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Anisotropic effects of columnar structure on attenuated total reflection experiments
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15 February 1997
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OPTICS COMMUNKATLONS Optics Communications 135 (1997) 257-263
Anisotropic effects of columnar structure on attenuated total reflection experiments Haiming Wang Centro de Investigaciones
en Optica, Apartado Postal I-948,37000
Leo’n, GTO, Mexico
Received 11 August 1995; revised version received 10 October 1996; accepted 23 October 1996
Abstract The attenuated total reflection (ATR) method is used to investigate an aluminum (Al) film covered with an obliquely evaporated magnesium fluoride (MgF,) overcoating. Light is tunneled into the MgF, film by a hemispheric coupler at two orthogonal incident angles to explore anisotropic effects of the MgF* film on the ATR experiments. Optical constants of the Al film are determined from the excitation of surface plasmou polaritons (SPPs) in the Al film with an accuracy better than five percent. A theoretical analysis of the measurement errors is given to illustrate the applicabitity of this method. Keywords:
Thin films; Surface plasmon polaritons; Attenuated total reflection
1. Introduction Metal mirrors have been extensively used in optical engineering and are usually coated with dielectric high-reflection layers. Characterization of these dielectric-metal layered media involves determination of optical constants for both metallic and dielectric films. Moreover, the dielectric overcoatings will exhibit some degree of birefringence when anisotropic crystalline media are deposited. The birefringence may also be induced by a columnar structure of a preferentially homogenous alignment in the dielectric films that are deposited under an oblique incidence of vapor flux [l-3]. Optical constants of dielectrics can be determined by many conventional optical techniques, such as the guided-wave method [4] or the reflection/transmission (R/T) measurement [S]. These techniques can also assess optical constants of anisotropic films. On the other hand, optical constants of metal films depend on the way they are prepared. The attenuated total reflection (ATR) technique, which excites surface plasmon polaritons [SPPs) at an interface of metallic and dielectric media, has demonstrated its capability as a powerful, simple, and accurate method. This method is particularly adequate for studying thin film assemblies comprising metal layers [&lo]. Some works have reported investigations on anisotropic layered media bordering metals, such as silicon dioxide (SiO,)
films obliquely evaporated on silver [ 111, aluminum mirrors coated with magnesium fluoride (MgFa) layers [12], and the excitation of SPPs by guided waves using a rutile (TiO,) prism [13]. However, effects of anisotropic media involved, particularly those related to the columnar structure, are not discussed in detail. The purpose of this paper is to study SPPs at a thin film assembly consisting of a metal bordering an anisotropic dielectric medium. Anisotropic effects, particularly those induced by the columnar structure, on the phase-matching condition for exciting SPPs are discussed. Experiments are carried out on Al films covered with MgFa overcoatings. Optical constants of these Al/MgF, film assemblies are retrieved from coupling angles at which the light power is deprived from an incident beam to excite SPPs.
2. Theory Fig. 1 depicts the experimental principle for exciting SPPs by the ATR method, in which a hemispheric coupler [14] is used to handle anisotropic effects. Levy et al. propose a method that uses a pyramid as a coupler and allows the measurement of reflectance in two perpendicular directions [ 1 I]. In their configuration, however, variation of reflectance with incident angles is difficult to
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258
H. Wang / Optics Communications 135 (1997) 257-263
Laser beam
Fig. 1. The principle of the attenuated total reflection (ATR) experiment.
compensate, especially when an anisotropic coupler is used. By using a hemispheric coupler, since an incoming light beam falls onto the coupler at a normal incidence, it is easy to direct the beam to the coupling spot and to compensate the change in reflectance. To study the anisotropic effects related to the columnar structure of films, we adopt a model proposed by Horowitz [3]: referring to Fig. 1, we describe the film assembly by the “laboratory” coordinates (x, y, z), in which the x-axis is perpendicular to the film surface and ( y, z) coordinates lie in the film plane. In turn the “physical” coordinates (x’, y’, z’) that are in the principal directions of the columnar structure induced anisotropy in each dielectric medium are formed by a rotation of angle @ around the y-axis, as shown in Fig. 1. These “physical” coordinates may vary in different media. In each anisotropic dielectric medium a material equation in the coordinates of principal axes (x’, y’, z’) is given by D = q,( x’+ E,, + y’qYt EYt +z)Ejz~Ez~), where E,, is the free space dielectric permittivity, and where E and D are the electric field and the electric displacement vectors, respectively. In the coordinates (x, y, z) the normal and tangential components of the displacement vector read:
Nullifying the determinant of Eq. (3) yields a dispersion relation for the normal and tangential components of the wave vector k,,=K,,K-
AK,
K,, = Wrl,)(k&:
AK=
nz = esx, sin2 @ + esz, cos* @,
(2)
sin @ cos @.
In the metal and the substrate (media 1 and 2) the three principal elements of the dielectric tensor are merely identical. The metallic film has a complex dielectric function l2 = E; + ie$. There are two mechanisms to excite SPPs at the dielectric/metal interface (interface 23): When the coupler is used in total reflection, if the dielectric layer is sufficiently thin so that an evanescent wave reaches interface 23, then SPPs may be directly excited at it. In the case the dielectric layer is thick, it may support several guided modes. If the tangential component of the guided waves match that of
I’*,
ks&x/n;).
The magnetic field is related to the electric optical admittance, Ys, Hsy = +YsEsz,
field by the
Ys = (Eo/CLo)“*(k,rl,2/K,,)K.
(5)
Introducing Eqs. (4) and (5) into boundary conditions results in a definition of propagation constant, /3. The tangential component of the wave vector remains constant across all boundaries, k,, = k,, = /3. The dispersion relation reads:
E2
rlz = ls+, (30s’ @+ esZ, sin’ @,
- k;$‘2,
K = [ 1 - &(wL)*]
(k;;e,_fi*)“*
where
nZX= ( egZ, - Q)
SPPs, again SPPs can be excited. For a thick metallic film, SPPs are sensitive to changes in optical constants near interface 23. They decay before reaching the metal/sub strate boundary (interface 21>, and therefore, are insensitive to the thickness of the metallic layer. In that case the metallic layer can be regarded as infinite. The wave equations in isotropic metallic media and their solutions are well known. For a p-polarized plane wave propagating in the x-z plane, in the metal the normal and tangential components of the wave vector are simply connected by a dispersion relation: k,, = (kze, - kg,)‘/*, where k, = 2?r/A. In the anisotropic dielectric, on eliminating the magnetic vector we have Maxwell’s equations expressed in the form
+ (k,$&.-P2)1’2
=
o
(6)
1731173zK
that defines an explicit phase-matching relation of the tangential component of SPPs, /3, to the dielectric functions of the materials adjacent to interface 23: P=ko[%d(‘?,2K*-
~~)/(7&7;K*-
E:)]“*.
(7)
When the dielectric medium is isotropic, Eq. (7) reduces to the same form as that presented in Ref. [lo]. To express the propagation constant, we define an effective index, N = /3/k,, with its real and imaginary parts, N’ and N”, calculated by N’ = (1/2)“‘(
) N 12 + ad) “*,
N” = (l/2)*/*(
1N ) 2 - ad) “*,
(8)
H. Wang/Optics
where their amplitudes
where a=
(
6;’ + E;~)(&,~K~
q:K’e;+ 2e’;‘(2e;2
Rzj=
- E;’ + E;‘)
-47,2K’[~;(q;q;+qTq;)+~;)(q;q;-qyq;)]
- $K’E;),
+ 2+:K2~;
I IE2121qj12+V~K41q212
+2,;K”[
l;(q;q;
+E;(q;q;
- q,lq;)]}-I,
- E;’ + E;‘)],
7;$K2
d= d/[(
- E;’ + E;‘)’
+ 4ei2ez2
1
1, (9)
= 2vZ"K"[ E;(q;q);
l;’ _ N'2 +N"2)'/2,
I q2
41; = ( 1/2)“2(
I q2 2 _ e;2 + N” - Nn2)
lq212=
[(+Nf2
l/2,
(10)
1, r/r
t N”‘)’
+ (e; - 2 N’N”)’
and where
q’j =
(1/2p2[I q3 I ’ + o:- (s,/a)“( N” -N”‘)] “‘, I”, (1/2y2[ I q3 I2 - T$ + (r/,/qJ”( N'* -N"')]
h/B)[(4- N12 + N”2)2 + 4N’*N,,‘]
“‘.
(11) For a thin metallic film, the condition for the existence of SPPs can be obtained by analyzing the reflection of plane waves at interfaces 23 and 21, which gives the following dispersion equation [lo]: rz3rz1 exp(i2k,,d,)
= 1.
(12)
Here the anisotropic effects of the dielectric are included in the reflection coefficients calculated by the optical admittance in the dielectric, Eq. (5), that in the metal, and that in the substrate, Y, = Y, = (~0/~o)1’2e~/k~,, The normal component of the wave (Eo/luo)“2+kr,. vector in the substrate can be calculated by a way analogous to that for K3x, i.e., k,, = k,(q’, + iq’$ where the calculation of q; and q’,’ can use Eq. (11) in which the dielectric functions are replaced by using nj = 7: = err rl = 0, and K = 1. This leads to an expression for the reZection coefficients: rzj=
j=
(Y,
1,3,
- q)/(
Y2 + q)
= Rt$* exp( -i2+2j),
(13)
- E;(q;q;
+ q;q;)]
’ (14)
Anisotropic effects of the hemispheric coupler are related to the phase-matching condition. The tangential component of an evanescent wave outside the base of the coupler is given by k,, = ke~~/~ sin 0,. Here the permittivity of the coupler, e4, depends on the incident angle 0,. This permittivity can be calculated from the dispersion relation in the coupler that relates the normal component to the tangential component of the evanescent wave vector: then the perk,, = k,,(qz/~q+)1’2[~4x - (k4z/k,,)2]1/2; mittivity is simply given by [ 151 l4 = (~~~e~~)/(
llx cos28, + l4z sin28,)
that leads to the tangential wave vector
141312=(17L/1)x)i[ n~-(N’+iN”)2]“212 =
- q;q;)
I~2121qj12-~~K41~212
From which the normal components of the wave vectors are expressed as k,, = k,q, = k&q; + iq’;), and K,, = k,q, = k,(q’, + iq’;), where q; = (1/2)“*(
+q;q;)
tan 242j
N I* = d(a2 + b’)“‘.
2+
and phases are defined by
1
X
q; =
259
Communications 135 (1997) 257-263
k,, = k,(
l4xe4z)1’2
component
(15) of the evanescent
sin 0,/( e4x cos2B4 + e4Z sin’0,)
l/2
.
(16) Making this match the real part of the tangential component of SPPs, k,, = k, N’ [16], we determine a coupling angle for the ATR to occur: sin @,,a = N’e:!‘/[
e4x~4z - N”( e4r - e,,)]
I’*.
(17)
This equation is applicable when N” -=ZZ 1. From Eqs. (8) and (9), it means b =K n, leading to a similar restriction on metals for the ATR method to be validated: E’ < - I, e” < I E’ I [8,17]. At the coupling angle SPPs excited near the metal/dielectric interface deprive the light power of an incident beam and a sharp drop of reflectance will be observed.
3. Experiments MgF, is the most widely used material for thin film engineering, mainly used to modify reflectivity of optical components. MgF2 films can be thermally evaporated from a tantalum or molybdenum boat, with substrates heated to a temperature about 200°C. Bulk crystalline MgF2 is
260
H. Wang/Optics
Communications 135 (1997) 257-263
Fig. 2. The experimental setup to record reflectance as a function of incident angles.
slightly anisotropic and has refractive indices n, = 1.370 and 12,= 1.381 at A = 550 nm for ordinary and extraordinary rays, respectively. MgF, films may exhibit lower refractive indices, since a 100% packing density in thin film deposition is impossible. Experiments are conducted on several MgF,/ AI/ BK7-glass thin film assemblies. The BK7-glass substrates are routinely polished, on which Al films are evaporated at nearly normal incident angles of vapor flux to avoid dichroism in them. The refractive indices of the MgFz over-coatings deposited on top of the Al films vary, depending on the packing density and on the incident angle of vapor flux in evaporation. As a result of this incidence, MgF, films may show some degree of birefringence. The refractive indices of the MgF, films can be identified by the R/T measurement [5]. The dielectric function of Al films is assessed by the ATR method, which identifies the coupling angles via the measurement of reflectance as a function of incident angles. Fig. 2 shows the experimental setup. A helium neon laser with the wavelength A = 0.6328 pm is used to illuminate the sample under test. The laser beam is modulated by a mechanical chopper. The modulation frequency is sent to a lock-in amplifier as a reference frequency that provides a way to suppress the stray light from environment. The sample is mounted on a stage consisting of two rotating motors driven by a motor controller. The angular position of the motors is transmitted to a PC. The laser is set on one of the rotating arms of the stage, whereas a photodiode is placed on another arm. The modulated laser beam impinges upon the coupler (that is not shown in the figure) and then is tunneled into the sample at the coupling angles. A synchronized rotation of these two motors enables us to change incident angles of the probe laser beam and to keep the photodiode always in the direction of specular reflection. Another photodiode is used to collect the light signal directly emitting from the laser before it reaches the sample (This photodiode is not shown in the figure.) The reflected signal in reference with the light
signal directly emitting from the laser gives the reflectance of the sample. The photocurrent signals from both photodiodes are directed to a preamplifier, then are amplified by the lock-in and sent to the PC. Combining the reflectance signals with the position signals of the motors, we can record the reflectance of the sample as a function of incident angles. Parameters of the sample can be recovered from Eqs. (17) and (7) or (12). However, a calculation of effective index from Eq. (12) is rather complicated. Here a method presented in Ref. [9] is adopted and the propagation constant is determined by the following approach: First the Al films are considered having an infinite thickness, the dispersion equation is then determined solely by interface 23, this solution is called PO. Then boundary 21 approaches boundary 23 and a slightly changed solution p = &, + A /3 is then obtained from Eq. (12). This procedure virtually is a perturbation approach, in which the initial estimation is given by PO; whereas Ap is a perturbation term, whose real part displaces the resonance value of PO and its imaginary part changes the damping of the film assembly. To clarify the effects of the columnar structure, we use three different incident angles of vapor flux: @’ = 5.00”, 30.00”, and 65.00”, to deposit the MgFz films in the MgF,/Al/BK7-glass thin film assemblies. The thicknesses of the MgFz films are controlled around 110 nm. This insures an effective tunnel of light power from the incident beam to the Al film at the coupling angles. The refractive indices and columnar orientations of the MgF, films are obtained from the R/T measurement [5] and shown in Table 1. In the ATR experiment, two perpendicular orientations of the coupler are used to explore its anisotropic effects, as
Table 1 The dielectric functions of MgF, /Al/BK7-glass thin film assemblies. The prism coupler was positioned with its optical axis both perpendicular (quantities with the subscript “ I “1 and parallel (quantities with the subscript “ II”) to the incident plane Sample #l
#2
#3
- 42.252 - 14.121 32.73 - 42.870 - 13.902 30.04 5.00 2.61 1.369 1.375 1.377 - 42.561 - 14.012
- 40.490 - 16.502 32.61 - 42.103 - 16.100 29.92 30.00 16.27 1.364
- 46.578 - 15.813 32.68 -48.162 - 15.210 29.98 65.00 47.54 1.361 1.373 1.374 - 47.370 - 15.512
1.373 1.375 - 41.297 - 16.301
H. Wang/Optics
-
Sample #l
.-----
Communications
Sampleti
.- .--..‘.Sample #2
1.0
8
0.8
g Gz d
0.8 0.4
33
32
Incident angle 0, (degree) Fig. 3. The measured reflectance versus incident angle curves for the MgF, /Al/BK7-glass thin film assemblies samples 1-3. Their optical properties are exhibited in Table 1. The coupler is posi-
tioned with its optical axis perpendicular to the incident plane.
it is described in Pq. (17). For each MgF,/Al/BK7-glass sample corresponding to each one of the coupler orientations, a dropped reflectance value is observed. Totally 100 times of reflectance measurements are carried out for each sample. Fig. 3 plots the reflectance versus incident angle curves averaged over one hundred measurements for the three MgF,/Al/BK7-glass film assemblies. Here the optical axis of the coupler is positioned perpendicular to the incident plane. In these experiments, thick Al films with thicknesses about 135 nm are examined. As a result, SPPs excited at the MgF,/Al interface are not sensitive to the dielectric function of the substrate. From these curves, we can clearly pinpoint the coupling angles (0,, I>, which are shown in Table I. Similar experiments are conducted for the coupler placed with its optical axis parallel to the incident plane. Accordingly, the averaged reflectance versus incident angle curves are plotted in Fig. 4. The coupling angles (@,,a ,,> again are identified and displayed in Table 1. From these curves alongside those shown in Fig. 3, we
_ 1 .O
8 5 ‘E( g I+!
-
Sample #I
-----
Sample #3
--.-.--..-Sample#2
0.8 0.6 0.4 0.2 0.0
’
28
29
30
Incident angle 6, (degree) Fig. 4. The same as Fig. 3. Here the coupler is positioned optical axis parallel to the incident plane.
with its
135 (1997) 257-263
261
recover the dielectric functions of the Al films, which are exhibited in Table 1, too. In principle the dielectric functions of the Al films recovered with two different coupler orientations must be identical, since Al films are not birefringent. However, from Table 1 we find that they indeed exhibit some degree of difference. A probable explanation to this difference could be the slight bireftingence caused by the incident angle of vapor flux deviating from the normal incidence. The Al film in sample 1 is deposited at normal incidence and therefore exhibits the smallest deviation in dielectric functions corresponding to two coupler orientations. Samples 2 and 3 are deposited at incident angles of vapor flux @” = f 5”, accordingly, the deviation in their dielectric functions is larger. However, even at normal incidence, the difference in the dielectric function still remains. One of the reasons for this difference is that the experiment method is not error-free. The principal refractive indices of the MgF, overcoatings are assessed by the R/T measurement. This method can only recover refractive indices with an uncertainty to the third decimal place. The error in the measurement of refractive indices of the MgFa overcoatings can be instantly transferred to the ATR measurement. In addition, the dielectric function of Al films can change from place to place. Nevertheless, since here a hemispheric coupler is employed, the probe laser beam can be more accurately directed to the coupling spot. At any rate, the difference between the dielectric functions obtained with two perpendicular coupler orientations indeed reveals the uncertainty of the measurement. In consequence we calculate some statistical parameters of the measurement to evaluate this difference. First, the mean value of the dielectric functions, ( l;) = (l/2)( l; I + e&l and ( l;I) = + E$ are taken as the nominal values of the 0/2xe;. Al films, which are shown in Table 1. Then the relative difference between the values of dielectric functions corresponding to the coupler perpendicular and parallel to the incident plane are then calculated referring to these mean values of the dielectric functions. In Table 2 it is shown that the difference varies over a range of about + 3.9%. Another important statistical parameter is the standard deviation of the measured values. From the central limit theorem, we can consider that the maximum error of the measured dielectric functions is three times their standard deviation values. In Table 2 it is shown that the standard deviation acquired from the data with the same coupling orientation, whether for a; and a; or a,; and o’;;, is obviously smaller than the statistical parameters obtained from the data with two different coupler orientations, a’ and a”. In fact, for sample 1 the maximum measurement error corresponding to the same coupling orientation, 2.98%, is significantly smaller than the error in the case of both coupling orientations, 4.00% - this is a difference exceeding 25%. For samples 2 and 3, the maximum errors for the case of same single coupler orientation is, still
262
H. Wang/Optics
Communications 135 (1997) 257-263
Table 2 The statistics for the dielectric function data of MgF, /Al/BK7glass thin film assemblies, the subscripts “ I ” and “ 11”indicate
the orientations of the prism coupler, as shown in Table 1 Sample
AE;/(E;)(%) Ae;/(<;> (%) fl: 0; 30; /E;A(%0) 3u’; /E;) L (%) 5, 2; ,E& (%I 3$/E,“,, (%I u’ 0” 3U’/(E;) (%o) 30*/(E;) (o/o)
#1
#2
#3
- 1.45 1.56 0.198 0.104 - 1.41 -2.21 0.210 0.138 - 1.47 - 2.98 0.476 0.187 -3.36 - 41x)
-3.91 2.47 0.221 0.148 - 1.64 - 2.69 0.234 0.168 - 1.67 -3.13 0.430 0.201 -3.12 -3.70
-3.34 3.89 0.230 0.100 - 1.48 - 1.90 0.189 0.191 - 1.18 -3.77 0.481 0.197 - 3.05 -3.81
smaller than those for both coupler orientations, however, the difference now is not so significant. Considering the experimental setup and method, we have four main factors causing inaccuracy: first, the noise in the reflectance detection is the most fundamental error source and is inevitable. This causes uncertainty in the measured reflectance values and can only be reduced by improving the detection unit, including the photodiodes and the amplifiers. Generally light intensity detection can reach accuracy better than one percent. Next, the angular positioning error in the incident angle measurement will also contribute to inaccuracy. This can be caused, for example, by the rotating stage and fine positioning errors. This factor is significantly smaller than the first one and can be reduced by the average process as it is described here. Another source of inaccuracy is the change of coupling spot when the coupler is shifted from one orientation to another. The use of the hemispheric coupler is to reduce this effect. This can also be reduced by the average process. Finally, the data processing technique may also induce some error. As it is described above, both in guided-wave and ATR experiments, optical constants of the film assembly under examination are obtained by finding coupling angles at which the incident light power is tunneled into the films. In the measured reflectance versus incident angle curves these coupling angles are observed as some sudden drops, as those shown here. Actually this process is a least-squares fit procedure that adjusts the optical constants of the films (in the experiments here the dielectric function of Al films) to match the measured reflectance curves. This process can be refined by an iterative operation. Even though the residual error always exists in the least-squares fit and it will also cause inaccuracy. In fact, in the ATR experiment
the data processing technique may contribute to certain degree of inaccuracy in measurements, as the theoretical calculation is rather tedious and complicated. From the results shown in Table 2, we find that the accuracy of the ATR measurement is better than five percent. This judgment also agrees with the studies presented in the literature. Indeed, aluminum is one of the mostly known metals. Its dielectric functions, both in the form of bulk crystals or films, have been extensively studied. For example, in Ref. [12] Fontana et al. examine an Al film with the thickness of about 120 nm deposited onto a Pyrex substrate and protected by a MgFa overcoating. Its dielectric function is assessed by the ATR method at A = 0.6328 pm, which gives E’ = -46.6 f 3.1 and E” = - 18.6 + 4.1. These values are quite close to the results obtained here. In addition, Schulz et al. [18] have measured several metals by the spectrophotometric method. They give dielectric functions versus wavelengths but do not give the values at A =0.6328 pm. We obtain the values at A = 0.6328 pm by an interpolation, which shows values E’ = -39.56 and E” = - 14.67 for aluminum, also close to the values shown here.
4. Conclusions In this paper the attenuated total reflection (ATR) method is applied to study aluminum (Al) films covered with anisotropic magnesium fluoride (MgF,) overcoatings comprising the columnar structure. Anisotropic effects of the MgF, overcoatings are explored by using a hemispheric coupler in two perpendicular orientations. The phase-matching condition for exciting surface plasmon polaritons (SPPs) in the Al films is examined under the effects of the anisotropic MgFa overcoatings. The effects of a birefringent titanium dioxide (TiO,) coupler on the phase-matching condition are also taken into consideration. In experiments, the reflectance is recorded as a function of incident angles. The coupling of the light power into SPPs is identified by observing a dropped reflectance value in the reflectance versus incident angle curves. The dielectric functions of the Al films are recovered from these curves with an accuracy better than five percent. In comparison with other more accurate techniques such as the prism coupling method for determining optical properties of thin films, this accuracy is not very satisfactory. The main emphasis of this work is to give a fine theoretical analysis. The experimental results are given to illustrate the applicability of the method. More accurate results may be pursued in future studies.
References [l] K.H. Guenther 2992.
and H.K. Pulker,
Appl.
Optics
15 (1976)
H. Wang/ Optics Communications
135 (1997) 257-263
[2] H.A. Macleod, in: Opticai Thin Films, ed. R.I. Seddon, Proc.
[3]
[4] [5] [6] [7] [8] [9]
[lo]
Sot. Photo-Opt. Instrum. Eng. Vol. 325 (1982) p. 21. F. Horowitz, Structure-induced optical anisotropy in thin film, PhD diss., University of Arizona, Optical Sciences Center ( 1983). F. Flory, D. Endelema, E. Pelletier and I. Hodgkinson, Appl. Optics 32 ( 1993) 5649. Haiming Wang, J. Opt. Sot. Am. All (1994) 2331. A. Otto, Z. Physik 216 (1968) 398. E. Kretschmann and H. Raether, 2. Naturforschung 23A (1968) 2135. E. Kretschmamr, 2. Physik 241 (1971) 313. H. Raether, in Physics of Thin Films, Vol. 9, eds. G. Hass, M.H. Francombe and R.W. Hoffman (Academic, New York, 1977) p. 145. G.J. Sprokel and J.D. Swalen, in: Handbook of Optical
[Ill ]12] [13] [14]
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Constants of Solids II, ed. E.D. Palik (Academic, Boston, 1991) p. 75. Y. Levy, M. Jmich and J.D. Swalen, J. Appl. Phys. 57 (1985) 2601. E. Fontana, R.H. Pantell and M. Moslehi, Appl. Optics 27 (1988) 3334. Y. Nakagawa, M. Kaneko and N. Terunuma, Appl. Phys. Lett. 55 (1988) 2141. A.E. Craig, G.A. Olson and D. Satid, Appl. Optics 24 (1985)
[151 Z-Wang, optics comm. 119 (1995) 75. [16] W.P. Chen and J.M. Chen, J. Opt. Sot. Am. 71 (1981) 189. [17] P. Pokrowsky, Appl. Optics 30 (1991) 3228. [ 181 L.G. Schulz, J. Opt. Sot. Am. 44 (1954) 257; L.G. Schulz and F.R. Tanherhni, J. Opt. Sot. Am. 44 (1954) 362.
15 February 1997
cQ% ELSEVIER
OPTICS COMMUNKATLONS Optics Communications 135 (1997) 257-263
Anisotropic effects of columnar structure on attenuated total reflection experiments Haiming Wang Centro de Investigaciones
en Optica, Apartado Postal I-948,37000
Leo’n, GTO, Mexico
Received 11 August 1995; revised version received 10 October 1996; accepted 23 October 1996
Abstract The attenuated total reflection (ATR) method is used to investigate an aluminum (Al) film covered with an obliquely evaporated magnesium fluoride (MgF,) overcoating. Light is tunneled into the MgF, film by a hemispheric coupler at two orthogonal incident angles to explore anisotropic effects of the MgF* film on the ATR experiments. Optical constants of the Al film are determined from the excitation of surface plasmou polaritons (SPPs) in the Al film with an accuracy better than five percent. A theoretical analysis of the measurement errors is given to illustrate the applicabitity of this method. Keywords:
Thin films; Surface plasmon polaritons; Attenuated total reflection
1. Introduction Metal mirrors have been extensively used in optical engineering and are usually coated with dielectric high-reflection layers. Characterization of these dielectric-metal layered media involves determination of optical constants for both metallic and dielectric films. Moreover, the dielectric overcoatings will exhibit some degree of birefringence when anisotropic crystalline media are deposited. The birefringence may also be induced by a columnar structure of a preferentially homogenous alignment in the dielectric films that are deposited under an oblique incidence of vapor flux [l-3]. Optical constants of dielectrics can be determined by many conventional optical techniques, such as the guided-wave method [4] or the reflection/transmission (R/T) measurement [S]. These techniques can also assess optical constants of anisotropic films. On the other hand, optical constants of metal films depend on the way they are prepared. The attenuated total reflection (ATR) technique, which excites surface plasmon polaritons [SPPs) at an interface of metallic and dielectric media, has demonstrated its capability as a powerful, simple, and accurate method. This method is particularly adequate for studying thin film assemblies comprising metal layers [&lo]. Some works have reported investigations on anisotropic layered media bordering metals, such as silicon dioxide (SiO,)
films obliquely evaporated on silver [ 111, aluminum mirrors coated with magnesium fluoride (MgFa) layers [12], and the excitation of SPPs by guided waves using a rutile (TiO,) prism [13]. However, effects of anisotropic media involved, particularly those related to the columnar structure, are not discussed in detail. The purpose of this paper is to study SPPs at a thin film assembly consisting of a metal bordering an anisotropic dielectric medium. Anisotropic effects, particularly those induced by the columnar structure, on the phase-matching condition for exciting SPPs are discussed. Experiments are carried out on Al films covered with MgFa overcoatings. Optical constants of these Al/MgF, film assemblies are retrieved from coupling angles at which the light power is deprived from an incident beam to excite SPPs.
2. Theory Fig. 1 depicts the experimental principle for exciting SPPs by the ATR method, in which a hemispheric coupler [14] is used to handle anisotropic effects. Levy et al. propose a method that uses a pyramid as a coupler and allows the measurement of reflectance in two perpendicular directions [ 1 I]. In their configuration, however, variation of reflectance with incident angles is difficult to
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258
H. Wang / Optics Communications 135 (1997) 257-263
Laser beam
Fig. 1. The principle of the attenuated total reflection (ATR) experiment.
compensate, especially when an anisotropic coupler is used. By using a hemispheric coupler, since an incoming light beam falls onto the coupler at a normal incidence, it is easy to direct the beam to the coupling spot and to compensate the change in reflectance. To study the anisotropic effects related to the columnar structure of films, we adopt a model proposed by Horowitz [3]: referring to Fig. 1, we describe the film assembly by the “laboratory” coordinates (x, y, z), in which the x-axis is perpendicular to the film surface and ( y, z) coordinates lie in the film plane. In turn the “physical” coordinates (x’, y’, z’) that are in the principal directions of the columnar structure induced anisotropy in each dielectric medium are formed by a rotation of angle @ around the y-axis, as shown in Fig. 1. These “physical” coordinates may vary in different media. In each anisotropic dielectric medium a material equation in the coordinates of principal axes (x’, y’, z’) is given by D = q,( x’+ E,, + y’qYt EYt +z)Ejz~Ez~), where E,, is the free space dielectric permittivity, and where E and D are the electric field and the electric displacement vectors, respectively. In the coordinates (x, y, z) the normal and tangential components of the displacement vector read:
Nullifying the determinant of Eq. (3) yields a dispersion relation for the normal and tangential components of the wave vector k,,=K,,K-
AK,
K,, = Wrl,)(k&:
AK=
nz = esx, sin2 @ + esz, cos* @,
(2)
sin @ cos @.
In the metal and the substrate (media 1 and 2) the three principal elements of the dielectric tensor are merely identical. The metallic film has a complex dielectric function l2 = E; + ie$. There are two mechanisms to excite SPPs at the dielectric/metal interface (interface 23): When the coupler is used in total reflection, if the dielectric layer is sufficiently thin so that an evanescent wave reaches interface 23, then SPPs may be directly excited at it. In the case the dielectric layer is thick, it may support several guided modes. If the tangential component of the guided waves match that of
I’*,
ks&x/n;).
The magnetic field is related to the electric optical admittance, Ys, Hsy = +YsEsz,
field by the
Ys = (Eo/CLo)“*(k,rl,2/K,,)K.
(5)
Introducing Eqs. (4) and (5) into boundary conditions results in a definition of propagation constant, /3. The tangential component of the wave vector remains constant across all boundaries, k,, = k,, = /3. The dispersion relation reads:
E2
rlz = ls+, (30s’ @+ esZ, sin’ @,
- k;$‘2,
K = [ 1 - &(wL)*]
(k;;e,_fi*)“*
where
nZX= ( egZ, - Q)
SPPs, again SPPs can be excited. For a thick metallic film, SPPs are sensitive to changes in optical constants near interface 23. They decay before reaching the metal/sub strate boundary (interface 21>, and therefore, are insensitive to the thickness of the metallic layer. In that case the metallic layer can be regarded as infinite. The wave equations in isotropic metallic media and their solutions are well known. For a p-polarized plane wave propagating in the x-z plane, in the metal the normal and tangential components of the wave vector are simply connected by a dispersion relation: k,, = (kze, - kg,)‘/*, where k, = 2?r/A. In the anisotropic dielectric, on eliminating the magnetic vector we have Maxwell’s equations expressed in the form
+ (k,$&.-P2)1’2
=
o
(6)
1731173zK
that defines an explicit phase-matching relation of the tangential component of SPPs, /3, to the dielectric functions of the materials adjacent to interface 23: P=ko[%d(‘?,2K*-
~~)/(7&7;K*-
E:)]“*.
(7)
When the dielectric medium is isotropic, Eq. (7) reduces to the same form as that presented in Ref. [lo]. To express the propagation constant, we define an effective index, N = /3/k,, with its real and imaginary parts, N’ and N”, calculated by N’ = (1/2)“‘(
) N 12 + ad) “*,
N” = (l/2)*/*(
1N ) 2 - ad) “*,
(8)
H. Wang/Optics
where their amplitudes
where a=
(
6;’ + E;~)(&,~K~
q:K’e;+ 2e’;‘(2e;2
Rzj=
- E;’ + E;‘)
-47,2K’[~;(q;q;+qTq;)+~;)(q;q;-qyq;)]
- $K’E;),
+ 2+:K2~;
I IE2121qj12+V~K41q212
+2,;K”[
l;(q;q;
+E;(q;q;
- q,lq;)]}-I,
- E;’ + E;‘)],
7;$K2
d= d/[(
- E;’ + E;‘)’
+ 4ei2ez2
1
1, (9)
= 2vZ"K"[ E;(q;q);
l;’ _ N'2 +N"2)'/2,
I q2
41; = ( 1/2)“2(
I q2 2 _ e;2 + N” - Nn2)
lq212=
[(+Nf2
l/2,
(10)
1, r/r
t N”‘)’
+ (e; - 2 N’N”)’
and where
q’j =
(1/2p2[I q3 I ’ + o:- (s,/a)“( N” -N”‘)] “‘, I”, (1/2y2[ I q3 I2 - T$ + (r/,/qJ”( N'* -N"')]
h/B)[(4- N12 + N”2)2 + 4N’*N,,‘]
“‘.
(11) For a thin metallic film, the condition for the existence of SPPs can be obtained by analyzing the reflection of plane waves at interfaces 23 and 21, which gives the following dispersion equation [lo]: rz3rz1 exp(i2k,,d,)
= 1.
(12)
Here the anisotropic effects of the dielectric are included in the reflection coefficients calculated by the optical admittance in the dielectric, Eq. (5), that in the metal, and that in the substrate, Y, = Y, = (~0/~o)1’2e~/k~,, The normal component of the wave (Eo/luo)“2+kr,. vector in the substrate can be calculated by a way analogous to that for K3x, i.e., k,, = k,(q’, + iq’$ where the calculation of q; and q’,’ can use Eq. (11) in which the dielectric functions are replaced by using nj = 7: = err rl = 0, and K = 1. This leads to an expression for the reZection coefficients: rzj=
j=
(Y,
1,3,
- q)/(
Y2 + q)
= Rt$* exp( -i2+2j),
(13)
- E;(q;q;
+ q;q;)]
’ (14)
Anisotropic effects of the hemispheric coupler are related to the phase-matching condition. The tangential component of an evanescent wave outside the base of the coupler is given by k,, = ke~~/~ sin 0,. Here the permittivity of the coupler, e4, depends on the incident angle 0,. This permittivity can be calculated from the dispersion relation in the coupler that relates the normal component to the tangential component of the evanescent wave vector: then the perk,, = k,,(qz/~q+)1’2[~4x - (k4z/k,,)2]1/2; mittivity is simply given by [ 151 l4 = (~~~e~~)/(
llx cos28, + l4z sin28,)
that leads to the tangential wave vector
141312=(17L/1)x)i[ n~-(N’+iN”)2]“212 =
- q;q;)
I~2121qj12-~~K41~212
From which the normal components of the wave vectors are expressed as k,, = k,q, = k&q; + iq’;), and K,, = k,q, = k,(q’, + iq’;), where q; = (1/2)“*(
+q;q;)
tan 242j
N I* = d(a2 + b’)“‘.
2+
and phases are defined by
1
X
q; =
259
Communications 135 (1997) 257-263
k,, = k,(
l4xe4z)1’2
component
(15) of the evanescent
sin 0,/( e4x cos2B4 + e4Z sin’0,)
l/2
.
(16) Making this match the real part of the tangential component of SPPs, k,, = k, N’ [16], we determine a coupling angle for the ATR to occur: sin @,,a = N’e:!‘/[
e4x~4z - N”( e4r - e,,)]
I’*.
(17)
This equation is applicable when N” -=ZZ 1. From Eqs. (8) and (9), it means b =K n, leading to a similar restriction on metals for the ATR method to be validated: E’ < - I, e” < I E’ I [8,17]. At the coupling angle SPPs excited near the metal/dielectric interface deprive the light power of an incident beam and a sharp drop of reflectance will be observed.
3. Experiments MgF, is the most widely used material for thin film engineering, mainly used to modify reflectivity of optical components. MgF2 films can be thermally evaporated from a tantalum or molybdenum boat, with substrates heated to a temperature about 200°C. Bulk crystalline MgF2 is
260
H. Wang/Optics
Communications 135 (1997) 257-263
Fig. 2. The experimental setup to record reflectance as a function of incident angles.
slightly anisotropic and has refractive indices n, = 1.370 and 12,= 1.381 at A = 550 nm for ordinary and extraordinary rays, respectively. MgF, films may exhibit lower refractive indices, since a 100% packing density in thin film deposition is impossible. Experiments are conducted on several MgF,/ AI/ BK7-glass thin film assemblies. The BK7-glass substrates are routinely polished, on which Al films are evaporated at nearly normal incident angles of vapor flux to avoid dichroism in them. The refractive indices of the MgFz over-coatings deposited on top of the Al films vary, depending on the packing density and on the incident angle of vapor flux in evaporation. As a result of this incidence, MgF, films may show some degree of birefringence. The refractive indices of the MgF, films can be identified by the R/T measurement [5]. The dielectric function of Al films is assessed by the ATR method, which identifies the coupling angles via the measurement of reflectance as a function of incident angles. Fig. 2 shows the experimental setup. A helium neon laser with the wavelength A = 0.6328 pm is used to illuminate the sample under test. The laser beam is modulated by a mechanical chopper. The modulation frequency is sent to a lock-in amplifier as a reference frequency that provides a way to suppress the stray light from environment. The sample is mounted on a stage consisting of two rotating motors driven by a motor controller. The angular position of the motors is transmitted to a PC. The laser is set on one of the rotating arms of the stage, whereas a photodiode is placed on another arm. The modulated laser beam impinges upon the coupler (that is not shown in the figure) and then is tunneled into the sample at the coupling angles. A synchronized rotation of these two motors enables us to change incident angles of the probe laser beam and to keep the photodiode always in the direction of specular reflection. Another photodiode is used to collect the light signal directly emitting from the laser before it reaches the sample (This photodiode is not shown in the figure.) The reflected signal in reference with the light
signal directly emitting from the laser gives the reflectance of the sample. The photocurrent signals from both photodiodes are directed to a preamplifier, then are amplified by the lock-in and sent to the PC. Combining the reflectance signals with the position signals of the motors, we can record the reflectance of the sample as a function of incident angles. Parameters of the sample can be recovered from Eqs. (17) and (7) or (12). However, a calculation of effective index from Eq. (12) is rather complicated. Here a method presented in Ref. [9] is adopted and the propagation constant is determined by the following approach: First the Al films are considered having an infinite thickness, the dispersion equation is then determined solely by interface 23, this solution is called PO. Then boundary 21 approaches boundary 23 and a slightly changed solution p = &, + A /3 is then obtained from Eq. (12). This procedure virtually is a perturbation approach, in which the initial estimation is given by PO; whereas Ap is a perturbation term, whose real part displaces the resonance value of PO and its imaginary part changes the damping of the film assembly. To clarify the effects of the columnar structure, we use three different incident angles of vapor flux: @’ = 5.00”, 30.00”, and 65.00”, to deposit the MgFz films in the MgF,/Al/BK7-glass thin film assemblies. The thicknesses of the MgFz films are controlled around 110 nm. This insures an effective tunnel of light power from the incident beam to the Al film at the coupling angles. The refractive indices and columnar orientations of the MgF, films are obtained from the R/T measurement [5] and shown in Table 1. In the ATR experiment, two perpendicular orientations of the coupler are used to explore its anisotropic effects, as
Table 1 The dielectric functions of MgF, /Al/BK7-glass thin film assemblies. The prism coupler was positioned with its optical axis both perpendicular (quantities with the subscript “ I “1 and parallel (quantities with the subscript “ II”) to the incident plane Sample #l
#2
#3
- 42.252 - 14.121 32.73 - 42.870 - 13.902 30.04 5.00 2.61 1.369 1.375 1.377 - 42.561 - 14.012
- 40.490 - 16.502 32.61 - 42.103 - 16.100 29.92 30.00 16.27 1.364
- 46.578 - 15.813 32.68 -48.162 - 15.210 29.98 65.00 47.54 1.361 1.373 1.374 - 47.370 - 15.512
1.373 1.375 - 41.297 - 16.301
H. Wang/Optics
-
Sample #l
.-----
Communications
Sampleti
.- .--..‘.Sample #2
1.0
8
0.8
g Gz d
0.8 0.4
33
32
Incident angle 0, (degree) Fig. 3. The measured reflectance versus incident angle curves for the MgF, /Al/BK7-glass thin film assemblies samples 1-3. Their optical properties are exhibited in Table 1. The coupler is posi-
tioned with its optical axis perpendicular to the incident plane.
it is described in Pq. (17). For each MgF,/Al/BK7-glass sample corresponding to each one of the coupler orientations, a dropped reflectance value is observed. Totally 100 times of reflectance measurements are carried out for each sample. Fig. 3 plots the reflectance versus incident angle curves averaged over one hundred measurements for the three MgF,/Al/BK7-glass film assemblies. Here the optical axis of the coupler is positioned perpendicular to the incident plane. In these experiments, thick Al films with thicknesses about 135 nm are examined. As a result, SPPs excited at the MgF,/Al interface are not sensitive to the dielectric function of the substrate. From these curves, we can clearly pinpoint the coupling angles (0,, I>, which are shown in Table I. Similar experiments are conducted for the coupler placed with its optical axis parallel to the incident plane. Accordingly, the averaged reflectance versus incident angle curves are plotted in Fig. 4. The coupling angles (@,,a ,,> again are identified and displayed in Table 1. From these curves alongside those shown in Fig. 3, we
_ 1 .O
8 5 ‘E( g I+!
-
Sample #I
-----
Sample #3
--.-.--..-Sample#2
0.8 0.6 0.4 0.2 0.0
’
28
29
30
Incident angle 6, (degree) Fig. 4. The same as Fig. 3. Here the coupler is positioned optical axis parallel to the incident plane.
with its
135 (1997) 257-263
261
recover the dielectric functions of the Al films, which are exhibited in Table 1, too. In principle the dielectric functions of the Al films recovered with two different coupler orientations must be identical, since Al films are not birefringent. However, from Table 1 we find that they indeed exhibit some degree of difference. A probable explanation to this difference could be the slight bireftingence caused by the incident angle of vapor flux deviating from the normal incidence. The Al film in sample 1 is deposited at normal incidence and therefore exhibits the smallest deviation in dielectric functions corresponding to two coupler orientations. Samples 2 and 3 are deposited at incident angles of vapor flux @” = f 5”, accordingly, the deviation in their dielectric functions is larger. However, even at normal incidence, the difference in the dielectric function still remains. One of the reasons for this difference is that the experiment method is not error-free. The principal refractive indices of the MgF, overcoatings are assessed by the R/T measurement. This method can only recover refractive indices with an uncertainty to the third decimal place. The error in the measurement of refractive indices of the MgFa overcoatings can be instantly transferred to the ATR measurement. In addition, the dielectric function of Al films can change from place to place. Nevertheless, since here a hemispheric coupler is employed, the probe laser beam can be more accurately directed to the coupling spot. At any rate, the difference between the dielectric functions obtained with two perpendicular coupler orientations indeed reveals the uncertainty of the measurement. In consequence we calculate some statistical parameters of the measurement to evaluate this difference. First, the mean value of the dielectric functions, ( l;) = (l/2)( l; I + e&l and ( l;I) = + E$ are taken as the nominal values of the 0/2xe;. Al films, which are shown in Table 1. Then the relative difference between the values of dielectric functions corresponding to the coupler perpendicular and parallel to the incident plane are then calculated referring to these mean values of the dielectric functions. In Table 2 it is shown that the difference varies over a range of about + 3.9%. Another important statistical parameter is the standard deviation of the measured values. From the central limit theorem, we can consider that the maximum error of the measured dielectric functions is three times their standard deviation values. In Table 2 it is shown that the standard deviation acquired from the data with the same coupling orientation, whether for a; and a; or a,; and o’;;, is obviously smaller than the statistical parameters obtained from the data with two different coupler orientations, a’ and a”. In fact, for sample 1 the maximum measurement error corresponding to the same coupling orientation, 2.98%, is significantly smaller than the error in the case of both coupling orientations, 4.00% - this is a difference exceeding 25%. For samples 2 and 3, the maximum errors for the case of same single coupler orientation is, still
262
H. Wang/Optics
Communications 135 (1997) 257-263
Table 2 The statistics for the dielectric function data of MgF, /Al/BK7glass thin film assemblies, the subscripts “ I ” and “ 11”indicate
the orientations of the prism coupler, as shown in Table 1 Sample
AE;/(E;)(%) Ae;/(<;> (%) fl: 0; 30; /E;A(%0) 3u’; /E;) L (%) 5, 2; ,E& (%I 3$/E,“,, (%I u’ 0” 3U’/(E;) (%o) 30*/(E;) (o/o)
#1
#2
#3
- 1.45 1.56 0.198 0.104 - 1.41 -2.21 0.210 0.138 - 1.47 - 2.98 0.476 0.187 -3.36 - 41x)
-3.91 2.47 0.221 0.148 - 1.64 - 2.69 0.234 0.168 - 1.67 -3.13 0.430 0.201 -3.12 -3.70
-3.34 3.89 0.230 0.100 - 1.48 - 1.90 0.189 0.191 - 1.18 -3.77 0.481 0.197 - 3.05 -3.81
smaller than those for both coupler orientations, however, the difference now is not so significant. Considering the experimental setup and method, we have four main factors causing inaccuracy: first, the noise in the reflectance detection is the most fundamental error source and is inevitable. This causes uncertainty in the measured reflectance values and can only be reduced by improving the detection unit, including the photodiodes and the amplifiers. Generally light intensity detection can reach accuracy better than one percent. Next, the angular positioning error in the incident angle measurement will also contribute to inaccuracy. This can be caused, for example, by the rotating stage and fine positioning errors. This factor is significantly smaller than the first one and can be reduced by the average process as it is described here. Another source of inaccuracy is the change of coupling spot when the coupler is shifted from one orientation to another. The use of the hemispheric coupler is to reduce this effect. This can also be reduced by the average process. Finally, the data processing technique may also induce some error. As it is described above, both in guided-wave and ATR experiments, optical constants of the film assembly under examination are obtained by finding coupling angles at which the incident light power is tunneled into the films. In the measured reflectance versus incident angle curves these coupling angles are observed as some sudden drops, as those shown here. Actually this process is a least-squares fit procedure that adjusts the optical constants of the films (in the experiments here the dielectric function of Al films) to match the measured reflectance curves. This process can be refined by an iterative operation. Even though the residual error always exists in the least-squares fit and it will also cause inaccuracy. In fact, in the ATR experiment
the data processing technique may contribute to certain degree of inaccuracy in measurements, as the theoretical calculation is rather tedious and complicated. From the results shown in Table 2, we find that the accuracy of the ATR measurement is better than five percent. This judgment also agrees with the studies presented in the literature. Indeed, aluminum is one of the mostly known metals. Its dielectric functions, both in the form of bulk crystals or films, have been extensively studied. For example, in Ref. [12] Fontana et al. examine an Al film with the thickness of about 120 nm deposited onto a Pyrex substrate and protected by a MgFa overcoating. Its dielectric function is assessed by the ATR method at A = 0.6328 pm, which gives E’ = -46.6 f 3.1 and E” = - 18.6 + 4.1. These values are quite close to the results obtained here. In addition, Schulz et al. [18] have measured several metals by the spectrophotometric method. They give dielectric functions versus wavelengths but do not give the values at A =0.6328 pm. We obtain the values at A = 0.6328 pm by an interpolation, which shows values E’ = -39.56 and E” = - 14.67 for aluminum, also close to the values shown here.
4. Conclusions In this paper the attenuated total reflection (ATR) method is applied to study aluminum (Al) films covered with anisotropic magnesium fluoride (MgF,) overcoatings comprising the columnar structure. Anisotropic effects of the MgF, overcoatings are explored by using a hemispheric coupler in two perpendicular orientations. The phase-matching condition for exciting surface plasmon polaritons (SPPs) in the Al films is examined under the effects of the anisotropic MgFa overcoatings. The effects of a birefringent titanium dioxide (TiO,) coupler on the phase-matching condition are also taken into consideration. In experiments, the reflectance is recorded as a function of incident angles. The coupling of the light power into SPPs is identified by observing a dropped reflectance value in the reflectance versus incident angle curves. The dielectric functions of the Al films are recovered from these curves with an accuracy better than five percent. In comparison with other more accurate techniques such as the prism coupling method for determining optical properties of thin films, this accuracy is not very satisfactory. The main emphasis of this work is to give a fine theoretical analysis. The experimental results are given to illustrate the applicability of the method. More accurate results may be pursued in future studies.
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and H.K. Pulker,
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135 (1997) 257-263
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