Experimental Study of Polarization Properties of Rough Surface

Singularities in behaviour of ellipsometric angles via functions of incidence angle of light were revealed experimentally under investigating four types of the samples manufactured on two materials: on dielectric-quartz and on metal-aluminium. Surface roughness was simulated by creation of artificial relief using a two-dimension orthogonal grating (random phase mask). Quadratic “defects” in the line were subjected to stochastic law of d istribution. The “defects” were produced by etching in depth up to 1 m and their sizes were equal to 25x25 m and 2.5x2.5 m on each of materials . The impact of sizes of art ificial “defects” and their density upon polarization of reflected light was investigated by the multip le-angles-of-incidence ellipsometric measurements at wavelength 0.63m. For the first time, a random phase mask was used for simulat ion of rough surface.


Introduction
The numerous attempts have been undertaken to find correlation between measurements of light scattering and the characteristics of surface roughness. Another but considerably more difficult approach is to solve the inverse scattering problem. A number of different empirical models of surface roughness have been used to characterize surfaces including sine grating, triangle g rating (echelette), and rectangle grating [1][2][3][4][5][6]. Almost all above models converge to effective mediu m model (EMA). In this paper, we use, for the first time, the random phase mask model. The random phase (RPM) is a two-dimension orthogonal grating with the stochastic law 1/2n+1 d istribution of square "defects" with sizes a×a. Our calculations of the polarizing characteristics of the RPM, and d iscuss the influence of each of parameters of defects on polarizing angles had been described by author [7]. The analysis was carried out for mu ltip le-angles-of-incidence (MAI) ellipsometric measurements. It is needed to reminder that ellipsometric angles are connected with well known Fresnel reflection coefficients for p-and s-polarized light by the following relationship: Rp/ Rs= tan exp (i), because Rp and Rs are co mplex values. Interest to similar experimental works had been caused by big quantity of investigations connected with plas mon effects [8][9][10][11].
Relat io nsh ips ob tain ed aut ho r [12][13] d iffer fro m formulae which were got by Hein z Raether [14] based on another theoretical prerequisites, although they look like as the similar. Two purposes of this report are to establish correlat ion relationship between a priory known parameters of a relief and ellipsometric angles via functions of incidence angle of light, i.e. ( 0 ) and ( 0 ) and to exp lain observable experimental singularities of ( 0 ) and ( 0 ), on the basis of the analytical formu lae for RPM model using results obtained earlier in [7,15] by summat ion of all part ial reflected and dissipated waves. Artificial roughness of surfaces was created by the etching through a two-dimension orthogonal grating with the stochastic law 1/2 n+1 distribution of square "defects" with a size a×а. The high "defects" in horizontal lines had been organized into domains of various configurations [13]. The total areas of high and low do mains are equal.

Experimental Techniques
In generally, reflected field fro m RPM is represented by superposition of the pictures which are turning out as a result of Fraunhofer diffract ion on square aperture with most probable size a [16]. All measurements were carried out at wavelength of light 0.63 m under incidence angles Changeover fro m one size of RPM cell to another (fro m 2525 m to 2.52.5 m) is equivalent to the changing in ~100 t ime of "defects" density on an investigated surface. Changeover fro m one material to another is equivalent to the changing at least in 20 t imes of intensity of a diffuse scattered light because the reflectiv ity of quartz is about ~4%, and of alu min iu m is ~ 90 % (at normal angle of incidence of the light and under the only one reflection).
Theoretical analysis clearly revealed, that intensity of the dispersed beams is insignificant also its contribution in mirror -co mponent is poorly appreciable, except for special situations, for instance, near at Brewster's angle where reflection coefficient for p-polarized light is min imal (for metal) or equal to zero (for dielectric -quartz) or when the minimu m of an interference of light reflected by the top and bottom sides is observed. Conditions of an interference are satisfied when  1the path-length difference of the beams reflected by the top and bottom sides, is multiple n . These conditions are valid at incidence angle  interferen .

Experimental Results
Samples have been situated on ellipsometer table so that a line of "defects" coincided with the plane of incidence of the light, and corresponded to theoretical analysis of model. It was found that depth of artificial relief dramat ically impact upon characteristics of reflected light.

Small Depth of Relief
Until depth of art ificial relief is small (d20110 n m) and mutual suppression of rays of light reflected by top-bottom sides does not occur, ( 0 ) and ( 0 )polarization characteristics of light reflected by surface with artificial "defects" slightly differ fro m the same characteristics of smooth surface, as shown in Figure 2. The density of "defects" essentially enhances these differences provided that relief depths fit in almost the same limits. For example, in case of Al, both minimal value of  at Brewster angle and value of p rincipal angle at =/ 2 are changed in bounds of a few degrees (Fig. 3).

Relief Depth Closed to /4
As soon as depth of artificial relief becomes sufficient for suppression of rays of light reflected by top-bottom sides, singularities in ( 0 ) and ( 0 ) -polarization characteristics are arose (Fig.4).
In case of d ielectric, the changing in  is larger than in  due to low reflectiv ity of quart z so graph used coordinates 0 and 0. Contrary, in case of metal the amp litude changing is larger than the changing in phase because reflectivity of Al is great; and coordinates 0 and 0 is used for graph.

Impact of "Defect" Density
The increasing of "defect" density results in proportional changes of both ( 0 ) and ( 0 ) as shown in Figure 5. Besides, in case of Al, peak widths in ( 0 ) and ( 0 ) are greater than in the event of fused quartz though sizes of "defect" are the same. The last experimental fact is result of great reflectivity of metal A l.

Large Depth of Relief
Further increasing of depth of relief leads to enhancing of shadowing effect and decreasing of intensity light dispersed in zero order (or mirror d irection), i.e. to the weaken ing of its contribution in photo-detector window. The shadowing becomes so large that contribution of light scattering by lateral and bottom sides takes negligib le small value and amp litude changing (in ellipsometric angle ) is almost no different in co mpare with smooth surface. While phase changing (in ellipsometric angle ) is so noticeable that they do not look like on all previous dependences of (0) as shown in Figure 6 (upper) and the largest changing of phase angle  is observed at small angles of incidence of light. Besides, as seen fro m the same Figure, the Brewster's angle brings noticeable distortion in polarization characteristics of light reflected fro m fused quartz because p-component becomes to zero. Situation is abruptly changed for analogical defects etched in alu min iu m: the changing in  and in increments of  are the same as shown in Figure 6 (lower). Influence of the Brewster's angle for metal also is apparent; the pseudo-Brewster's angle for alu miniu m is appro ximately equal to 81 and there are singularities in  -the amp litude polarizat ion angle near 81 as shown in Figure 6 (lower). Unfo rtunately, it is impossible to prepare artificial roughness with cell size 2.52.5 m by etching into depth up to 1m. Because, size of top side of every cell is decreased to value of double depth provided that etching is isotropic. Figure 6. Dependences of phase angles of complex relative refractive index on incidence angle of light (0) when depth of relief exceeds quarter of wavelength of light. The changing in amplitude angle  due to artificial roughness is negligible and it is not shown here. Depths of defects in fused quartz are following: d1=690 nm, d2=904 nm, d3=1005 nm. The behavior of extrema of higher order is shown in details on the insert. Size of elementary cell is 2525 m. Brewster's angle is noted by dot line; Dependences of amplitude angles (0) and increments of phase angles (0) of complex relative refractive index on incidence angle of light 0 when depth of relief exceeds quarter of wavelength of light. The changing in  due to artificial roughness is comparable with changing in the increment of . Depths of defects in aluminium are following: d1=572 nm, d2=791 nm, d3=1005 nm. Size of elementary cell is 2525 m Therefore the same relief differently impacts upon the polarization characteristics of light reflected fro m a surface of metal and dielectric, as apparently fro m a Figure 3

Experiment Confirms Theory
Summarizing all experimental measurements to validate correctness of theoretical analysis [7] consider how way values of depth of artificial defects impact on polarization of light reflected fro m rough surface in Figure 7, i.e. when singularities occur in ( 0 ) and ( 0) dependences.
Experimental measurements carried out on two sets of sample consisting from different materials: dielectric -fused quartz and metal-alu miniu m are very closely coincided with t wo curves calculated fro m interference conditions and with taking account of defect's depths which were measured by independent technique  The random phase mask is coarse model of a rough surface: size of squire "defects" is great (a»), though distribution of defects is subjected to the binomial law of distribution, but their depth is identical, therefo re interference effects are brightly exh ibited if defect's depth exceeds quarter of wavelength, i.e. at d> /4. However, even within the limits of such coarse model for s mall depth of artificial relief RPM (d </ 4), the changes of experimental dependences ( 0 ) and ( 0 ) overlap with the similar dependences obtained on polished surfaces [17,18]. As for author opinion, similar approach could be applied to some metamaterials [19]; where, fo r examp le, experimentall y was demonstrated independence of energy position (near 2 eV) of false absorption peak in transparency region of silicon after films of titaniu m and titanium nitride were deposited. While the magnitudes of  1 and  2 are undergone to great deformations; these changing are impossible in the case for surface plasmons.

Conclusions
Random Phase Mask which was suggested here as model of rough surface, has allowed to exp lain and to predict singularities in experimental polarization dependences of ( 0 ) and ( 0 ).
Conditions of the maximal influence of a diffused light on polarizing characteristics of rough surface are found out: a) when interference condition is fulfilled or/and b) when reflection of p-polarized light is vanished for dielectric (or is minimal for metal) at Brewster's angle. Presence of extrema in dependences of ( 0 ) and ( 0 ) is co mmon phenomenon for both dielectric and metal materials, it is caused by an interference of light reflected and light diffused; and only defect's depths define positions of these extrema. The differences (magnitude and width of extrema and their amount) in a state of polarization of light reflected fro m an artificial rough surface with irregular relief are defined by reflect ivity of material only. The basic co mmon behaviours and their differences are theoretically predicted and experimentally verified for dielectric and metal.
These results can be useful to definit ion of the sizes of test structures in integrate circuit, for instance.