Description of a scattering apparatus: application to the problems of characterization of opaque surfaces
C. Amra, C. Grezes-Besset, P. Roche, and Emile Pelletier
We show how the complexity of a micropolished optical surface can be investigated in detail by measurement of the distribution of scattered light. We deal with problems of roughness anisotropy and uniformity together with cleaning problems. Experimental results concern numerous black glasses from different polishing shops and allow a determination of the polish inhomogeneity in a same glass set. After that, we present a detailed study of the apparatus function of the scatterometer, and we determine the limits of validity of our optical characterization method.
I. Introduction Minimization of scattering in optical multilayers
such as Fabry-Perot filters and laser mirrors is an important problem today because of the stringent demands of many optical applications.1 Recent results have shown that reduction of scattering requires the use of supersmooth substrates,2 and polishing skills have now advanced to the point where, for certain materials, surface roughness does not exceed some 0.2 nm.
Characterization tools have improved in parallel with this increase in the quality of optical surfaces. Among the different methods3 that give access to the roughness of an optical micropolished surface, measurement of light scattered by the sample in each direction of space is particularly well adapted, because this technique yields additional information on the anisotropy of the surface roughness.4
We developed in Marseilles a scatterometer capable of measuring the distribution of scattered light in all directions of space5 as well as a vector theory of light scattering to interpret the experimental results.5 In the case of opaque surfaces, these combined tools of
The authors are with Ecole Nationale Superieure de Physique de Marseille, Laboratoire d'Optique des Surfaces et des Couches Minces, CNRS U. A. 1120, Domaine Universitaire de St. Jerome, 13397 Marseille CEDEX 13, France.
theory and experiment provide the roughness spectrum and autocorrelation function (roughness and autocorrelation length) of substrate defects as well as the uniformity and anisotropy of the roughness. The detection level of the apparatus must be very low since we know that the total integrated scattering (TIS) (scattering losses integrated in the whole space) from a supersmooth surface can be less than ~10 - 6 of the incident flux. A short description of the apparatus is given.
With this experimental setup, we have been able to undertake a systematic study of the quality of black glasses produced by different optical shops; in particular, we studied the polish homogeneity within several series of samples. Results on cleaning process and sample degradation are also presented.
Since scattering levels and residual roughness in multilayer stacks are often due to the reproduction of substrate defects,6 it is essential to specify which defect periods our optical method is sensitive to. We are also interested in the problem of approximating a roughness spectrum with analytic functions. Such results would permit a better comparison between our characterization method and others.
II. Experimental Setup This apparatus has already been described4,5; thus
we will only include a short description of it, pointing out some complementary information.
Figure 1(a) is an example of the spatial distribution of the light scattered from an optical surface. Our apparatus has been devised so that we can measure this light distribution I(θ) in the incident plane whatever the illuminating incidence. In addition, it permits measurement of the scattered flux I(θ,φ) out of this plane [Fig. 1(b)], when the sample is illuminated under normal incidence with nonpolarized light.
Fig. 1. Scattering distribution from rough surface: (a) Polar representation in the incident plane (φ = 0) of the angular scattering curve of a glass surface calculated with 5-nm roughness. The specular beams are drawn, and Oz is the normal to the surface. The radius of the polar curve is the scattered flux BRDF • cosθ per unit of surface and solid angle normalized to the incident flux. (b) Angles θ and φ describing a scattering direction. The surface with normal Oz is illuminated at incidence i (no reflected or transmitted beam is shown here). σ is the spatial frequency of the grating responsible for scattering in direction (θ,φ). Thus we can see that the polar representation of (a) concerns the variations of scattered intensity
I(θ,φ) in the incident plane φ = 0.
A. Experimental Procedure In the case represented in Fig. 2(a), the sample is
illuminated under quasinormal incidence (i = 1.5°) with the unpolarized light of a He-Ne laser (0.6 μm; 1 mW). Light wells eliminate all the parasitic light due to the specularly reflected and transmitted beams.
A moving arm, 1 m long, bears a photomultiplier which carries out a set of 100 measurements I(θ) in the incident plane π with a sampling interval ∆θ ≅ 1.8°. A plane section of the scattering curve is recorded in that way in the angular range 3.3° ≤ θ ≤ 176.7° (θ is the angle between the normal to the sample and the measurement direction).
The recording of the scattering curve I(θ,φ) out of the incident plane is based on the fact that it is equiva-
Fig. 2. Principles of the apparatus and scattering representations: (a) principle of the apparatus where the incident and reflected beams have been superimposed. In fact, the sample ∑ of normal ON is illuminated at i = 1.5°. In these conditions, the minimum angle for scattering measurements is θ = 3.3° with respect to ON, which is 1.8° with respect to the reflected beam. The receiver records 100 measurement data points 7(0) in the incident plane π. Moreover, the sample can rotate around its normal ON so that we can measure for each direction 0 the value I(θ,φ) for 250 rotations ∆φ of the sample. (b), (c) Polar representations of the function BRDF • cos0(θ,ø). In the case of (b), the polar radius is the BRDF • cosθ value, while in (c) the polar radius is the scattering angle 0. The interest of either curve depends on the dynamic range of scattered
flux.
lent here to measuring the scattering I(θ,φ) for a given position of the sample or to measuring in the incident plane the scattering Iθ,φ = 0) when the sample is rotated an angle φ around its normal.
For that the sample is mounted on a rotating stage in such a manner that, for each measurement direction θ, we can record 250 values of the scattered flux, corresponding to 250 rotations of the sample around its normal. The sampling interval is ∆ø = 1.44°. Thus the scattering curve is described in the whole space with these 25,000 measurement points which are stored on a floppy disk for further data processing.
It is then usual to study the function BRDF cosθ(θ,φ), which represents the flux scattered per surface and solid angle unit normalized to the incident flux. In addition to the classical representations that give for a given angle φ the variations of scattering vs angle θ, it is convenient to use two polar representations [Figs. 2(b) and 2(c)] both of which display the anisotropy of surface defects. For each of these
Fig. 3. Scatterometer response when there is no sample in the measurement room. The angular ranges (0 → 90°) and (90° → 180°), respectively, correspond to scattering by reflection and trans
mission.
curves, the deviation from concentric circles represents the roughness anisotropy.
B. Detection Limit of the Scatterometer For a surface of good quality, we measure in a partic
ular direction scattered flux of ~10 - 1 1 of the incident flux. Thus we must be vigilant concerning errors that could be introduced by the presence of parasitic light in the neighborhood of the specular beams one million times more intense than the scattered beams.
When there is no sample in the system, we measure a BRDF • cosθ of ~10 - 7 in the 3.3° ≤ θ ≤ 174° angular range (Fig. 3), correlated with the detectability threshold of the receptor. However, due to the parasitic flux in the neighborhood of the transmitted beam, this value slightly increases further than some 10~5 between 174 and 176.7°, which leads to a TIS of 3 × 10-6. If now a sample is put in the apparatus, this parasitic light will be found again by reflection and transmission in the neighborhood of the specular beams with intensities proportional, respectively, to the R and T reflection and transmission factors of the sample. For example, in the case of black glass, we have T:0 and R = 0.04, so that BRDF • cosθ becomes smaller than 10~6 in these angular ranges, a value that we have chosen as the origin of our curves. Integrated in the whole space, the parasitic light then leads to a TIS of 0.12 × 10-6. We are thus certain of measuring a scattering that originates from the roughness of the surfaces, even if they are supersmooth (TIS = 2 ppm = 2 × 10-6).
III. Application to Problems Posed by the Characterization of Micropolished Surfaces
The first difficulty to overcome when characterizing a surface concerns cleanliness conditions. The room where the measurements are taken is under slight overpressure and equipped with a blowing ceiling to eliminate dust particles in the air. Moreover, we must be sure that the scattering actually measured originates from surface roughness and not from residues of polishing or cleaning processes.
With these precautions, we can then study the problems of roughness uniformity and anisotropy together
with the problems inherent in our characterization method, the principle of which is recalled now:
With Bousquet's theory5 leading to an expression such as I(θ,φ) = a(θ,φ)y(θ,φ) for the intensity scattered in a direction (θ,φ) of the space, we have access to the roughness spectrum γ of the surface defects by dividing the measured value I(θ,φ) by the ideal coefficient a(θ,φ) given by theory. (This coefficient depends only on illumination and observation conditions.) The roughness spectrum γ is the Fourier transform of the autocorrelation function of the surface defects, and it contains all the information related to these defects. Roche and Pelletier4 have shown that the spectrum obtained in this way was invariant when the illumination and observation conditions varied, which shows that it is a characteristic specific to the surface.
A. Surface Cleaning Even if it does not lead to a surface degradation,
cleaning remains a critical problem. We use a mixture of pure alcohol and ethylic ether (no improvement has been noted by using collodion in solution), and the samples are systematically observed after cleaning with an interference contrast microscope (Nomarski). As seen in Tables I, II, and III, we have measured the
Table I. Total Integrated Scattering DE of Six Black Glasses Measured After Two Cleanings with an Interval of One Year Between Them: We
Observe no Variation of Scattering (1 ppm = 10 - 6)
Table II. Partially Integrated Scattering (for Different Angular Ranges) of One Sample G5N After Two Cleanings with an Interval of One Year Between Them: Angular Scattering Remains Practically the Same
Table III. Total Integrated Scattering DE of Five Black Glasses Measured After Two Cleanings with an Interval of One Year Between Them: There
scattering of several samples after two cleanings with one year between them.
Two cases can occur: either the surface is clean on the first attempt and we
find again the same scattering value regarding both the whole scattering (Table I) and the angular repartition (Table II);
or cleaning product residues or dust particles remain on the glass, easily observed with the interference contrast microscope. In this case, several cleanings are necessary and the polish can be damaged (Table III). The difficulty is with the rubbings electrically charging the surface, which tends to attract dust particles and hold the cleaning product residues.
We can also note that a good cleaning does not modify surface anisotropy: anisotropy curves recorded before and after cleaning are practically always identical. In particular, the direction of the last cleaning process (marked on the substrate) does not give rise to any scattering in a plane perpendicular to this direction. All these experiments show that the surface polish does generate the measured scattering.
B. Roughness Uniformity With our apparatus, the diameter of the illuminat
ing spot is 3.7 mm, so that we measure the scattering from a surface 2 of 11 mm2. Although such a characterization is sufficient in the case where the substrate will be coated to serve as a laser mirror, in many other applications we must study the quality of the polish on the whole surface of the sample.
For this purpose, the sample is mounted on two X-Y moving stages which permit carrying out the measurement of scattering from seventeen areas ∑ of the surface, uniformly distributed on a disk 25 mm in diameter. Table IV gives the complete front scattering of six black glasses of different roughnesses measured on two zones Σ 8 mm apart. The isoscattering curves recorded for these two zones (Fig. 4) are very similar in all cases: the small parallel scratches responsible for the directions of intense scattering are, when present, distributed on the whole surface. Only the amplitude of the scratches can vary from one region to another.
C. Roughness Anisotropy The anisotropy maps are characteristic of the sur
face, better than a whole scattering. Figure 5 shows examples of isotropic and anisotropic surfaces. Due to
Fig. 4. Level curves [see Fig. 2(c)] of scattering for two different zones ∑1 and ∑2 of the same sample.
Table IV. Total Integrated Scattering DE1 and DE
2 of Six Black Glasses Measured in Two Different Zones ∑ 1 and ∑ 2
Fig. 5. Examples for isotropic and nonisotropic surfaces.
roughness anisotropy, scattering can perceptibly vary from one scattering plane φ to another [Fig. 6(a)], which leads to the determination of a roughness spectrum for each scattering plane. However, it would be more convenient to characterize a surface with few
the mean roughness spectrum in the measurement range.
But it is also important not to be limited to a qualitative value of the anisotropy. For that, starting from the measurements BRDF cosθ (θ,φ), we calculate an angular autocorrelation function of the surface defects (Fig. 7), details of which will be given elsewhere. The main interest in using this function is that it allows defining an isotropy degree for the surface defects by considering the minimum of the curve; in this way, we obtain a quantitative value to display the roughness anisotropy.
D. Surface Index Determining the roughness by our optical method
implies precise knowledge of the superficial index of the substrate. However, it is well known that, according to the type of glass and polishing process used, the surfacic index can be different from the bulk index. We have found a variation of 0.2 nm for the roughness depending on whether we consider for the index the extreme values of 1.52 and 1.45 given in the literature.7
Fig. 6. Plane sections of the angular scattering curve: (a) angular scattering measured for different scattering planes φ; (b) mean plane section of the angular scattering curve over 250 measurement planes
IV. Homogeneity of the Surface Quality in a Same Glass Set
The homogeneity of the polish quality in a same set of glasses remains a basic problem.
For this purpose, we have asked several optical shops to supply us with five sets of black glasses (diameter φ = 35 mm, thickness e = 5 mm) polished on both faces to λ/4 and λ/2. For each sample, we measured the mean plane of the scattering curve and plotted the anisotropy curves. Each roughness spectrum has then been approximated with the Hankel transform γ of the sum Γ of an exponential and Gaussian function8:
Fig. 7. Angular autocorrelation functions F(a) of surface defects of two samples with their anisotropy curves. Curve F(a) with a high decrease (isotropy degree = 0.25) corresponds to the anisotropic surface, while the constant curve (isotropy degree = 1) corresponds to the isotropic surface. Each function F(a) is normalized to its zero
value F(0).
parameters. Thus, we prefer to give a mean plane section of the scattering curve determined on 250 measurement planes φ [Fig. 6(b)]; the sample is then characterized by analytical functions which allow fitting
with σ = (2π/λ) sinθ. We can obtain an accurate value (see Sec. V) of the
roughness δ and the autocorrelation length L of the surface defects by δ = (δe
2 + δg2 )1/2 and L ≅ Lg .
Let us recall that the exponential function characterizes low spatial frequency defects responsible for scattering at small angles (between 0 and 10° approximately), while the Gaussian one characterizes high spatial frequency defects responsible for scattering at larger angles. Some results are presented in the following figures and tables and allow testing the homogeneity of the polish quality in the same set of glasses:
For the first series of glasses (Fig. 8), we find an important inhomogeneity of the polish, since the global scattering ZE varies from 5.4 to 182 ppm. The roughness of some samples is given in Table V.
The second series (Fig. 9) is also very inhomogen-eous; all the scattering varies from 5.9 to 119 ppm. Seven samples have a global scattering of ~14 ppm with the following characteristics: (δe; Le) = (1 nm; 4500 nm) and (δg; Lg ) = (0.5 nm; 200 nm). Samples (c) and (d) have roughness spectra very different from
Fig. 8. Mean angular scattering curves of four black glasses from the first set of glasses with their anisotropy curves. Each experimental curve has been superimposed with a theoretical curve calculated with a roughness spectrum that is the Hankel transform of the sum of an exponential and Gaussian function. Agreement between calculated and measured curves is very good here. In the case of sample d, we could use a third analytic function for a better agree
ment for 0 > 65°.
those of classical polishes, and it is not possible to fit them with an exponential and Gaussian curve; their scattering is connected with defects of a very different nature, probably due to cleaning or polishing residues. In these cases, we find a quasi-isotropic scattering.
On the other hand, the polish homogeneity is very good for the third and fourth series of glasses (Fig. 10), since we have
11 ppm ≤ DE ≤ 18ppm for set 3,
12 ppm ≤ DE ≤ 25 ppm for set 4.
For fourteen samples of set 3, we find the following characteristics:
(δe; Le ) = (1 nm; 3000 nm) (δg; Lg ) = (0.5 nm; 200 nm).
For fifteen samples of set 4, we find (δe; Le ) = (1 nm; 4000 nm) (δg; Lg ) = (0.7 nm; 200 nm).
For these two series of glasses, the good polish homogeneity implies a small deviation of the sample parameters δ and L with respect to the given values ∆δ ≤ 0.2 nm and ΔL = 30 nm.
In all cases, homogeneity is better for low spatial frequency defects (small scattering angles).
Concerning the last set of glasses, it is quite inhomo-geneous and its mean polish is not very good: 21 ppm
Table V. Parameters for Gaussian and Exponential Functions that Characterize Defects of Four Samples from the First Set of Glasses
(samples a, b, c, and d of Fig. 8)
Fig. 9. Mean plane sections of angular scattering curves of four black glasses from the second set of glasses with their anisotropy
curves.
Fig. 10. Mean plane sections of angular scattering curves of different samples of the third set of glasses. The polish quality is very
homogeneous for this glass series.
≤ DE ≤ 76 ppm. As an example, we find for a sample (δe; Le ) = (1.7 nm; 2300 nm) and (δg; Lg ) = (0.7 nm; 200 nm).
From all these experimental results, it comes out that in a general way the high spatial frequency defects can be characterized with a Gaussian curve of autocorrelation length L ≅ 200 nm.
V. Domains of Application and Limits of Validity of Our Optical Characterization Method
Insofar as there are other characterization methods of optical surfaces and to allow a better comparison of the results with these different methods, it is important to specify the domain of application and the limits of validity of our optical method. Let us recall first that theory limits us to the characterization of surfaces, the roughness of which is low compared with the illuminating wavelength, and the defect slope is far from unity.
In addition, to allow a detailed analysis of the experimental results, we must know accurately the apparatus function of the scatterometer and specify to what extent knowing the roughness spectrum in the range of measurable spatial frequencies allows us to reach the autocorrelation function of the surface defects.
A. Apparatus Function of the Scatterometer Let h(r) and Γ(r) be the surface profile and its
autocorrelation function and h(σ) and γ(σ) their respective Fourier transform. Thus the roughness spectrum can be written as 7 (σ) = | h (σ) |2. We will separate here the influence of the solid angle of the receptor (which directly acts on the spectrum 7) from the other effects (which act on the function h).
Provided that the reflection coefficient of the surface remains constant for each plane wave of the incident flux, we can show that the differences between the theoretical assumption and experimental conditions can be taken into account by introducing the following convoluting operators:
where ∆σ represents the halfwidth of the corresponding functions.
Thus we can see that the function h is smooth on a maximal spectral width ∆σ of ~0.01 μm - 1 , which assigns a frequency or angular resolution limit (∆θ ≅ 0.05°) independently of the aperture of the receptor used. Moreover, it is essential to take into account the fact that we obtain, after inverse Fourier transform of the measured spectrum γm (σ) = |h(σ) * ƒ(σ)|2, the autocorrelation function Γm (τ) of the product hf. Thus the autocorrelation function Γm to which we have access characterizes the variations h(r) of the surface only in the range |τ| « (1/∆σ) = 100 μm.
Concerning the solid angle of the receptor (ΔΩ = 6.8 × 10 - 5 sr), it leads to spectrum averaging on an angular width ∆θ ≅ 0.5°. After inverse Fourier transform of the measured spectrum, we obtain the autocorrelation function
where g has a halfwidth ∆σ = 0.09 μm - 1 . The function g assigning a decrease, a measured
autocorrelation length L will be meaningful only if we have L « (4/0.09 μm - 1) ≅ 40 μm.
We will retain this condition which is the most important.
Let us specify here that the measured autocorrelation lengths practically never exceed 2 or 3 μm, which is satisfying. In the same way, we measure roughnesses δ always smaller than 2 or 3 nm, so that we have always (δ/λ) « 1.
B. Measurable Spatial Frequencies At quasinormal incidence (i = 1.5°), the receptor
explores by reflection the angular range 3.3° ≤ θ ≤ 90°, so that the range of measurable spatial frequencies is 0.31 μ m - 1 ≤ σ ≤ 9.67 μm - 1 .
We thus characterize gratings that have periods between 0.65 and 20 μm, and the roughness can be calculated as
with k = (2π/λ)(l - sin1.5°) = 9.67 μ m - 1 and ε = (2π/ λ)(sin3.3° - sin1.50) = 0.31 μm - 1 .
The measured spectrum γm being limited on both sides by these extreme values ε and k, the inverse Fourier transform gives access only to an autocorrelation function Γm smoothed in the following way:
with
where J is the Bessel function of order 1. The convolution by the function ƒ imposes the later
al resolution for Γm , since it masks the defects of spatial periods lower than the wavelength (light does not see these defects). This is because we measure only external scattering without taking account of evanescent waves.
Concerning the function g, its action stretches on a larger range but is negligible with respect to the preceding one, since the order of magnitude of g/f is (ε/k)2
≅ (θm — i)2 ≅ 1 × 10 - 3 , where θm is the minimum angle for scattering measurements. This shows the interest of succeeding in measuring scattering curves up to angles θm as small as possible.
C. Use of Analytic Functions We have seen how we can approximate the spectrum
γm in the measurement range with the Fourier transform γα of the sum Γα of an exponential and a Gaussian curve. The interest in this method is that it leads directly to the analytical expression Γα of the autocorrelation function, and the statistical parameters of the surface are then given by
and L ≅ Lg (taking account of the fact that in general Lg ≤ Le /10).
However, this implies that the function γa approximates the spectrum γm on a spatial frequency range going from 0 to infinity, which is not the case. Strictly speaking, the autocorrelation function of the defects characterizable with our method should be written as Γm = Γα * (ƒ - g). Although this remark is not always taken into account, this leads to an ambiguity in the determination of parameters δ and L of the surface due to the measurement accuracy; in some cases we have the choice between several analytical functions (Γ1 and Γ2, for example) to fit the spectrum in the measurement range. This ambiguity disappears if we take account of the preceding remark: Γ1 * (ƒ — g ) = Γ2 * (ƒ — g ) = Γm. However, we have noted that the roughness values obtained by
provided that we choose for Le the smaller acceptable value.
Further information is given later.
VI. Conclusion Measuring the spatial distribution of light scattered
in the whole space allows a detailed investigation of the complexity of micropolishes in modern optics. It gives access to the global and angular scattering, the roughness spectrum, the autocorrelation function (roughness and autocorrelation length), and the anisotropy and uniformity curves. Anisotropy can be quantitatively determined by defining an anisotropy degree starting from an angular autocorrelation function.
We also dealt with cleaning processes. Applied to several sets of glasses, our characterization method allowed us to show that the homogeneity of the polish quality in a same glass set remained a crucial problem.
Finally, a detailed study of the apparatus function of the scatterometer permits us to determine the fields of use and the limits of validity of our optical characterization method.
This paper is based on one presented at the Fourth Topical Meeting on Optical Interference Coatings, Tucson, 12-15 Apr. 1988. References 1. C. Grezes-Besset, C. Amra, B. Cousin, G. Otrio, E. Pelletier, et R.
Richier, "Etude de la diaphonie d'un systéme de démultiplexage par filtres interférentiels. Conséquences de la diffusion de la lumière par les irrégularités des surfaces optiques," Conf. présentee à TELEMAT 87, Marseille, 1-5 juin 1987, Ann. Télécommun. FRA 43, 105-000 (1988).
2. P. Roche, C. Amra, and E. Pelletier, "Measurement of Scattering Distribution for Characterization of the Roughness of Coated or Uncoated Substrates," Proc. Soc. Photo-Opt. Instrum. Eng. 652, 256-000 (1986).
3. J. M. Bennett, "Scattering and Surface Evaluation Techniques for the Optics of the Future," Opt. News 11, 17 (July 1985).
4. P. Roche and E. Pelletier, "Characterizations of Optical Surfaces by Measurement of Scattering Distribution," Appl. Opt. 23, 3561-3566 (1984).
5. P. Bousquet, F. Flory, and P. Roche, "Scattering from Multilayer Thin Films: Theory and Experiment," J. Opt. Soc. Am. 71, 1115-1123 (1981).
6. C. Amra, P. Roche, and E. Pelletier, "Interface Roughness Cross-Correlation Laws Deduced from Scattering Diagram Measurements on Optical Multilayers: Effect of the Material Grain Size," J. Opt. Soc. Am. B 4, 1087-1093 (1987).
7. E. Casparis-Hauser, K. H. Guenther, and K. Tiefenthaler, "Spectrophotometry and Ellipsometric Study of Leached Layers Formed on Optical Glass by a Diffusion Process," Proc. Soc. Photo-Opt. Instrum. Eng. 401, 211 (1983).
8. 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).
