## Abstract

We present a white light scatterometer operating at a unique scattering direction. Mechanical motions and wavelength scans are removed. The technique provides an immediate flexible characterization of roughness with no loss of resolution.

© 2018 Optical Society of America

## 1. Introduction

Light scattering has been widely used for decades to analyze the microroughness of optical surfaces and coatings [1–16]. These techniques have today reached a high degree of complexity, and one can find facilities which provide one angular BRDF pattern per surface pixel, incident wavelength and illumination angle, polarization state [17–23].

However as often when complexity is satisfied, there comes a demand for a gain of simplicity, and this is strongly connected with the range of applications (textiles, paints, cosmetics, living tissues, precision optics and microelectronics …). Actually numerous users would take great benefits to reduce cost, acquisition time and volume data in scattering facilities, which requires to minimize mechanical motions and wavelength scans, and work with single parameters. Furthermore flexibility of the scattering systems should be guaranteed for a number of applications where the samples cannot be displaced, positioned or manipulated.

Within this framework, we here present an opportunity to think scatterometers in an alternative way. Indeed angular mechanical movements can first be replaced by a wavelength scan at a unique scattering direction, with no loss of optical resolution. Furthermore the wavelength scan can be replaced by a white light illumination on the sample. The result is an instantaneous white-light scatterometer (W.L.S.) which collects a unique data at one single direction. We show how the introduction of specific optical filters makes this data proportional to the root mean-square of roughness, hence providing immediate on-line wafer characterization. In a last step additional filters can be used to retrieve the complete spectrum of roughness. The procedure is valid for optical surfaces, that is, quasi-specular surfaces which satisfy the small roughness approximation. Advantages and limitations of the characterization technique are discussed.

## 2. Former TIS scatterometry and the resolution problem

Former techniques are based on the well-known Total Integrated Scattering formula (T.I.S.) which gives the total losses as a function of a roughness-to-wavelength ratio, that is:

_{R}the total scattering losses integrated in the whole space of reflection, R the ideal reflection coefficient of the smooth sample, n

_{0}and i

_{0}the real refractive index and the illumination angle in the incident medium, λ the working wavelength in vacuum, and δ the surface quadratic roughness (or root mean square) given as:with z = h(

**r**) = h(x,y) the surface sample profile under study over the illumination area Σ. Formula (1) is valid for low roughness-to-wavelength ratios (δ/λ << 1), and excludes the specular beam. Though such formula is approximate and global, its related techniques based on integrated spheres (Fig. 1) have provided great help in surface characterization for years. Indeed such techniques deliver immediate data and a unique roughness value, which reduces both time acquisition and complexity in the scattering analysis. Notice that a similar formula yields for transmission, that is:

However surface analysis based on these scalar Eqs. (1) and (3) suffers from a number of weaknesses that can no more be accepted in high-precision optics. Indeed angular scattering is not predicted with these formulaes, whereas spatial frequencies carry major information on surfaces and films microstructure, and drive the energy balance at specific directions of optical systems. Furthermore surface anisotropy cannot be analyzed with (1-3).

Above all the bandwidth (BP) of spatial frequencies is a key point since roughness has no sense unless the optical resolution is specified [24–30], which applies to any physical quantity to be measured. This is recalled in Fig. 2 where a surface profile z = h(x,y) is registered with a sampling step Δx = Δy. The resulting roughness is given as:

with N the number of data points over the scan length L = N Δx. Equation (4) recalls that roughness is step dependent and its value varies with the measurements techniques, depending on how these techniques spread from the microscopic (AFM, PSTM, STM) to the macroscopic (optical microscopy, scattering, profilometry) ranges, as illustrated in Figs. 2(a)-2(b).

Such resolution problem is most often discussed in the Fourier plane. Indeed in Fig. 3 the minimum (inverse scan length 1/L) and maximum (inverse step 1/Δx) frequencies are clearly identified and the square of roughness is given by the integration of the power spectrum γ (or roughness spectrum) in this bandwidth, as:

**ν**= (ν

_{x}, ν

_{y}) = ν (cosϕ, sinϕ) is the 2D spatial frequency, ν = |

**ν**| its modulus in the range [1/L, 1/Δx], and ĥ the Fourier transform of the h surface profile. This point will be further developed in the next section devoted to angle resolved scatterometers.

## 3. Single-wavelength angle-resolved scatterometers

Since the roughness is bandwidth dependent, it is first necessary to identify which bandwidth is required for the working application (mechanical adhesion or hardness, hydrophobic properties, microscopic growth, optical losses…). In optics when the energy balance is concerned at one wavelength λ_{0} (the case of gyrolasers for instance), single wavelength angle-resolved scatterometers (ARS) give an ideal solution. Indeed these ARS set-ups record the angular scattering pattern and allow extraction of the roughness spectrum in the adequate bandwidth BP(λ_{0}), that is, the bandwidth of the radiative waves which carry energy in the far field, hence disregarding the range of evanescent waves that do not intervene in the energy balance of transparent media. At normal illumination (i = 0°) this one-wavelength bandpass is given as:

_{min}to a maximum angle 90°. Oblique illumination allows to extend the bandwidth up to a half-wavelength but raises more complexity in the experiment. Notice that the minimum frequency plays the role of the inverse scan length (1/L) of the previous section, in the sense that the larger period which can be analyzed is of the order of λ

_{0}/sinθ

_{min}. To reduce this minimum angle the incident beam should be well collimated and parasitic light should be cancelled at best.

The formula to extract the roughness spectrum of an isotropic surface illuminated at normal illumination with non-polarized light is given by [5,31]:

where the subscript (-) and ( + ) are for angular scattering by reflection and transmission respectively, C^{±}an optical factor which does not depend on the topography, and I

^{±}the measured angular intensity, that is, the scattered flux per unit of solid angle normalized to the incident flux. This last quantity is often denoted as BRDF.cosθ (for I

^{-}) or BTDF.cosθ (for I

^{+}), where BRDF is analogous to a luminance.

Equation (9) is given by first-order electromagnetic theory [5,31] under the assumption of a low roughness-to-wavelength ratio. The optical factors are calculated at normal illuminaton (i = 0°) as:

_{S}and q

_{P}polarization factors:

_{0}and θ

_{s}the scattering directions in the superstrate (usually air) and the substrate, respectively.

When the spectrum has been extracted following (9), the roughness is given as previously (see Eq. (5)) by an integration over the bandpass (8), that is:

It is also often useful to know that the optical factors given in (10)-(11) show slight variations with direction, so that a further approximation C(θ) ≈C(0°) allows to turn them in a more simple formula as:

To be complete, final integration of these angular patterns over the normal angle θ allows to retrieve the global TIS Eqs. (1) and (3) of the previous section, under the condition cosθ ≈1 in the angular range [32].

At this step we keep in mind that the spectral dispersion of roughness δ(λ) is related to the integration bandpass given in Eq. (8) and depends on the spectrum shape in this bandpass. Since the bandpass is wavelength dependent, so is the roughness (see Fig. 4). This is a real difficulty when the energy balance must be analyzed at several wavelengths or in a broad-band region, and this has been the motivation to develop multi-wavelength scatterometers [22,23,25] so as to broaden the spatial frequency range. Some attempts have been made to give a general overview from the mid-infrared roughness (similar to planeity or waviness) to the UV range. Others have extracted the spectra over several decades of spatial frequencies [24–29], allowing the connection between optics (ARS) and near field microscopy (AFM), which is possible under the condition Δx ≈λ. To go further in the bandwidth extension (up to X rays for instance), fractal assumptions were also used to turn the spectrum into a power of inverse spatial frequency, hence allowing analytical calculation. Since total scattering losses can also be predicted by the roughness-to-wavelength ratio, these assumptions allowed to show that a ν^{−2} behavior would lead to an achromatic roughness (with the result that losses increase at short wavelengths), whereas a ν^{−4} behavior leads to an achromatic scattering [29] (same scattering from the UV to the X rays).

## 4. Motionless (one-angle) scatterometer

Multiwavelength scatterometers most often cover the range of applications of optical coatings. Here we consider a wavelength range around the “extended visible”, that is, from 250nm to 1.7μm, in adequation with low-cost receivers. Notice that much narrower ranges are also commonly used, and still present key interest in the analysis of optical coatings whose spectral variations are known to be highly contrasting.

The first idea to reduce the complexity is to give up the angular mechanical motion classically used to scan the spatial frequencies. Indeed owing to the property ν = n_{i}sinθ_{i}/λ given in Eq. (14), one can replace the angular scan [θ_{min}, 90°] at one wavelength λ_{0} by a wavelength scan [λ_{1},λ_{2}] at one direction θ_{0}. The resulting bandpass would be at normal illumination:

_{1}< λ

_{2}, and this one-angle bandpass BP(θ

_{0}) must be compared to the previous one-wavelength bandpass BP(λ

_{0}) given in (8). Notice here that no added complexity is brought due to the wavelength range, since the dispersion law of refractive indices is well known for substrates in a moderate band.

Let us first compare the dynamics of the two bandpasses given in (8) and (18), that is, the ratio of the maximum to the minimum spatial frequencies. The condition for identical dynamics is straightforward and given by:

For a minimum scattering angle around 5°, this wavelength ratio is around 11, which is prohibitive. Indeed this would require a broad-band source and a broadband detector (or several detectors). However with a minimum scattering angle around 10°, the wavelength ratio is around 6 and this becomes acceptable, for instance with a source in the range (300nm- 1.8μm). Notice here that optimal single-wavelength scatterometers may allow a minimum angle of 0.1°, provided that parasitic scattering is removed with high accuracy. This would give a wavelength ratio close to 500 that cannot be reached with this technique. However as stated in the introduction, a large number of applications do not require to consider such low spatial frequencies; furthermore these low scattering angles cannot be reached with samples that cannot be positioned in a scatterometer.

Let us now consider the optical resolution given by the inverse of the maximum spatial frequency. Following again relations (8) and (18), resolution is not modified under the condition:

Hence in order to hold the resolution of a one-wavelength scatterometer (1WS) operating at λ_{0} = 633nm, one would use a one-angle scatterometer (1AS) working at θ_{0} ≈30° with a minimum wavelength λ_{1} ≈300nm. This minimum wavelength increases to λ_{1} ≈400nm with a working angle θ_{0} ≈40°, which makes no difficulty. This means that one can easily adjust the θ_{0} scattering angle to retrieve the resolution of a classical 633nm 1WS scatterometer.

The last and key comparison concerns the absolute bandpass, now including both dynamics and resolution. Results are given in Figs. 5(a)-5(c) where the bandpass BP(θ_{0}) of the one-angle scatterometer (1AS) is plotted versus its working scattering angle θ_{0}, and compared to the classical (1WS) one-wavelength bandpass BP(λ_{0}) given on the left scale at λ_{0} = 633nm with θ_{min} = 5°. In this Fig. 5, the one-angle scatterometer operates with different wavelength ranges [λ_{1},λ_{2}]. Figure 5(a) (top left) is for the visible range [400nm, 800nm]. The comparison with 1WS shows moderate success since a high working scattering angle (around 75°) would be required to obtain similar widths in the bandpasses, all the more than these bandpasses would be differently positioned. Figure 5(b) (top right) is given for the range [350nm, 1μm] and shows similar bandwidths at θ_{0} ≈60°, which is much better. The positioning is different but greater bandwidths are reached at higher angles. Notice however that an exact positioning is not necessarily a constraint since the classical 1WS may involve different wavelengths (λ = 633nm is not a reference). The last Fig. 5(c) (bottom-centered) is plotted for the range [300nm, 2μm] and shows much better performances than the classical 1WS, and this remains true in a large range of working scattering angles θ_{0}.

To be complete, another comparison of bandpasses is given in Fig. 6 and now takes into account the fact that the working wavelength of the classical scatterometer (1WS) can also be tuned. The right figure is given for the 1WS and shows the BP variation with wavelength, while the angular range is fixed to [5°;85°]. In this right figure the colors are given to identify the working wavelengths, so that BP(633nm) appears in red. The left figure is given for comparison and concerns the 1AS scatterometer. One can see the BP variation versus the working angle, and the colors again allow to identify the wavelengths. Such Fig. 6 is helpful to adjust the bandpass depending on the application and on the scatterometer (1AS or 1WS).

At this step one can say that a one-angle (motionless) scatterometer (1AS) operating in a wide wavelength range offers performances which could replace those of classical (1WS) scatterometers operating at one wavelength. Such conclusion is valid for low-roughness surfaces, and depends on the application and on the working bandpass. One should also notice that roughness is most often quasi-stationary over overlapping bandpasses, so that a perfect bandpass positioning is not necessary. Furthermore for non-optical applications, although the bandpass must be known, its position and width do not have to be necessarily predetermined as we did.

Figure 7 is given to conclude this section with an experimental validation of those principles. For that we used a facility (SALSA) that we recently developed [18,21,22] to measure the angular patterns of light scattered from optical coatings in a continuum visible range. The source is a supercontinuum laser operating in the spectral range [λ_{1} = 415 nm, λ_{2} = 2µm] and the detection is performed with a CCD array efficient in the range [400 nm-1000 nm], so the available spectral range is [415nm;1000nm]. The incident power is unpolarized and illuminates the sample at quasi-normal incidence (i_{0} ≈0°). The scattered light can be measured at all wavelengths in the angular range [θ_{min} = 1°; 90°], with an angular step Δθ = 0.1° provided by a mechanical motion. The wavelength step is Δλ = 2 nm, and this spectral scan is controlled by a laserline tunable filter which is a volume filter that guarantees a spectral purity δλ ≈1 nm around each wavelength λ_{i}. Thanks to this facility we were able to extract a detailed 2D intensity mapping I(θ,λ) of the scattered light, and so to mimic both cases of the one-wavelength scatterometer (1WS) and the one-angle scaterrometer (1AS) here discussed. Figure 7 is relative to a polished black glass sample. In this figure we extracted the roughness spectrum γ(θ,λ) from the I(θ,λ) mapping, on the basis of Eq. (9). In a first step we selected a working angle θ_{0} and plotted the curve γ(θ_{0},λ) in the wavelength range, which characterizes the one-angle scatterometer procedure (1AS). In a second step we selected a working wavelength λ_{0} and plotted the curve γ(θ,λ_{0}) in the angular range, which characterizes the one-wavelength scatterometer procedure (1WS). However since we had all data at diposal with the SALSA facility, we iterate the procedure for all wavelengths λ_{0} and angles θ_{0}.

Results are plotted in Fig. 7 with 516 scattering data. For these measurements the wavelength step was 50nm in the range [400nm;1000nm] and the angular step was 2° in the range [2°;88°]. As predicted, similar spectra data are obtained at the same frequencies *ν(λ,θ)*, whatever the angles or wavelengths that allow to reach this frequency. Hence all data emphasize an accurate overlapping of the spectrum at the intersection of bandwidths, which is a clear validation.

## 5. White light scatterometer

One can go further in the gain of simplicity, and this consists in removing the wavelength scan. For that the sample is now illuminated with white light (that is, with a broad-band source), whereas (white) scattering is still measured at one single direction θ_{0}. This technique provides an immediate unique data integrated over all wavelengths, and we will see below under which conditions this data can be proportional to the roughness square. Notice here that we give up the Fourier spectrum and limit the surface characterization to the root mean square of roughness. This is not a weakness since most users qualify the surface with a roughness data, even if this roughness data results from a spectrum integration. Furthermore the next section (6) will address how to recover the spectrum with white light.

Let us consider the angular scattering by reflection in the incidence plane (ϕ = 0°), and write again this in-plane scattering as:

This relation is given at normal illumination for unpolarized light. Now assume that the incident light is a broad-band source delivering a power spectral density F(λ) = dP/dλ. Scattering measured at one direction θ_{0} will deliver a voltage V(θ_{0}) proportional to a wavelength integration:

At this step one would like to transform the integral form (22) into another one similar to the roughness integral. Following (5) we know that roughness is already given for an isotropic surface (γ* = γ) as a spectrum integral:

Now introducing the modulus of spatial frequency for a change of variables, we obtain:

Comparison of (24) and (22) shows that the average signal V(θ_{0}) will be proportional to the roughness square provided that the following condition is satisfied:

In order to satisfy such condition one can introduce an optical interferential filter in the system (Fig. 8). This filter can be placed either on the incident beam or on the scattered beam, which leads us to replace in (25) the product F(λ)K(λ) by F(λ)K(λ)T(λ), with T(λ) the transmission factor. We obtain:

Then relations (22), (24) and (25) yield:

^{2}, as required. Notice here that we wrote the roughness as δ

^{2}(θ

_{0}) so as to remind that the bandpass is θ

_{0}dependent as predicted by (18). The calibration procedure is discussed further.

Let us now discuss the filter condition (27) which is the key point to reach this result. In fact this constraint can be rather easy to satisfy with thin film multilayers whose design techniques are known to be highly advanced [33], provided that the power spectral density of the source F_{λ}, and the spectral response of the receiver K_{λ}, are beforehand characterized. Examples are given further in this section. However to go further in the analytic investigation, one can first use the simplified expression of the C factor given in Eqs. (11) and (10), that is:

The result is a transmission filter now proportional to:

This relation emphasizes the fact that the proportionality factor of the filter varies with the measurement direction and solid angle. However such dependence can be easily be controlled with the calibration procedure (see further in the text), since these parameters do not vary with wavelength. We also notice that the reflection factor R was assumed to be constant in the wavelength range. Indeed it is common to neglect the dispersion law of substrate refractive index in moderate bands. Otherwise this reflection factor should also be introduced in (30) in the form F(λ) K(λ) R(λ). Eventually one could also keep the exact formulation of the scattering coefficient (C) throughout the whole procedure.

At this step on the basis of (30) one can use thin film softwares to design an optical filter from the knowledge of the source and the receiver spectra, and use these data to fit a curve proportional to λ /[F(λ) K(λ)]. For that the proportionality factor β must respect:

Hence this β factor is arbitrary but bounded. The maximum β value would lead to 100% transmission at one wavelength, but this is optional.

Figure 9 is given to emphasize the feasibility of the filter. For that we used the power spectral density of the super continuum source (NKT Photonics Power EXB-6) and the spectral response of a CCD sensor. The ratio λ /[F(λ) K(λ)] is plotted in blue versus wavelength in the range (400nm- 1000nm). This function was then approximated with a multidielectric stack in the range (450nm- 900nm). The stack involves 73 layers of Ta_{2}O_{5} and SiO_{2} thin film materials. As it can be seen in Fig. 9, the agreement is excellent since the two curves are superimposed in the range (450nm- 900nm). Additional layers would also allow to extend the range of agreement, when necessary. However one should take care of not using a range where the filter transmission is too low, in order to hold a high performance system detectivity. Eventually an edge-pass filter must be used to block the wavelengths shorter than λ_{min} and greater than λ_{max}.

Let us now come back to (26) and consider a calibration procedure. The proportionality to roughness would allow an instantaneously characterization of samples with an extreme simplicity. However the proportionality factor ${\eta}_{0}({\theta}_{0})$must be known. One solution may consist in using a surface etalon whose roughness δ_{E} is known by other techniques. If we denote by *V _{E}* the white scattering signal of this etalon sample, the sample roughness can be directly extracted as:

However difficulties arise from the accuracy of the etalon roughness measured by other techniques, all the more than roughness is bandwidth dependent. To overcome these difficulties another calibration technique must be used, which involves white scattering measurements on a standard lambertian sample. Indeed the lambertian signal V_{L} will be given as:

Now using (31) we obtain:

This last Eq. (36) gives the proportionality factor η_{0}. Hence it allows to determine the sample roughness from (28), so that the calibration procedure is completed.

To conclude this section we notice that it would also be practical to give more flexibility to the key transmission filter T(λ), in case where the spectral density of the source (or detector) would shift versus time. Actually another solution can be found in adjustable filters that can be built with micro-mirror matrices and gratings [34], that we introduce in the next section.

## 6. Retrieving the spectrum complexity

Now that white light scattering (WLS) at one angle has been shown to be proportional to the root mean square of roughness, one can wonder whether it is still possible to recover the full spectrum of roughness. Actually this task is possible and should be proceeded as follows. Indeed the roughness given in (23) can be seen as a zero-order moment of the 2D isotropic spectrum, which leads us to investigate the quantities:

In terms of wavelengths this k^{th} moment can be rewritten as:

Therefore these moments δ_{k}^{2} can be measured with the same technique described until now, provided that a series of filters T_{k}(λ) are at disposal. One can check that these filters would satisfy:

_{k}a proportionality factor:

Under these conditions we obtain:

_{0}with the T

_{k}filter.

To go further the simplified factor given in (16) can be used again, which modifies (39-40) as:

As previously the calibration procedure invoves a lambertian sample and allows the determination of the proportionality factor in (41), that is:

_{Lk}the voltage recorded for the lambertian sample at direction θ

_{0}with the T

_{k}filter.

The last step consists in retrieving the spectrum from these measurements. For that we first write the radial autocorrelation function as the inverse Fourier transform of the 2D isotropic spectrum:

_{0}the zero-order Bessel function:

Then we develop the exponential term of the Bessel function in a series:

To conclude, relation (48) allows to build the autocorrelation function Γ(τ) from the measured even moments δ_{2k}^{2}, so that the spectrum is known from the inverse Fourier transform of Γ. Actually here the major difficulty comes from the series of transmission filters T_{k}(λ), and from the number of filters that are necessary to build the Γ function. For that reason one must replace the thin film solution by another solution based on micro-mirror matrices [34] to produce such adjustable filters, as discussed before.

Notice that in a way similar to the zero^{th} moment (the roughness), these filters must respect the condition:

Eventually numerical calculation shows that a moderate number of terms can be enough in the series given in (48), which reduces the number of filters.

## 7. Conclusion

We have introduced alternative solutions to build original scatterometers able to deliver instantaneous roughness data. These solutions involve no angular mechanical motion nor wavelength scan. Indeed the angular scan was first replaced by a wavelength scan at one scattering direction, and then the wavelength scan was replaced by a white light illumination at one incidence angle. Based on these two principles, the one-angle white light scatterometer (WLS) delivers immediate roughness data with no loss of optical resolution. The bandpass of spatial frequencies can be adjusted with the choice of the unique scattering direction, and the system performance depends on the broad-band source. The key point of the system consists in the introduction of a specific interferential filter which follows a function of the spectral properties of the source and receiver. We have shown that such filter can be designed and produced by classical thin film techniques. To summarize, everything happens as if white light replaced an integrated sphere.

To go further we also showed how the frequency spectrum of roughness could also be retrieved. For that a series of filters is required, and these filters can be produced with another technique base on micromatrix mirrors. Each filter allows to measure a k^{th} moment of the spectrum, so that the autocorrelation function of the topography can be rebuilt. To summarize, everything happens as if the filters replaced a spectrophotometer.

In a general way these white light techniques would provide great help to analyze samples which cannot be manipulated or introduced in a scatterometer (large samples, samples which cannot be separated from a system…). Notice that since a unique direction is involved at both the entrance and the ouput of the WLS system, optical fibers can be used to carry and collect light, which provides an extreme flexibility to analyze samples. Also, the roughness anisotropy could be analyzed via a sample rotation around its normal.

Compared to the best classical single wavelength scatterometers, the performance of the white light technique is lower because of a narrower bandpass. However there is a large number of applications where roughness has only to be known in a specific bandwidth where low spatial frequencies can be ignored. Moreover, many users might be interested in a system with lower performances but at a lower cost and which would fit their band-pass requirements.

Eventually, one must keep in mind that the technique works within the framework of first-order theories, which mainly concerns polished surfaces. However depending on the required accuracy, the model can be extended to roughness-to-wavelength ratios of the order of λ/20, which allows the WLS technique to address fields outside precision optics.

## Funding

SATT Aix Marseille Université

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