Micron-sized structures on flat substrates supporting sub-micron defects are analyzed by means of a parameter based on integrated backscattering calculations. This analysis is performed for different particle and defect sizes, optical properties and for two different configurations (defect on the microstructure or on the substrate). Calculations in the far field are complemented by some near field results. It is shown that information about the defect (presence, size and optical properties) can be obtained from the proposed backscattering parameter.
© 2007 Optical Society of America
During the last decades, researchers on light scattering by surfaces have focused on the electromagnetic problem of particles on substrates. Their results have generated non-invasive light scattering techniques for particle sizing with applications in different fields[1, 2, 3]. In previous works[4, 5], the authors have extensively studied light scattering by particles on substrates from both numerical and experimental points of view.
Backscattering detection is a kind of far-field scattering configuration that has proved itself very sensitive to small variations in the geometry and/or optical properties of scattering systems with structures comparable to the incident wavelength[6, 7].
Backscattering patterns can be looked upon as the result of selecting a single value corresponding to the incident direction from each full-scan scattering pattern obtained for every angle of incidence. Though backscattered light could be the only possible measurement for a non-accessible sample, its advantages make of it a useful approach in other situations, as has been shown experimentally.
In a recent work, it was described how a Gaussian defect located on a cylinder changes the backscattered intensity and how an integration of the backscattering intensity over either the positive or negative quadrant (corresponding to the defect side or the opposite one, respectively) yields to a parameter which allows not only to deduce the existence of a defect but also to provide some information about its size and location on the cylinder. Such study was carried out for a homogeneous system, where substrate, cylinder and defect were supposed perfect conductors. The relative variations of the integrated backscattering intensity will be used here and will show its potential as a very sensitive tool in defect assessment. The objective of the present work is to go deeper into the analysis of the efficiency of the proposed method by considering a more realistic system: a defect, either metallic or dielectric, located near a cylindrical microstructure, either on it or beside it, lying on the flat substrate underneath (see Fig. 1). The defect is considered also cylindrical rather than Gaussian. The reasons for this choice will be discussed in the following section.
Of course, other geometries may be preferred for the practical interest or for a similarity with a particular problem, but we think that the possibilities involved in the proposed frame cover the generality of the problem.
This paper is organized as follows. Section 2 is devoted to describe the geometry and the numerical method proposed to solve the problem. Section 3 contains the main results and their corresponding discussion. Finally Section 4 summarizes the main conclusions of this research as well as a proposal for future work.
2. Scattering geometry and numerical method
The scattering geometry is similar to that described in a previous work, i.e, a cylindrical metallic microstructure of diameter D located on a flat conducting substrate, and supporting a much smaller defect. Its shape will be assumed cylindrical with diameter d. The main reason for this choice is that a cylindrical shape can be modeled easier than a Gaussian one, offering the possibility of using approximate solutions. Besides, calculations considering either cylindrical or Gaussian shapes lead to very similar results in terms of scattering patterns and relative scattering cross sections. Two situations will be analyzed: Configuration A) The defect is located on the cylinder and its position is given by the angle φ, which is considered always in the right side, φ>0, with no loss of generality. Configuration B) The defect is located on the substrate nearby the cylinder. In principle, we want to show the differences, if any, appearing between these two configurations in order to extract the most important conclusions leading to a possible distinction between them. This could give more insight in the solution of the inverse problem.
The scattered field in each medium is obtained by numerically solving the Maxwell’s integral equations using the Extinction Theorem formulation applied to the 2-D geometry for multiple connected domains. To perform the numerical calculation we need a discrete surface contour profile (substrate, cylinder and cylindrical defect). It is important to have a partition fine enough to assure a good resolution in the strong curvature regions of the surface (lower part of the cylinder and the defect). Furthermore, and due to obvious computing limitations, the surface has to be finite. In order to have a uniform illumination over the structure and to avoid edge effects, we will assume an incident Gaussian beam of suitable width 2w 0. It is assumed monochromatic of wavelength λ and linearly polarized perpendicular to the plane of incidence (S-polarized). The length of the substrate, L, has been fixed to 60λ and 2w 0=6λ. The results shown in the following section are referred to the backscattering direction (θs=-θi) for any value of θi, ranging from -80° to 80° (to avoid grazing incidences).
The near field plots shown later on for these configurations have been obtained using a finite element calculation software.
3.1. Configuration A. Defect on cylinder
Figure 2 shows the backscattered intensity pattern, Iback as a function of the incident angle θi, for a metallic cylinder of size D = 2λ considering two different types of defect materials: silver (ε=-17.5+0.5i for λ=0.6μm) and glass (ε=2.25) both sized d=0.15λ. For a better visualization of the patterns, glass defect cases have been shifted upwards by one order and only three defect positions have been considered. These positions produce the strongest variations of our backscattering integration, as will be shown later on.
For backscattering angles θs <0, that is, when the incident beam comes from the opposite side to that containing the defect, the backscattering pattern has nearly the same shape, in terms of intensity values and minima positions, regardless of the nature of the defect. Perhaps, the small differences are observed better when the defect is metallic. However, for θs̎0, a difference in the backscattered intensity can be noticed when comparing silver and glass results. If we observe a minimum in close detail (see zoomed regions), we see how Iback increases with respect to the cylinder without defect when this is made of silver. In the case of glass located at the same position, Iback decreases. These differences can be easily assessed by using the integrated backscattering parameter, σb, as defined by
This function integrates separately the backscattering values obtained at either the defect side (σb +) or the opposite side (σb -). One way of dealing with this magnitudes is defining an “incremental integrated backscattering” parameter, σbr, as defined by
where subscript 0 stands for the perfect cylinder configuration (no defect).
Figure 3 shows a comparison of σbr as a function of the angular position of the defect on the main cylinder for the silver and glass defect cases. It can be seen that the maximum value of σbr + has an approximate linear dependence with the defect size d. As an example, we have fitted its evolution for the case of D=2λ and metallic defect , finding the following approximate equation,
with a regression coefficient of 0.99 and d expressed in units of λ
For d∈ [0.05λ, 0.2λ] and cylinder size comparable to λ, it is found that, surprisingly, the position [σbr]max and [σbr]min is independent of the cylinder size. Another characteristic of the evolution of σbr + is the presence of a minimum or a maximum around φ = 90° for metallic and dielectric defects respectively. An analysis of the behaviour of this minimum, allows us to conclude that [σbr]min also follows a linear law with the defect size. However, the slope is no longer independent of the cylinder size.
The most interesting feature shown in this comparison is that, in all cases considered, σbr for a glass defect has an opposite behaviour to that observed for a silver defect. That is, for silver defect positions where σbr takes the maximum increase with respect to the perfect case, the dielectric defect finds the maximum decrease of the same parameter, and viceversa. This behaviour suggests a way to discriminate metallic from dielectric defects.
We shall focus now on the evolution of parameter σbr + with the optical properties of the defect and in particular for a dielectric defect around the regions where the oscillating behaviour of σbr reaches the maximum amplitude, that is, φ=50° and φ=90°. Figure 4, shows three curves of σbr +(50°) as a function of the dielectric constant, ε (ranging from 2.5 to 17), for three different defect sizes. For each defect size, σbr +(50°) starts negative and with negative slope; it reaches a minimum (-0.22) and then takes a positive slope to reach a maximum (aprox. 0.45), passing through a zero value. This means that, for each size, there is a value of ε high enough as to produce values of σbr +(50°) similar to those obtained for silver defects. The zero value would correspond to a situation where σbr +(50°) is not able to discriminate the original cylindrical microstructure for that holding the defect. When the former analysis is repeated for φ=90° a similar behaviour is found, though σbr +(90°) has an opposite sign, as expected after observation of Fig. 3. As an example, σbr +(90°) for a defect sized d=0.1λ is included in Fig 4. Similar calculations have been carried out for different values of D, ranging from λ to 2λ, leading to similar results (same σbr(ε) with zero values obtained for different values of ε). The region shadowed in Fig. 4, typically a glass defect, can be linearly fitted therefore providing with a direct estimation of the dielectric constant of the defect. As an example, σbr(50°) = -0.03∙ε + 0.02 for the case d=0.1λ.
Finally, we would like to point out that a dielectric substrate would provide with similar qualitative results as for the metallic case shown in this research. Dielectric substrates would modify the visibility factor of the lobed structure of the backscattering patterns, without modifying the general behaviour of σbr. In Fig. 5 we show a comparison of σbr for dielectric (ε=2.25) and metallic (silver for λ=0.6μm) substrates.
3.2. Configuration B. Defect on substrate
The small defect is now located on the substrate, close to the main cylinder (1 or 2 wavelengths) and both made of silver (ε=-17.5+0.5i for λ=0.6μm). Again, in Figure 6 we see that there is still a clear difference in the backscattering pattern obtained at each of the two quadrants, thus making possible to predict the side where the defect is located beside the main cylinder. Results plot in the figure correspond to a cylinder of D=2λ and two defect positions: x=1.2λ, 2λ, while the defect size is fixed to d=0.2λ. It is observed that the backscattering pattern is more affected in the side where the defect is. This behavior is quite similar to that observed in the case of a defect on the cylinder. However, if we look at the side opposite to the defect, the backscattering pattern changes due to the presence of the defect, if this is located outside the “shadow” cast by the big cylinder. When the defect is close to the cylinder (x=1.2λ), that is, inside the “shadowed” region, the scattering pattern of the left hand side does not change for any incident angle. Nevertheless, for a longer distance from the cylinder (x=2λ) the change in the left hand side can be noticed for incidences as high as θi=40°.
Figure 7 shows the evolution of σbr for two different cylinders, D=1λ,2λ and for three defects d=0.1λ, 0.15λ and 0.2λ. The shadowed area represents defect positions “under” the cylinder, not considered in the calculations. The smaller “shadow” produced in the D=λ case causes oscillations in σbr - for small values of x.
It is worth noticing that, for a given cylinder size, the presence and location of a defect like this, can be monitored to some extent. For a cylinder D=λ and x as far as 2λ, σbr - produces values as big as 10% increase in integrated backscattering for a defect of d=0.2λ. When the defect is closer than x=1.5λ, σbr + takes negative values while σbr - is not significant. Finally, when x<λ, σbr + is very sensitive and strongly tends to zero. In the case of a cylinder D=2λ, the most interesting feature is the combination of high absolute values and the strong oscillation of σbr + for x comprised within the interval [λ,2λ]. Here the absolute value of |σbr +| indicates the proximity of the defect, and the sign would show the location within such interval.
Although the size of the defect does not change this behaviour, it is interesting to notice that when comparing both cases, σbr + is sensitive to the defect size and also dependent on the size of the micron cylinder, something that did not happen in the former configuration (defect on cylinder). In fact, configurations A and B (see Fig. 1) can be looked as intrinsic different scattering problems: while in configuration B, there are two distinct scattering particles, in configuration A the defect is only slightly modifying the shape of the cylinder and consequently the overall scattering pattern. To illustrate this comparison, Figure 8 shows some examples of the near field and far field pattern produced by Configurations A and B for different defect positions and for normal incidence. Figure 8(a) represents the near field plot of the perfect cylinder and it will be used as a reference in all the following cases. Figures 8(b) and 8(c) correspond to the case of a metallic defect on the cylinder. It can be seen how the defect does not change significantly the shape of the near field when compared to the perfect cylinder case, whereas, if we look at Fig. 8(d), we observe a new spatial distribution around the micron-sized particle located in what initially was a maximum of the local field produced by the main cylinder. The same fact is found in the far-field plot, where this case also corresponds to the maximum change with respect to the non-defect case. Finally, Figure 8(e) corresponds to the case of a metallic defect on the substrate and close to the cylinder, very close to the position shown in Fig. 8(c) for configuration A. As expected, both cases are almost undistinguishable, and give the same near field pattern.
Again, if we repeat the calculations for configuration B, but this time considering a dielectric substrate, σbr - presents the same flat response, whereas σbr + is slightly more sensitive for the case of dielectric substrate, presenting a similar pattern but higher values of σbr (see Fig. 9).
In this work a system structured at two different levels (a micron cylinder lying on a flat substrate and contaminated by another sized only some tens of nanometres), has been analyzed by studying its scattering properties in the backscattering direction.
A parameter σb ± defined as the integration of the backscattered intensity over a given quadrant allows the evaluation of overall backscattering variations, for instance, by measuring the relative variation with respect to a starting backscattering object, σbr ±, as has been done in this work. Other useful magnitudes might be: “asymmetrical integrated backscattering”, |σb +-σb -|, or the adimensional “backscattering asymmetry index”, 2∙|σb + - σb -|/(σb + - σb -), presumably suitable for experimental situations. Parameter σb itself can be experimentally obtained through different widely used set-ups.
The possibilities of a parameter like σbr ± depend very much on the scattering problem and on the previous knowledge we may have about it. When applied to our double cylinder case, it has been shown that: i) in configuration A (contaminant on cylinder), parameter σbr ± depends on the defect size, for a given defect position, while in configuration B (contaminant on substrate), σbr ± is also dependent on the main particle size, D, for each defect position; ii) in configuration A the defect nature (metal/dielectric) may be identified by checking the sign of σbr +, for a given defect position, iii) A remarkable increase of σbr - is characteristic of configuration B, not being found in any case of configuration A, iv) the strong oscillations in σbr + observed in configuration B for small values of x, identify very precisely the position of the defect, provided we know that such configuration is the actual geometry of the system, v) The fact of using metallic or dielectric substrate does not affect significantly the behaviour of σbr and therefore it does not present a limitation for this parameter.
This is not the first time that changes in the backscattering pattern obtained form a basic configuration allows the recognition of some geometrical or material changes produced in the scattering object, but we think the procedure shown here is applicable to a wide range of 1-D structures (whose defects often require fast inspection).
Of course the natural extension of these results to 3D geometries would enlarge the range of practical situations and the scope of this work, but it would also require more powerful computing techniques and tools applied over an extensive casuistic. This research can constitute the basis for analyzing more complex geometrical systems.
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