrate permittivities @see Eq. ~6b!#for observation of angle uB2. As a result the directapplication of this approach is limited. Fortunately,

Smdaiwhco

fifaaf

Istta

0

a

o

a

p

=

v

this limitation may be easily removed by use of acylindrical lens as an ambient with ε0 . 1, so that the

BA method becomes effective. It is important toention that under such an approach each SBA is

etermined not as the crossing point of the sample’snd the substrate’s reflectances but as the intercross-ng point in the angular interference patterns of aedge-shaped or a growing film, so there is no need toave an uncovered sample surface. This fact be-omes essential for the monitoring of the parametersf films in situ during deposition.

2. Shadow Brewster Angle Approach

The intensity reflection coefficient R~u! forp-polarized light of an ambient–film–substrate sys-tem has the well-known form4

R~u! 5 U r01 1 r12 exp~2id!

1 1 r01r12 exp~2id!U2

, (1)

where r01 and r12 are Fresnel reflection coefficients atthe ambient–film and the film–substrate interfaces,respectively, d 5 ~4pdyl!~ε1 2 ε0 sin2 u!1y2, l is theree-space wavelength; d is the film thickness, and us the angle of incidence in the ambient. It followsrom Eq. ~1! that there are two particular incidencengles, uB1 and uB2, where the reflection coefficientst the ambient–film and the film–substrate inter-aces become zero:

r01~uB1! 5 0, (2a)

r12~uB2! 5 0, (2b)

and the dependence of R on the thickness d ceases.n other words, in the reflection coefficient R~u! of aample there appear two points of inflection wherehe amplitude of the oscillations of the angular pat-erns becomes zero. Then, according to Eqs. ~1!, ~2a!nd 2~b!, one immediately obtains

R~uB1! 5 ur12~uB1!u2, (3a)

R~uB2! 5 ur01~uB2!u2. (3b)

Since these equations are satisfied for any thicknessd of a film, they are valid also for d 5 0, when R~u, d 5! [ ur02~u!u2, where r02~u! is the ambient–substrate

Fresnel reflection coefficient. As a result, we con-clude that the Abeles relations5 are valid for bothngles,

R~uB1! 5 ur02~uB1!u2, (4a)

R~uB2! 5 ur02~uB2!u2. (4b)

Therefore the curves of the interference patternsfrom all different places of a film with variable thick-nesses at these angles should intersect one another.The explicit forms for the amplitudes r01~u! and r12~u!are4

r01~u! 5ε1 cos u 2 @ε0~ε1 2 ε0 sin2 u!#1y2

ε1 cos u 1 @ε0~ε1 2 ε0 sin2 u!#1y2 , (5a)

r12~u! 5ε@ε1 2 ε0 sin2 u#1y2 2 ε1@ε 2 ε0 sin2 u#1y2

ε@ε1 2 ε0 sin2 u#1y2 1 ε1@ε 2 ε0 sin2 u#1y2 ,

(5b)

whereas the amplitude r02~u! may be obtained by thechange ε13 ε in Eq. ~5a!. From Eqs. ~2! and ~5! onebtains the following expressions for the SBA:

sin2 uB1 5ε1

ε0 1 ε1# 1, (6a)

sin2 uB2 5ε1ε

ε0~ε1 1 ε!# 1. (6b)

Expression ~6a! has a solution for any film and am-bient media, whereas inequality ~6b! is satisfied onlyin a rather narrow limited region of the film’s and thesubstrate’s dielectric permittivities below solid curveε0 5 1 ~see Fig. 1!. For a given ε0 the two-SBA regionlies between the left-hand ε1 5 ~ε0εyε0 2 ε! ~dashedcurves! and the right-hand ε1 5 ~ε0εyε 2 ε0! ~solidcurves! branches, which tend to the asymptote ε 5 ε0for ε1 .. 1. All left-hand branches are situatedbove the bisectrix ε1 5 ε. For ε1 , ε there remains

only the right-hand boundary. The dashed areaaround the bisectrix corresponds to the conventional~with one Abeles angle! approach, generalized for thecase ε0 Þ 1.

Two characteristic ~one-SBA and two-SBA! exam-les of the interference patterns of light incident from

Fig. 1. Diagram showing the SBA regions as a function of ε andε1 for the different values ε0. Curve E ~ε1 5 Eε, where E 5 1 1

2! corresponds to the maximum possible difference 9.88° be-tween the angles uB1 and uB2 ~uB1 5 49.94°, uB2 5 40.06°! for anyalues of ε and ε1, with the optimum value ε0 5 ε0

max 5 ~EyE 2 1!5 1.707, thus implying ε0 , ε. For ε0 . ε the maximum diver-gence of the SBA is reached when ε3 ` and ε13 1, so that uB13~py4! and uB2 3 ~py2!.

1 July 1999 y Vol. 38, No. 19 y APPLIED OPTICS 4165

Smc

cwo

fils h

u

1

s

4

a vacuum ambient ~ε0 5 1! are given in Figs. 2 and 3.The angular dependencies of the reflection coefficientR~u! for different film thicknesses are shown in Fig. 2.Only expression ~6a! is satisfied, so that an angle ofmerely uB1 is observable. The case in which inequal-ity ~6b! is valid, thus making both angles uB1 and uB2observable, is given in Fig. 3. Owing to the closenessof uB2 to ~py2!, intersections of the interference pat-terns take place at acute angles so that the experi-mental finding of the angle uB2 is hampered. Belowwe show how to shift angle uB2 to the left-hand sideand thus avoid this obstacle.

In the usual planar geometry @Fig. 4~a!# the two-BA approach is applicable only for a narrow class ofaterials. If, however, one chooses as an ambient a

ylindrical lens with ε0 . 1 bearing the film @Fig.4~b!#, the solution of expression ~6b! acquires a phys-ical sense in the entire region below solid curve ε0 5onst in Fig. 1. With an increase of ε0 the regionhere this equation has a solution extends through-

ut almost all the physical area.

3. Generalization of the Abeles Approach to the Caseof an Arbitrary Difference «1 2 «

Under the conventional Abeles approach with uni-form film on a partially covered substrate the limita-tion un 2 n1u , 0.3 ~or uε1 2 εu # 1! on the refractiveindices1,6,7 arises because of the need to compare the

lm’s and the substrate’s reflected intensities. Iteads to the requirement R~uB1! ' 1023 ,, 1. Themallness of R~uB1! implies the nearness of the angle

uB1 to the substrate’s Brewster angle uB0. Therefore

Fig. 2. One-SBA interference patterns of a wedge-shaped film~ε1 5 2.25! on a substrate ~ε 5 16! for four different values ofparameter dyl: ~a! 5, ~b! 5.125, ~c! 5.25, ~d! 5.375. Expression~6a! has the solution uB1 5 56.31°. Curve s corresponds to theambient–substrate reflection, and for ε . ε1 it is the bending of allthe pattern curves from above for u , uB1 and from below for u .

B1.


for ε0 5 1 @Fig. 4~a!# only those values of ε1 that fall inthe interval ε1

2 , ε1 , ε11 for a given ε can be mea-

sured with sufficient precision ~as shown in Fig. 5!.The angle uB1 exists below the curve ε0

max ~verticalatching!, whereas uB2 is observable above the curve

ε0min ~horizontal hatching!. At the boundaries ε0

max

Fig. 3. ~a! Patterns similar to those in Fig. 2 for the film with ε1 5.44 and substrate ε 5 2.25. Now both expressions ~6! have so-

lutions uB1 5 50.19° and uB2 5 69.56°. Unfortunately, underlarge values of angle uB2 intersection of all the curves takes placeat acute angles so that the measurement of this angle becomesdifficult. By use of a cylindrical lens @Fig. 4~b!# with ε0 . 1 one canhift angle uB2 to the left-hand side. Curves a–d correspond to the

same thicknesses as in Fig. 2. ~b! Region of the SBA is shown ona larger scale.

oce

u

bb

oWonpv

tph

ld~

D

r

S

and ε0min the angles uB1 and uB2 become uTR ~the angle

f total reflection! and ~py2!, respectively; i.e., theyease to exist. In the cross-hatched area both SBA’sxist. The line ε0 5 ε corresponds to the convergence

of these angles, whereas lines ε1 5 ε and ε1 5 ε0correspond to the convergence of the angles uB1 or

B2, respectively, with the substrate’s Brewster angleuB

0 @in which case the reflectances R~uB1! or R~uB2!become zero#. The existence of an angle uB2 allowsone to avoid this limitation on the difference uε1 2 εuy arranging for R~uB2! to be small for any relationetween ε and ε1. To accomplish this, as is clear

from Fig. 5, it is necessary to choose a lens with ε0 'ε1, which will guarantee the closeness of angle uB2 touB

0 so that the reflectance R~uB2! may be made arbi-trarily small.

Therefore the existence of the second SBA allowsone to extend the applicability of the usual Abelesmethod on films with arbitrary permittivities. Evenmore possibilities arise when both of the shadow an-gles are observable.

4. Transparent Film of Variable Thicknesson an Unknown Isotropic Substrate: Two-Shadow-Brewster-Angle Method

For a transparent film of variable thickness one mayplot the experimental curves R~u! at different placeson the film and over a certain range of angles ofincidence and find the angles uB1 and uB2, where thescillations cease and all the curves intercross.hen the angles are determined by the intercrossing

f the interference curves, the requirement of small-ess of the reflectances at these points becomes su-erfluous. From Fig. 5 it follows that for the givenalues ε and ε1 ~ε1 . ε! the suitable values of ε0 are in

the interval ε0min , ε0 , ε0

max, where

ε0min 5

εε1

ε 1 ε1, ε0

max 5εε1

ε1 2 ε. (7)

When ε1 , ε there remains only the lower restrictionon the value ε0. In this way, for any chosen ε0, ex-pressions ~6! may be inverted as

ε1 5 ε0 tan2 uB1, ε 5ε0 sin2 uB2

1 2 sin2 uB2 cotan2 uB1. (8)

Fig. 4. Possible geometries of the experiment: ~a! conventialsetup with ambient ε0 5 1, ~b! setup with a cylindrical lens ε0 . 1.

The precision of determination of the shadow an-gles’ position strongly depends on the difference ininclinations of the intercrossing curves that corre-spond to different thicknesses d. For a film withvariable thickness the maxima of these differences,labeled as g1 and g2, do not depend on d. They aregiven in Appendix A. It is clear that for excessivelysmall values of g1 or g2 ~acute intersecting angles! thewo-SBA approach becomes inoperable. It takeslace in the vicinity of the boundaries of the cross-atched area, where uB23 ~py2! or uB13 uTR ~Fig. 5!.

In these limiting cases, g1 or g2 becomes zero anddetermination of ε and ε1 becomes impossible. Toavoid such closeness to the boundaries, the optimalchoice of ε0 is

ε0opt <

ε0min 1 ε0

max

2<

εε12

ε12 2 ε2 .

When ε1 , ε this optimal choice for the material of aens is ε0 5 3ε, which corresponds to the maximumifference in SBA: sin2 uB2 2 sin2 uB1 5 1y2ε1yε0!3y2@~ε0 2 ε!y~ε 1 ε1!#. Although this approach

does not employ any intensity measurements, theexperimental inaccuracies duB1, duB2 of the determi-nation of the SBA positions are proportional to theinaccuracy DR in the reflectance duB1g1 ' duB2g2 '

R. Since, usually, DR is of the order of 1023, theinaccuracies du are of the order of 1023 as well. Dif-ferentiating Eqs. ~8!, we obtain the following rela-tions between the inaccuracies dn1, dn ~n 5 =ε! of theefractive indices and duB1, duB2:

dn1

n15

2sin 2uB1

duB1,

dnn

5 2cotan uB1

sin2 uB1~1 2 sin2 uB2 cotan2 uB1!duB1

1cotan uB2

~1 2 sin2 uB2 cotan2 uB1!duB2. (9)

Thus the two-SBA method allows, in principle, for thedetermination of the dielectric permittivities of a filmand a substrate with an arbitrary ratio of their per-mittivities.

5. Anisotropic Film ~«1 3 «1! on a Known Isotropicubstrate

The presence of the second SBA allows for the deter-mination of the vertical and the horizontal anisotro-pies of a film when the permittivity of the substrate εis known.

A. Uniaxial Vertical Anisotropy

In the case of a uniaxial anisotropic film with theoptic axis normal to the film–substrate interface thepermittivity tensor is ~ε1!xx 5 ~ε1!yy Þ ~ε1!zz ~further inthis section we omit index 1!. Here the coordinateaxes ~x and y! are assumed to lie parallel to the


T

isb

pdu

f

a

I

mascw=

p

4

interface, whereas the third ~z! lies perpendicular toit. Then expressions ~6! take the form

sin2 uB1 5εzz~εxx 2 ε0!

εxxεzz 2 ε02 , sin2 uB2 5

εεzz

ε0

~εxx 2 ε!

εxxεzz 2 ε2 .

(10)

he inversion of Eqs. ~10! for direct determination ofεxx and εzz in terms of the angles uB1 and uB2 is givenin Appendix B.

In the case of a small vertical anisotropy, bzx 5~εzzyεxx! 2 1 ,, 1, it is convenient to perform themeasurements near the line ε1 5 ε0, where angle uB1is always close to py4 and uB2 is close to the sub-strate’s Brewster angle uB

0 . As is evident from Fig. 5,this may be performed only in the cross-hatched re-gion for ε1 , 2ε, where ε1 5 2ε is the intersection pointof the line ε1 5 ε0 with the curve ε0

max. In this lim-ting case, u~py4! 2 uB1u ,, 1, Eqs. ~B1! acquire aimple form and the expression for a weak anisotropyzx reduces to the relation

bzx 5 4Sε 1 ε0

ε D2 SuB1 2p

4D ~sin2 uB2 2 sin2 uB0 !. (11)

B. Planar Anisotropy ~«xx Þ «yy!

Sometimes small planar anisotropy byx 5 ~εyyyεxx! 21 has even greater practical interest than does uni-axial vertical anisotropy. Such anisotropy is of a

Fig. 5. Domains of existence of the SBA uB1 and uB2 in the ~ε1, ε0!lane for a given substrate ε 5 3.


principal importance for common-index thin-film po-larizers for a light at normal incidence,8 fabricatedfrom a single evaporant material deposited from twodifferent angles and in two mutually perpendicularplanes. Usually, in situ measurement of principalrefractive indices of such films is performed by arather complicated two-angle ellipsometry method.9The most convenient variant for determining such ananisotropy in our approach consists of performingmeasurements in the vicinity of the line ε0 5 ε, i.e.,uε0 2 εu ,, ε, where the angles uB1 and uB2 are close toeach other. In this case from Eqs. ~B1! one immedi-ately obtains

byx 5~ε0 1 ε!~ε2 2 ε0

2!~sin2 uB2x 2 sin2 uB2

y !

ε0ε2FSε0 1 εε D2

sin2 uB2x 2

ε0

ε G, (12)

where uB2x and uB2

y are the values of the second SBAfor two orientations ~zx and zy, respectively! of the

lane of incidence. Obviously, for an isotropic me-ium, but even for a very weak anisotropy ~say, 1%!,B2x 5uB2

y produces a shift of almost 1° in the SBApositions, thus this technique is quite competitivewith waveguide and reflectometric approaches formeasurement of an optical anisotropy.10–12

In addition to the angular measurements one canalso use the intensity measurements and determineplanar anisotropy from the difference of the reflec-tances R~uB2

x ! and R~uB2y !. By use of Eqs. ~4! and ~B1!

or small anisotropy one finds

bxy 5εxx 2 εyy

εyy5

~ε0 2 ε!~ε1 2 ε0!

2ε1ε0dRxy,

dRxy 5R~uB2

x ! 2 R~uB2y !

R~uB2y !

. (13)

In this case, with the value ε0 chosen to be between εnd ε1, the precision of determining an anisotropy bxy

is even better than the precision of the intensity mea-surements.

6. Weakly Absorbing ~«(1 ,, «*1! Film on a Knownsotropic Substrate

The influence of absorption on determination of theoptical constants of films and its measurement is awell-known problem.13–16 Possibly, the most usefuladvantage of the SBA approach is extenuation of thereflectometric ~but without any absolute intensity

easurements! methods to the measurement of weakbsorption of the film deposited on a transparent,trongly absorbing, or even opaque substrate. Byalculating the reflection from a sample with aeakly absorbing film, ε01 ,, ε91 ~or k1 ,, n1, whereε1 5 n1 5 n1 2 ik1!, with a thickness dyl .. 1, in

Eq. ~1!, one should take account of an absorption ε01 in

HI

wt

tsR

is

c

s

only the expression for the phase d, where a smallparameter ε01 is multiplied by a great factor dyl:

d 5 4pdl

~ε91 2 sin2 u 2 iε01!1y2

< 4pdl

~ε91 2 sin2 u!1y2 2 2pdl

iε01~ε91 2 sin2 u!1y2 .

(14)

ere we assume vacuum to be the incident medium.n Eqs. ~5! for the amplitudes r10 and r12 the absorp-

tion of a film may be neglected, and therefore in thisapproximation expressions ~6! for the SBA remainunchanged. On the contrary, the Abeles relations,Eqs. ~4!, cease, generally speaking, to be valid. Asfollows from Eq. ~1! and expression ~14!, the relation~4a! becomes

R~uB1! 5 ur02~uB1!u2 expF24pdl

ε01~ε91 2 sin2 uB1!

1y2G5 ur02~uB1!u2 expF28p

dl

n1 k1

~n12 2 sin2 uB1!

1y2G ,

(15)

hereas Eq. ~4b! remains valid with an accuracy upo the ~ε01!2 terms.

C. Transparent Substrate

If the value n1 is unknown, one can determine ε91 fromexpression ~6b! by measuring the angle uB2 and thencalculating angle uB1 using expression ~6a!. Cer-ainly, with a known n1 this procedure is unneces-ary. Further, one should measure the reflection~uB1! of a sample at the angle calculated. The

needed reflectance of the substrate ur02~uB1!u2 isreadily calculated, since the dielectric permittivity εis known. As a result, the parameter ε01d is given by

ε01 d 5 2l

4p

ε91~ε91 1 1!1y2 ln q,

k1 d 5 2l

8p

n1

~n12 1 1!1y2 ln q, or (16)

q 5R~uB1!

ur02~uB1!u2. (17)

After determining the thickness d by use of one of thenterference methods, we immediately find the ab-orption ε01 ~or k1! from Eqs. ~16!.The easily recognizable distinction between the in-

terference patterns of a transparent and a weaklyabsorbing film is demonstrated in Fig. 6 by numericalexample. Even a small absorption completely ruinsthe crossing point uB1 ~compare with Fig. 2! of allurves ~a–d! with the substrate’s reflection curve~s!

by spreading it into an 8° interval @Fig. 6~a!#. Inother words, even a weak absorption completely ex-cludes the possibility of applying the conventionalAbeles method. However, the angular position of

the intercrossing point, i.e., the point of cessation ofthe oscillations—which for a transparent film coin-cides with the crossing point of the sample’s and thesubstrate’s reflectivities ~see Fig. 2!—does not shift.At the same time the absorption strongly influencesthe value of the sample’s reflectance R~uB1!. It isimportant that the spread in values of R~uB1! for allcurves is small ~much less than a shift from its valuefor a transparent film!. This allows one to introduceR~uB1! independently of the film thickness and deter-mine ε01d from Eqs. ~16!. Under all imaginable val-

Fig. 6. ~a! Blurring of the intersection point uB1 5 56.31° ~theame as in Fig. 2! in the case of a weakly absorbing film n1 5 1.5 2

i0.002 on a transparent substrate ε 5 16. Curves a–d correspondto the same thicknesses as in Fig. 2. ~b! Vicinity of the intercross-ing point shown for transparent ~n1 5 1.5, upper bunch of curves!and absorbing ~n1 5 1.5 2 i0.002, lower group of curves! films onthe same transparent substrate ε 5 16.


23

w

wt

twt

u

ioefi

4

ues of the refractive index, a precision of ;10 in Rproduces a precision of ;1022 in k1. The best accu-racy will be attained for the values 1 2 q ; 1, i.e.,

hen the exponent is of the order of unity.The above-mentioned procedure is applicable onlyhen both SBA’s are observable. Even so, some-

imes it is difficult to measure angle uB2 exactly anddetermine ε91. Fortunately, the horizontal invari-ability of the inflection point of the interference pat-terns prompts a much simpler procedure fordetermining the film’s absorption without finding theangle uB2. Since, as we saw, the shift of the inter-crossing point relative to the angle’s position is neg-ligibly small, we may assume this position to be a realvalue uB1; after that we can calculate ur02~uB1!u2 athis angle and find the parameter q. In such a waye can find the absorption of a film even in the ver-

ically hatched area ε0 , ε0min, where angle uB2 does

not exist ~see Fig. 5!.

D. Opaque Substrate

Finally, it is extremely important that this methodmay be easily generalized to the case of an absorbing


or even an opaque substrate. Since the result de-pends, in fact, on only the ratio of the reflections forclean and overfilmed substrates, this approach worksequally well for an absorbing substrate. This is il-lustrated by the numerical example in Fig. 7, wherethe interference patterns of the same weakly absorb-ing film are shown in the cases of transparent ~k 5 0!and opaque ~k 5 4! substrates with n 5 4. Althoughthe influence of the substrate absorption is significantand shifts of the inflection points in the vertical di-rection are great, the value q for transparent andopaque substrates is practically the same ~0.861 and0.860, respectively!; i.e., a value k1 is reproducible.Hence for this approach the substrate’s absorption isnot a limiting factor, and thus even metal substratesare allowable. This advantage is essential when thetransmittance experiment is impossible.

7. Conclusions

First, we have generalized the Abeles method for de-termining the permittivity of a film to a case of thearbitrary ratio of the film’s and the substrate’s per-mittivities. Second, a new approach to the differentsituations with two unknown parameters—unknownpermittivities of a film and a substrate, an anisot-ropy, or a complex refractive index of a film—hasbeen developed, thus making it necessary to deter-mine two special points ~SBA’s! in the interferencepatterns of a film with a variable thickness or awedge form. These angles are almost always acces-sible to observation in an experiment with a cylindri-cal lens as an ambient @Fig. 4~b!#.

One possible practical application of this techniqueis the monitoring of the film’s homogeneity in situduring deposition. At present, monitoring of the in-homogeneity of the refractive index of dielectric filmsis performed by means of sensitive methods of spec-trometric ellipsometry.17,18 The present methodmay be used as a complementary approach. Afterthe first two measurements at the different thick-nesses of the film during deposition are made, oneshould find the intercrossing point of their angularpatterns in the region of small oscillation amplitudes,i.e., angle uB1 or uB2. For a homogeneous film all thesubsequent experimental curves during growthshould pass through this point. For the case of anabsorbing film of known refractive index the deter-mination of an absorption may be made by measure-ment of the parameter q at the angle and uB1, i.e., thefall in the reflectance at this angle. As is clear fromthe examples in Figs. 6 and 7, even a very smallabsorption, ε01 ,, ε91 or k1 ,, n1, may be detected byuse of sufficiently thick films with dyl .. 1. Thepossibility of using this method in the case of opaquesubstrates is of principal importance.

Appendix A

The precision of determination of the angles uB1 andB2 strongly depends on the intersection angles of the

interference curves at these points, i.e., the differenceof the angular derivatives dRydu for films with dif-

Fig. 7. Vicinity of the SBA uB1 for an absorbing film n1 5 1.5 20.001 deposited on transparent ~n 5 4, lower bunch of curves! andpaque ~n 5 4 2 i4, upper bunch of the curves! substrates. Inach case the position of the intercrossing point for a transparentlm n1 5 1.5 is shown.

i

g

g

T

ε

ε

Is

A

1

1

1

1

ferent thicknesses. For a film of variable thicknessthe phase d in Eq. ~1! changes over all intervals @0,2p#, and the extrema slopes correspond to d equal to0 and p ~see Figs. 2 and 3!. The maximum differ-ence g of these slopes already does not depend on dand is the signature of all the curves intercrossing atthis shadow angle. From Eqs. ~1! and ~5! and ex-pression ~6! for

g1 5dR~u, d 5 0!

duU

uB1

2dR~u, d 5 p!

duU

uB1

,

g2 5dR~u, d 5 0!

duU

uB2

2dR~u, d 5 p!

duU

uB2

,

t follows that

1 58ε~ε 2 ε0!~ε 2 ε1!~ε1

2 2 ε02!@εε1 1 ε0~ε 1 ε1!#

1y2

ε13/ 2 ε0

1/ 2$ε 1 @εε1 1 ε0~ε 2 ε0!#1y2%4 ,

(A1)

2 58ε0~ε 2 ε0!~ε1 2 ε0!~ε2 2 ε1

2!@εε1 2 ε0~ε 1 ε1!#

ε3/ 2 ε13/ 2$ε0 1 @ε0~ε 1 ε1! 2 εε1#

1y2%4 .

(A2)

he values of the parameters g1 and g2 depend on therelations among ε, ε1, and ε0, i.e., position in the planeε1, ε0, for a given substrate ε. In the characteristicregions ~see Fig. 5! they take the form

ε1 < ε, g1 5 Sε0

ε D5y2 ε 2 ε1

2ε$ 1,

g2 5 32ε 2 ε1

ε< 1, (A3a)

0 < ε, g1 < g2 512 Sε1

ε D3y2 ε 2 ε0

ε# 1,

(A3b)

0 < ε1, g13 0 for ε03 εmax,

g2 512 Sε1

ε D1y2 ε1 2 ε0

ε< 1. (A3c)

Appendix B

The inversion of Eq. ~10! has the form

εxx 5ε2ε0~sin2 uB2 2 sin2 uB1!

ε2 sin2 uB2 cos2 uB1 2 ε0 sin2 uB1~ε 2 ε0 sin2 uB2!,

εzz 5 ε0

ε2 cos2 uB1 sin2 uB2 2 ε0 sin2 uB1~ε 2 ε0 sin2 uB2!

ε2 cos2 uB1 2 εε0 1 ε02 sin2 uB2

.

(B1)

n the limiting case u~py2! 2 uB1u ,, 1 these expres-ions reduce to Eq. ~11!.

This study was supported by the Fundacao demparo a Pesquisa do Estado de Sao Paulo, Brazil.

References1. F. Abeles, “Methods for determining optical parameters of thin

films,” in Progress in Optics, E. Wolf, ed. ~North-Holland, Am-sterdam, 1968!, Vol. 2, pp. 251–289.

2. R. Swanepoel, “Determination of surface roughness and opticalconstants of inhomogeneous amorphous silicon films,” J. Phys.E 17, 896–903 ~1984!.

3. T. Pisarkiewicz, “Reflection spectrum for a thin film with non-uniform thickness,” J. Phys. D 27, 160–164 ~1994!.

4. R. M. Azzam and N. M. Bashara, Ellipsometry and PolarizedLight ~North-Holland, New York, 1977!.

5. M. Hacskaylo, “Determination of the refractive index of thindielectric films,” J. Opt. Soc. Am. 54, 198–203 ~1964!.

6. J. M. Bennett and H. E. Bennett, “Polarization,” in Handbookof Optics, W. G. Driscoll and W. Vaughan, eds. ~McGraw-Hill,New York, 1978!, pp. 10–17.

7. O. S. Heavens, Optical Properties of Thin Solid Films ~Dover,New York, 1991!.

8. I. Hodgkinson, Q. H. Wu, and C. Rawle, “Common-index thinfilm polarizers for light at normal incidence,” in Optical Inter-ference Coatings, Vol. 9 of 1998 OSA Technical Digest Series~Optical Society of America, Washington, D.C., 1998!, pp. 173–175.

9. I. Hodgkinson, J. Hazel, and Q. H. Wu, “In situ measurementof principal refractive indices of thin films by two-angle ellip-sometry,” Thin Solid Films, 313–314, 368–372 ~1998!.

0. I. Hodgkinson, Q. H. Wu, and J. Hazel, “Empirical equations forthe principal refractive indices and column angle of obliquelydeposited films of tantalum oxide, titanium oxide, and zirconiumoxide,” Appl. Opt. 37, 2653–2659 ~1998!; I. Hodgkinson and Q.Wu, Birefringent Thin Films and Polarizing Elements ~WorldScientific, Singapore, 1997!, Chap. 8, pp. 147–149.

1. G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Z.Vitlina, and V. Baranauskas, “An interference method for thedetermination of thin film anisotropy,” Thin Solid Films 279,119–123 ~1996!.

12. G. I. Surdutovich, R. Z. Vitlina, A. V. Ghiner, S. F. Durrant,and V. Baranauskas, “Three polarization reflectometry meth-ods for determination of optical anisotropy,” Appl. Opt. 37,65–78 ~1998!.

13. O. S. Heavens and H. M. Liddell, “Influence of absorption onmeasurement of refractive index of films,” Appl. Opt. 4, 629–630~1965!.

4. H. A. Macleod, “Monitoring of optical coatings,” Appl. Opt. 20,82–89 ~1981!.

5. Rusli and G. A. J. Amaratunga, “Determination of the opticalconstants and thickness of thin films on slightly absorbingsubstrates,” Appl. Opt. 34, 7914–7924 ~1995!.

16. K. Lamprecht, W. Papousek, and G. Leising, “Problem of am-biguity in the determination of optical constants of thin ab-sorbing films from spectroscopic reflectance and transmittancemeasurements,” Appl. Opt. 36, 6364–6371 ~1997!.

17. C. K. Carniglia, “Ellipsometric calculations for nonabsorbingthin films with linear refractive-index gradients,” J. Opt. Soc.Am. A 7, 848–856 ~1990!.

18. G. Parjadis de Lariviere, J. M. Frigerio, J. Rivory, and F.Abeles, “Estimate of the degree of inhomogeneity of the refrac-tive index of dielectric films from spectroscopic ellipsometry,”Appl. Opt. 31, 6056–6061 ~1992!.


Shadow-Angle Method for Anisotropic and Weakly Absorbing Films

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