Reflectometry of absorbing uniaxial multilayer media

1 September 1994

OPTICS COMMUNICATIONS Optics Communications 110 (1994) 485-491

ELSEVIER

Reflectometry of absorbing uniaxial multilayer media R6bert Klug, -. Em6ke Liirincz Department of Atomic Physics, Technical University of Budapest I I I I Budapest, Budafoki U.S., Hungary Received 25 March 1994

Abstract

A novel reflection technique for index determination of isotropic or uniaxial absorbing multilayer on a lower refractive index substrate is presented. Above the angle of total reflection at the layer-substrate interface characteristic minima occur in the reflectivity which are suitable for high accuracy fitting of the layer parameters. The 4 x 4 transfer matrices of uniaxial multilayer are given for computer evaluation. Refraction index determination of a poled polymeric layer on an indium-tin-oxide coated glass is demonstrated. Advantages of the new method are outlined.

1. Introduction In the field of integrated optics, where one works with optical isotropic and/or anisotropic layers, the parameters of which are often electrically modified by post-preparation treatments and the thickness of which are comparable to the wavelength, the common problem of the refractive index determination with required accuracy may be a non-trivial task. The nonlinear optical (NLO) polymer films used in integrated optics are often poled making them electro- optically active and uniaxial with the optical axis normal to the film surface at the same time [ 11. The determination of the film refractive indices needs explicit expressions of electromagnetic propagation in uniaxially anisotropic layered media. A general description of anisotropic, optically active, stratified media has been given by Berreman [2]. The primary use of his differential 4 x 4 matrix method is when the refractive indices vary continuously in the media. Homogeneous layers are much easier to handle by the general 4 x 4 dynamical matrices introduced by Yeh [ 31, the exact expressions of which, however are not available in the literature for the case of uniaxially anisotropic absorbing layers. Reflectometry offers a suitable tool for the determination of the optical parameters (complex refractive indices and layer thicknesses) of thin films [4], however inevitable light intensity variation causes significant error in the evaluated data [ 51. Determination of the optical parameters by far much higher precision is possible by those techniques, where characteristic angles are to be determined instead of light intensity, e.g. optical waveguiding [6] and attenuated total reflection (ATR) [ 71. The accuracy of these latter methods reaches the third decimal in the refractive index. Optical waveguiding, however, demands special skill and sophisticated experimental setup. As a drawback, ATR methods can be used only for layers with an overall thickness less than the wavelength of the illuminating light. The reflectometry of a multilayer on a lower refractive index substrate offers an alternative of the above methods with comparable accuracy in the optical parameter measurement. Due to the total reflection at 0030-4018/94/$07.00 @I 1994 Elsevier Science B.V. All rights reserved SSDIOO30-4018(94)00329-S

R. Klug, E. Mincz /Optics Communications I IO (1994) 485-491

486

the layer-substrate interface, i.e. the appearance of characteristic minima in the reflectivity, a high accuracy determination of the layer parameters is possible. The case of absorbing uniaxial (or isotropic) multilayers is considered here with the optical axis perpendicular to the surface. In Sect. 2 wave propagation is discussed in such media and explicit expressions of the 4 x 4 Yeh matrices are given in a convenient form for computer evaluation. Advantages of the method are experimentally demonstrated on a b&layer sample, a poled uniaxial NLO layer on an indium-tin-oxide (ITO) coated glass in Sect. 3.

2. Theory - Wave propagation in uniaxial absorbii

multilayers

Following Yeh [3], the transfer matrix formalism will be used for the description of stratified structures. We begin with the description of plane wave propagation in a uniaxial medium. Let the optical axis be parallel to the z axis. The dielectric properties of such a medium can be introduced by [8]

(1) in a matrix form, where E = (E,, Ey, E,) is the electric vector, D = (Dx, D,,, D,) is the electric displacement, se is the dielectric constant of free space, no the ordinary, n, the extraordinary complex refractive index, respectively. The electric field has the form of E = Eeexp[i(ax + By + yz - or)] with a complex wave number k = (a, 8, y ) in the medium, which is assumed to be absorbing. The wave equation in a non magnetic material (i.e. ,k = 1) ~n~-jP-

afi

y* kini

4 w

- a2 - y*

k&t: - CY*- /I*

BY

I[ 1 EX Ey Ez

=O,

(2)

where kc is the wave number of free space. y can be derived from the nontrivial solution of Eq. (2) for given aand/

or shortly: yi = Gnz -e,,rc*

(a = 1,2,3,4),

(4)

where el,r = ni/nf, e3,4 = 1 is the ellipticity of the dielectric constants in the direction of E and K* = a* + /I*. In the isotropic case (no = ne) the two roots of y are equivalent and the equation of k$n* = a* + 8* + Y: is valid. The polarization of the waves for each wave number vector is given by

i,= No

(K*-

aYU e,tc*)

Pro

[ -edc*

1 ,

(5)

where N0 is a normalization constant such that i, .i, = 1. It can be seen, that if e = 1, i.e. E oscillates in such a plane, where the refractive indices of different directions are the same, the polarization vector is the null vector in Eq. (5) with optional direction. Let the X-Z plane be the plane of incidence of a plane wave, i.e. /V = 0. The polarization of the electric vector takes the form

487

R. Klug, E. L&incz /Optics Communications 110 (I 994) 485-491

i,,r = Ni.2

fY1 0 -ela [

1

(6)

, where Ni.2 =

The polarization of the magnetic vector is obtained by using Maxwell’s equations and is given by i, (R,/opo) x ii,, where .UOis the magnetic permeability of free space. From Eq. (6) follows

=

(7) Consider the case of a uniaxially anisotropic layered medium sandwiched between two isotropic ambient and substrate media. The polarization vectors have to be calculated in each layer. Let us consider the special case, when the optical axis is perpendicular to the interfaces in each layer, so the interfaces are parallel to the x-y plane. Each layer can be characterized by two complex refractive indices, n,(j) and ne (j). If the polarization vectors are known in a given jth layer, then the electric and magnetic field can be written, respectively E = k&(j)

.P,(j)

exp{i[ax + BY + yO(j) (z - zj) - otl},

(8)

Cl=1

(9) 0=l

where zj is the z-coordinate of the interface between the (j - 1)th and the jth layer. These A(j) amplitudes are not independent of each other, they are related through the continuity conditions at the interfaces. Following the notations used by Yeh [3] the stratified media will be described by the so called dynamical D(j), propagation P(j) and transfer Tj,j+ i 4 x 4 matrices (Eqs. (17)-(21) in Ref. [3]). In our special case the dynamical matrix D(j) in the jth layer

r

Nl (jh

(i)

NWk%(j) D(i)

WO 0

=

-N(jh

(j)

N(j)k%(j)

0

0

o

o

WO 0

1

1

(10)

1

-y3(j) --

y3W

WO

WO

1

has a block-diagonal form, because there is no mode coupling at the interfaces. It is obvious from Eqs. (3), that at o = 3,4 the ellipticity of the refraction index vanishes (es,4 = 1 ), i.e. the electric vector E oscillates in such a plane, where the refractive indices are equal in different directions (x-y plane, case of s polarization). The upper left and lower right 2 x 2 block of the dynamical matrix, D belongs to the p and s polarization, respectively. Since ys is only dependent on the ordinary refractive index, the medium can be considered isotropic in case of s polarization. The inverse of the dynamical matrix, D-’ is given by 1

WO 2Ni(j)k$,2(j) 2N1(~~l(j)

0

0

WPO 2Ni(j)k,@(j)

0

0

W(jh(j)

D-‘(j)

=

0

0

0

0

-1 -c-W0 2 2y3W 1 wo --

2 %3(j)

(11)

488

R. Klug, E. Ldrincz /Optics Communications 110 (1994) 485-491

The product of D-‘(j)

. D(i + 1) yields [ Fj,j+l + Gj,j+i -&j+i

D-‘(j)

.D(j

-FI;.,j+l + Gj,j+r 0

+ 1) =

0

I

+ Gj,j+l

Fj,j+l + Gj,j+l 0

0

0

0

0

0

Qj,j+* i-Qj,j+l f - Qj,j+l i + Qj,j+,

t +

1

1’

(12)

where (13) The transfer matrix is defined as Tj,j+i = D-‘(j)

*D(j + 1) *P(j + 1).

(14)

Eq. (14) transfers the four amplitudes from the jth layer to the (j + 1)th layer. Eqs. (12)-(14) give the useful formulas for computer evaluation in a uniaxial layered medium. The reflectance and transmittance of the uniaxially anisotropic layered medium can be determined in terms of the above defined matrices. In our special case there is no mode coupling so it is sufficient to calculate two complex amplitudes each associated with the reflection and the transmission, for the s and p polarizations, respectively. Assume that the light is incident from the ambient media (j = 0) to the layered structure, which consists of N layers on a substrate (j = S). Let As, A,,, B,,BP and G, C, be the incident, reflected, and transmitted electric field amplitudes, respectively. By employing the transfer matrices an overall transfer matrix can be calculated for any given anisotropic layered structure such that

[I

AP BP = To,ITI,~T~,~.. .TN-I,NTN,S

(15)

AS BS

Because Of the block-diagonality Of Eq. ( 12), hf13 = hf14 = h!f23 = hi24 = 0 and kf31 = I% = M41 = &2 = 0, therefore the reflection and transmission coefficients are expressed in terms of the matrix elements in the following way r,=-_=-,

BS

M43

AZ

M33

s

M33

r,=-_=-, BP AP

*s+L,

J421

(16)

Mlt

*p+c’. P

(17)

MI

The reflectances and transmittances of the multilayer for the two polarizations can be expressed by R, = (rs12 z-., =

and R, = Irp12,

Y3W y30

2 its’

and T, = $f$

(18) Itpj2.

(19)

In Eq. (19), the change in the cross-section of the incident and transmitted light and the difference in the dielectric constant of the ambient and substrate is taken into account by y. Since the definition (3) of y contains no circular functions, the reflectance and transmittance can be calculated as a function of the angle of incidence or that of the wavelength with Eqs. ( 16 )- ( 19 ).

R. Hug, E. Liirincz /Optics Communications110 (1994) 485-491

3. Experimental

- Reflectometry

of absorbing

multilayer

489

media

As an application, first let us consider the case of a monolayer sandwiched between a higher refractive index, no ambient medium and a lower refraction index, ns substrate. Calculated reflectance and transmittance versus the angle of incidence are shown in Fig. 1 for no = 1.83 and ns = 1.45. Light arrives from the ambient medium, passes the layer of no = 1.650 + O.O016i, ne = 1.700 + O.O016i, d = lpm. At 19s and 80 total reflection occurs at the back- and front-surface of the layer, respectively. The absorption in the layer causes characteristic minima in the reflectivity at angles where the phase shift within the layer is equal to 2nm (m is integer). These minima are referred to as discreet substrate modes in a study of absorbing KC1 films [ 91. This phenomenon is utilized here for the experimental determination of complex refractive indices and thickness of the layer or multilayer. The overall optical thickness of the layer(s) to be determined must be greater or equal to the wavelength of light in the layered medium to yield one minimum in the reflectivity, however higher number of minima results in higher accuracy of the layer parameters in the computer tit. The lower the refractive index of the substrate is, the larger the angle interval, 13s- 00 can be. The optimum angle interval depends on the overall thickness of the layered sample (i.e. on the number of minima). The feasibility of the method was already proved in Ref. [ lo], where complex refractive indices of different isotropic polymer mono- and bi-layers were determined. By measuring the reflectivity a prism of high refraction index is used as a suitable ambient for light projection and the angle of incidence to the prism is used as angle of incidence (Fig. 2). These angles are meaured by means of a goniometer with an accuracy of 6 min. The error in the measured value of the reflectivity depends on the stability of the laser source (typically < 1O”h for a HeNe laser). An example of experimental index determination in the anisotropic case of a b&layer is given at Iz = 0.633 pm in Figs. 3 and 4, where measured and fitted reflectivity are shown for s and p polarization, respectively. The sample is a poled uniaxial polymeric layer of 2.07 f 0.03 pm thickness on an indium-tin-oxide (ITO) coated glass substrate. The complex refractive index of the isotropic (non-poled) NLO film is known from a previous reflectivity measurement of a non-poled part of the same sample which is used as initial value in fitting. The transparent electrically conducting IT0 layer serves as an electrode in poling. The parameters of the IT0 layer are only approximately known: Re(nrro) = 1.9 - 2.1, Im(niro) < 0.01, dmo = 0.2-0.35 pm. The statistical examination of the errors [ 111 shows that the precise knowledge of the optical parameters of the IT0 layer is not necessary for a successful evaluation of the measured reflectivity of such a bilayer with seven unknown parameters. The ordinary and extraordinary refractive index of the NLO layer is determined with high accuracy from the reflectivity measurement for s and p polarization, respectively: Re(n, ) = 1.7 185 (5 ) and Re (n, ) = 1.7462 (5 1. The fitting of the measured reflectivity curve for p polarization

0.2..

Angle cd inddenca

Fig. 1. Calculated reflectances (R,, RP) and transmittances (T,, TP) ofamonolayer (n, = 1.65+0.0016i, n, = 1.7+0.0016i, d = lpm) for s and p polarizations. Light arrives from the ambient (no = 1.83) and propagates towards the substrate (ns = 1.45). 0~ and 80 are the angle of total reflection at the layer-substrate and the ambient-layer interface, respectively. Fig. 2. Experimental

arrangement

for the reflectivity

measurement.

R. Klug, E. LMncz /Optics Communications 110 (1994) 485-491

490

16

26

35

46

66

Angle of incidence, a

15

25

35

45

66

Angle of incidence, a

Fig. 3. Measured and fitted reflection curves of a bi-layer sample (for s polarization). 1.7185(5).

Fitted real part of nc(NL0)

=

Fig. 4. Measured and fitted reflection curves of a bi-layer sample (for p polarization). 1.7462(5).

Fitted real part of ne(NLO)

=

gives both refractive indices, but the accuracy of the ordinary refractive index is not sufftcient. The accuracy in the third decimal requires reflectivity measurement for both polarizations [ 111. The fitted imaginary part of the refraction index is Im( nois ) = 0.0016 ( 10) for both polarizations. The statistical error of these can be much better when only one absorbing layer is present. The accuracy of the thickness tit (typically fO.0 15 pm) is within the standard deviation of the alternative thickness measurement methods, such as surface profiling. The IT0 layer can serve also as a suitable auxiliary layer in refractive index determination of non-absorbing isotropic/uniaxial layers. E.g. the same NLO polymeric films are non-absorbing in the near infrared (at 2 = 1.3 and 1.5 ,um) but depositing them on an IT0 coated glass the presented refraction index determination method can be used with the same accuracy in the near infrared region [ 111. Advantages of the reflectometry for optical parameter determination: Simple experimental setup with a non-stabilized laser source and non-sophisticated angle adjustment, accuracy in the third decimal in the real part and an order of magnitude estimation in the imaginary part of the refraction index and a layer thickness determination within 1% accuracy.

4.

summary

In conclusion, light propagation in stratified absorbing isotropic or uniaxial optical media is investigated. Explicit formulae of the 4 x 4 Yeh transfer matrices are given for the case when the optical axis is perpendicular to the surface. We have shown that a simple reflectometric determination of optical layer parameters, such as complex refractive indices and thicknesses is possible by fitting the characteristic variation of the reflectivity between the angles of total reflection of the ambient-layer and layer-substrate interfaces with suitably chosen substrate and ambient (projecting prism) indices. An accuracy in the third decimal can be achieved for the real part of the refraction index. As a practical example in integrated optics the method is demonstrated experimentally on mono and b&layered stratified structures of nonlinear optical polymer films.

Support and samples of NLO polymer films from Siemens ZPE, as well as helpful discussions with dr. P. Riihl, Siemens ZPE Erlangen are gratefully acknowledged.

R. Klug, E. L&incz /Optics Communications 110 (1994)4885-491

491

References [ 1] [2] [3] [4] [5] [6] [7] [8] [9]

[lo] [ 111

E. Van Tomme, P.P. Van Daele, R.G. Baets and P.E. Lagasse, IEEE J. Quantum Electron. 27 (1991) 778. D.W. Berreman, J. Opt. Sot. Am. 62 (1972) 502. P. Yeh, J. Opt. Sot. Am. 69 (1979) 742. Handbook of Optical Constant of Solids II, Ch. 3, Methods for Determining Optical Parameters of Thin Films (Academic, New York, 199 1). P.J. Travers and L.S. Miller, Thin Solid Films 208 (1992) 55. W.L. Barnes and J.R.Sambles, Thin Solid Films 143 (1986) 237. Handbook of Optical Constant of Solids II, Ch. 4, The Attenuated Total Reflection Method (Academic, New York, 1991). M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1959). T.H. Koschmieder and J.C. Thompson, J. Modem Optics 38 ( 1991) 2095. E. Liirincz, R. Klug, P. Richter and P. Riihl, Proc. SPIE 1983 Optics as a Key to High Technology (1993) p. 292. E. L&incz and R. Klug, Refractometry of weakly absorbing multilayers via reflectometty, oral presentation at R-metry Conference organized by SPIE/PL Chapter at Warsaw 16-20 May, 1994 (to be published in SPIE Proceedings).