Determination of interface roughness cross-correlation properties of an optical coating from measurements of the angular scattering. P. Roche, P. Bousquet, ...
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Rocheet al.
Determination of interface roughness cross-correlation properties of an optical coating from measurements of the angular scattering P. Roche, P. Bousquet, F. Flory, J. Garcin, E. Pelletier, and G. Albrand Ecole Nationale Superieure de Physique, Centre d'Etude des Couches Minces (Associ6au Centre National de la Recherche Scientifique), Universit6 de Saint J6rOme,13397Marseille Cedex 13, France Received February 16, 1984; accepted June 12, 1984
Measurements have been made of angular scattering from smooth glass substrates before and after the deposition of an optical coating.
The experimental results are compared with theory.
In the particular case of a single layer,
we derive information on the relation between the degree of roughness of the substrate and the correlation between layer interfaces.
measurements permit the reconstruction of the spatial-scat-
INTRODUCTION Nowadays it is important to be able to manufacture optical multilayer coatings of high quality and with extremely small scattering losses. To meet these requirements, many laboratories are performing experimental and theoretical studies of the scattering phenomenon. In this paper we discuss progress in our laboratory in Marseille, emphasizing a new and
significant aspect.
tering curve. Level curves will permit us to give a plane rep-
resentation of this spatial-scattering curve: we plot, in polar coordinates, the projections on the plane (0, a) of different level curves BRDFj cos 61= constant, 0 being the vector radius and a the polar angle. The surfaces, the roughnesses of which are isotropic, would give here a set of concentric circles. We have observed that the angular-scattering curves of a normally
illuminated sample frequently are not surfaces of revolution. Figure 3 shows the light forward scattered [i.e., the light in the
DESCRIPTION RESULTS
OF APPARATUS AND SOME
The apparatus that has been constructed permits the measurement of the complete angular-scattering curve in the form of successive plane sections 1 (Fig. 1). The test sample can be
illuminated at any specified angle of incidence by means of a helium-neon laser. With a spatial filter, we have a light beam of an angular width of 3', and the diameter of the illuminated spot on the sample is 3.8 mm. The photomultiplier is situated 1 m from the test sample, and the solid angle of
scattered light received by the receptor is limited to 7 X 10-5 sr. The arm bearing the photomultiplier permits 3600 rotation around the test sample. A record of a plane section is shown in Fig. 2. The angle of incidence is 1030', and the
scattering diagram is recorded in the angular range 6 from 3020' to 176050'. We note that the angular field 0 < 0 < 900 corresponds to the half-space containing the specular reflection direction and that 0 > 900 corresponds to transmitted scattering. A stepping motor controlled by computer rotates the sample in its plane. a is the angle of rotation, and for each scattering angle 0, the 3600 rotation of a requires 40 sec. Whatever value
a takes, the studied surface always remain the same. It takes 2 h to obtain all 180 sections of the angular-scattering curve (Oa = 20); each section consists of 100 measurements in 0. For each of the 180 values of a, we can plot a plane section of BRDF cos 0 as a function of 0 in the angular ranges
0-90° (forward scattering) and 900-180° (backscattering). (BRDF is the bidirectional reflectance distribution function.)
In the case of illumination under normal incidence, these 0740-3232/84/101028-04$02.00
half-space containing the specular reflection direction (0 A 6 < 900)1,and great defects of symmetry can be observed. Figure 3(A) represents the spatial distribution of forward scattering for a substrate of superpolished silica. The total amount of scattered light is 4 X 10-6, and the symmetry defects are due to polishing defects of the substrate, which consist of a series of fine, parallel scratches. Figure 3(B) represents the forward scattering of the same substrate covered with a multidielectric coating, that is to say, a mirror consisting of high- and low-index layers, each of optical thickness equal to Xo/4. The integrated scattering in this same half-plane is now 140 X 10-6. The defect of symmetry is obvious; it plays an important role in the great increase in scattering after coating. CALCULATION OF SCATTERED LIGHT Different theoretical techniques, for predicting the angular dependence of scattering exist.14
Our method consists of a
vector theory that uses first-order perturbation to describe the optical behavior of slightly rough surfaces. Each of the surfaces and interfaces of the thin-film multilayer is replaced by a perfect one with an associated distribution of surface current. This causes further discontinuities of the electromagnetic field in the classical calculation of propagation in a stratified medium, and so we obtain expressions for the fields
scattered on either side of the multilayer. The angular distribution of the scattered field is put into the form of the BRDF: p(O, 0), where plcos OjdQ is equal to the ratio between the flux scattered by a surface element in the solid angle dQ around the direction (0, 0) and the flux © 1984 Optical Society of America
Vol. 1, No. 10/October 1984/J. Opt. Soc. Am. A
Roche et al. absorber I
chopper
specimen
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us to predict the scattering phenomenon, we should attempt to relate theory with experiment. Unfortunately, the multiplicity of parameters raises many problems. It is obvious that the example of laser mirrors is a rather bad
one for the development of precise studies of correlation between measurement of scattering diagrams and their theoretical interpretation.
We must be able to fix the values of
some parameters, such as the autocorrelation length, and an important task is to verify in detail the validity of the assumptions made on the autocorrelation and the cross-correlation functions. We therefore choose to analyze in detail the scattering of Fig. 1.
Optical arrangement of the apparatus used for measurement
of the angular scattering. BRDF cos
e
the substrate coated with only one layer. In this simple case, the BRDF can be written as P
4i
0 [COCO7oo + CCiyll
+ 2]9e(C 0 C-y
0
j)],
10-2
10-3
1o-4 C
10-i
0-6
\
0
30
60
e (degrees)
90-l170* 120
130
(A)
Fig. 2. One section of the angular scattering curve of a narrow-band Fabry-Perot filter. Design: glass (HL) 3H 18L (HL) 3H, where H (zinc sulfide) and L (cryolite) are quarter-wave layers for X = 632.8 nm. The two curves correspond to the measurements for pp and ss polarization. The angular field 0 < 0 < 900 corresponds to the halfspace containing the specular reflection direction. Conversely, 900 < 0 < 180° is the half-space of transmitted light.
incident upon the surface element.' So __
d(d) d,
4")I cos OoldQ 47 -
M
I
N0 ()
Ii=Oj=O ZOT Cv-Cj *-yij[K(d)-
K(i)]
ad
for reflection scattering, where dk(d) is the flux scattered in the solid angle dQ, 0) is the incident flux,
(Icos 00/cos iol Icos io/cos 0ol
No No(0
)|
11/cos
io cos Oo
A|cos io cos Oo
for polarization
ss
900
.I
for polarization pp for polarization
sp
for polarization
ps
8
.
and yii(K) is the Fourier transform of the autocorrelation function (i = j) or of the cross-correlation function (i # j) of the surface roughness. The form of the expression consists of sums in which each
term is the product of a factor that depends only on the characteristics of the ideal nonscattering multilayer and of a factor yij(K) that describes the irregularities of the surfaces. Since we are able to measure the spatial distribution of scattered light and since the theoretical mirror design enables
Fig. 3.
Anisotropy of the forward-scattered
light (0 < 0 < 90°).
(A)
Superpolished silica. The different curves correspond to the following values of the BRDF cos 0: a, 1.0 X 10-5; b, 1.5 X 10-6; c, 1.0 X 10-6. (B) The same substrate covered with a multidielectric mirror (HL) 7 H, 15 quarter-wave for X = 632.8 nm: a, 1.0 X 1O-3; b, 2.0 X 10-4; c, 7.0 X 10-5; d, 5.0 X 10-5; e, 3.0 X 10-5; f, 1.0 X 10-5
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Rocheet al.
where Ci are known coefficients and yij(K) are three unknown functions. In order to approach the problem of determination
of these functions, it is necessary to have a simplifying hypothesis concerning the form of autocorrelation and intercorrelation functions. Bennett and her colleagues,particularly Elson,2 give techniques for determining the autocorrelation function of a bare substrate.
A good representation
200-
- 35
Zinc sulfide
150 - 30
is
given by the Hankel transform (HT) of an exponential function: yoo(K) = HT [ao exp(
j GO (A)
Do 4
(ppm)
X,9
1,2 100_ f 25
-
LO)]
3,1
1,2 20
50 -
4,7
6,7 4,7
-15
where ao is the rms of the roughness and Lo is the autocorrelation length. We must now choosethe autocorrelation function between the interface and the air and the intercorrelation function between the two surfaces that delimit the layer. Here, we examine only the two limiting cases: uncorrelated interfaces (Qyo = 0) and completely correlated interfaces (Yol = 1), and we try to explain the scattering of a substrate with a single layer.
- 10
4,2 0
2,6
l
0
4,0 3,8
7,1 ,
,
X
4
A
2
8
2
Fig. 4. Evolution of the ratio ? = DC/Do of total before (Do) and after the deposition of a single The theoretical results are - = 7 if yoi = 1 and quarter-wave layer and 7 = 1 if -Yo, = 1 and ?) half-wave layer.
D
ne
scattering measured layer of zinc sulfide. 7 = 4 if -Yo1= 0 for a = 16 if yo, = 0 for a
for a given substrate roughness, we also see a slow decrease in
the degree of correlation with an increase in the layer thickness. All things considered, we can say that, for the layers of
INTERCORRELATION BETWEEN THE TWO INTERFACES OF A LAYER In our theory, the scattering of a layer of thickness e and re-
fractive index n is completely described by four parameters. These are the mean quadratic values of the roughness of the two interfaces; the autocorrelation length, which is known; and
the coefficient of intercorrelation between the two surfaces. From the calculations, this last coefficient seems to be a key parameter. We have calculated systematically the relative effect of each
of the parameters that play a role in the scattering phenomenon for the case of high-index and low-index layers on a glass
substrate. The sample is illuminated at normal incidence. We consider total scattering D in the half-space located on the side of the reflected beam, and, in an attempt to remove de-
pendence on the roughness of the substrate, we consider the ratio -q= DC/Doof total scattering measured before (Do)and after (D,) the deposition of the single layer. The results are particularly interesting with a high-index layer since n depends considerably on the intercorrelation between the two surfaces. With a quarter-wave layer and the hypothesis that al = ao, 77= 4 for entirely decorrelated surfaces, whereas q = 7 for correlated surfaces. Under the same
conditions with half-wave layer, the corresponding values of
optical thickness X/4 or X/2, the intercorrelation between the two interfaces of the layer is approximately one, that is, the layer reproduces and followsthe exact shape of the substrate, including the defects. However, if we consider the shape of the scattering diagram, we can see that this intercorrelation is, in fact, a function of the spatial frequency of the defects, and some experimental results are given elsewhere.4 Multilayer Systems With multidielectric mirrors of quarter-wave layers, we obtain the following conclusions.
The values of -y(K)deduced from the measurements made with pp and ss polarization show a clear decrease in the degree
of correlation when the substrate is isotropic. Calculation shows that the roughness of the interfaces that are farthest from the substrate is the origin of the scattering phenomenon but that the interface roughness depends on the defects of isotropy of the substrate. The defects of isotropy observed on the bare glass have pronounced effects on the spatial distribution of the scattered light observed after multilayer coating. The present state of knowledge of y(K) is insufficient for an accurate and detailed
interpretation of experimental results for appreciably anisotropic surfaces.
q are 16 and 1, respectively.
ACKNOWLEDGMENTS EXPERIMENTAL RESULTS Single Layers Figure 4 is a review of results obtained with a classical material, zinc sulfide. The experimental values of 7 are given as
We thank H. A. Macleod of the University of Arizona for his
help in the translation of this work. This paper was presented at the 1983 Annual Meeting of the Optical Society of America, New Orleans, La., October 17-20.
a function of the layer thickness and of the substrate roughness. We can summarize our results as follows. We see a decrease in the degree of correlation between the interfaces as the roughness of the substrate decreases, and this result is valid whatever the thickness of the layer. However,
REFERENCES 1. P. Bousquet, F. Flory, and P. Roche, "A scattering from multilayer
thin films: theory and experiment," J. Opt. Soc. Am. 71, 11151123 (1981).
Roche et al. 2. J. M. Elson and J. M. Bennett, "Relation between the angular dependence of scattering and the statistical properties of optical surfaces," J. Opt. Soc. Am. 69,31-47 (1979); J. M. Elson, J. P. Rahn,
and J. M. Bennett, "Relationship of the total integrated scattering from multilayer coated-optics to angle of incidence, polarization, correlation length, and roughness cross correlation properties," Appl. Opt. 22, 3207-3219 (1983).
Vol. 1, No. 10/October 1984/J. Opt. Soc. Am. A
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3. S. J. Gourley and P. M. Lissberger, "Optical scattering in multilayer thin films," Opt. Acta 26, 117-143 (1979).
4. J. Garcin, "Diffusion de la lumiete: etude experimentale et representation theorique des rugosites des interfaces de filtres multidielectriques," Doctoral Dissertation (Universitk d'Aix Marseille
III, Marseille, France, 1982,unpublished).
