- Email: [email protected]
Detection and recognition of local defects in 1D structures
1 September 2001
Optics Communications 196 (2001) 33±39
www.elsevier.com/locate/optcom
Detection and recognition of local defects in 1D structures ~a, F. Gonz Jose M. Saiz *, J.L. de la Pen alez, F. Moreno Grupo de Optica, Departamento de Fõsica Aplicada, Universidad de Cantabria, 39005 Santander, Spain Received 26 March 2001; accepted 5 June 2001
Abstract The use of a scattering model, based on the double interaction model, to explore microdefects in 1D microstructures on ¯at surfaces is proposed. Due to their high sensitivity, backscattering experiments have been performed for the case of a cylindrical ®ber resting on a ¯at substrate. They show that some local defects may be identi®ed by comparing the predictions of the model with the experimental results. The possibilities of this procedure for quality control in semiconductor industry are envisaged. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Light scattering; Backscattering; Surface defects
1. Introduction Eorts devoted to the scattering inverse problem for surfaces (knowledge of the properties of the scattering surface from the scattered light) are constant in the optics literature [1±4]. There has been a number of approaches to this problem, some general and some more speci®c. They always involve some kind of approximation and, in general, some assumption about the scattering surface is necessary to reach particular solutions. Some models are clearly addressed to a restricted kind of objects, wavelengths, etc., if not directly to a particular application [5]. In the case of surfaces composed of scattering objects on or above regular substrates there are two situations of special interest: (i) the case of a known object above an unknown surface, which
*
Corresponding author. Fax: +34-942-201402. E-mail address: [email protected] (J.M. Saiz).
has proved itself useful to model the surface exploration with a small tip in near ®eld microscopy [6,7] and (ii) the case of an unknown object on a well known surface. Here, the interest lies either in the recognition of the object features (shape, size, composition) or in the mere detection of such object changing the scattering pattern of the surface alone (contamination of electronic circuits [2,8], or optical surfaces, performance of solar cells [9]). In addition, the recognition of the scattering pattern produced by such surfaces (particles on substrates) helps in detecting changes produced on it, which may be useful in the case that some microstructure might be repetitive. Therefore, in order to carry out such analyses, it is needed a certain knowledge of the parameters of the problem (i.e. to know the surface characteristics) and a good theoretical model, able to reproduce the scattering patterns from these surfaces. The meaning of good in the former sentence is not clear, as it depends on the nature of the problem and on our requirements. Good may be exact, not so exact but
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 3 6 7 - 0
34
J.M. Saiz et al. / Optics Communications 196 (2001) 33±39
θs θi R εp
‹
h
εs
‹
fast, or not so exact but very versatile. The model that we propose in this work, the modi®ed double interaction model (MDIM), belongs to this last kind, in the sense that it is transparent (the components of the ®eld in a particular direction are easy to identify), plastic (many changes in the model ± accounting for dierent scattering situations ± are easy to implement), and fast. These properties confer this model the ability of suggesting interesting surface solutions to some particular scattering pattern observed. Once a solution is proposed, and if more accuracy is required, it may be checked afterwards with any other more exact method. However, for our purposes, MDIM has proved itself very reliable for surfaces composed of particles on substrates. In this research we apply this model to a particular kind of surface, consisting on a regular 1D metallic structure (a cylinder of approximately 1 lm diameter), whose basic backscattering pattern, when illuminating small lengths, is very sensitive, repetitive, and easy to reproduce with many models. However, the experimental pattern may be altered either by geometrical changes in the ®ber or by the presence of neighboring defects, possibilities that will be examined by means of the MDIM. 1D geometry surfaces are found for instance in wafers used in the semiconductor industry, whose level of contamination and defects must be explored as part of the quality control process. This paper is organized as follows: In Section 2 the model and the experimental setup are described brie¯y, and some results will be shown for the basic surface. In Section 3 various anomalous results are presented, together with some others obtained by means of the model, suggesting the possible origin of such anomalies. Finally, the main conclusions are summarized.
Fig. 1. Scatter geometry considered in the double interaction model. R: radius of the particle; h: height; ^es , ^ep : dielectric constants of substrate and particle; hi , hs : incidence and scattering angle; k: wavelength; A: incident amplitude.
the double interaction model [13], four ®elds contribute to the total ®eld, each calculated from the Mie solution for the isolated cylinder [14] and weighted to account for the dierent shadowing of each component produced in the illumination and observation of the image particle. Fresnel coecients are applied at the re¯ection on the ¯at surface and phase shifts corresponding to the dierent optical paths are introduced. The total scattered ®eld in a given direction may be written as: ET
4 X j1
Es j
1
with b Es 1 A0 A p b i Es A0 A h 2
b i Es 3 A0 A h b Es A0 A p 4
1
hs
hi
hs br hs 1 hs br hi 1
1=2
exp fid hs g
1=2
exp fid hi g
S hs S hi
hs br hi br hs 1
hi
1=2
S hi
S hs
1=2
exp fid hs g exp fid hi g 2
2. Cylinder-surface geometry 2.1. The modi®ed double interaction model For this geometry, whose pro®le is depicted in Fig. 1, and for a metallic scatter system, a model has been developed and described in previous works [10±12]. Brie¯y, since this model is based in
where hs is the scattering angle, measured from the normal to the substrate and positive towards the specular direction, hi is the angle of incidence, A0 b is the incident amplitude, A h is the complex scattered amplitude in direction h given by Mie calculation for the isolated cylinder, br h is the Fresnel complex re¯ection coecient for incidence h and the considered polarization, d h the phase
J.M. Saiz et al. / Optics Communications 196 (2001) 33±39
35
due to the additional path of each component, given by d h
2p 2R cos h k
3
where R is the radius of the cylinder and k is the incident wavelength. In Eq. (2), shadowing eects are characterized by the factor S h, which for the cylindrical geometry is given by [10,11] S h 1
sin jhj
4
In the pure backscattering direction (hs Eq. (2) takes the following form: b Es 1 A0 A p b i br hi 1 Es A0 A 2h 2
Es 3 E s 2 b Es A0 A pb r 2 hi 1 4
1=2
S hi
hi ,
exp fid hi g
S hi exp fi2d hi g 5
The ®t of the theory to the experiment has allowed us to ®nd, among other things, an estimate of the mean size [11,12,15] and size polydispersity [16] of the ®ber. When observing the pure backscattering (hs hi ), its sensitivity to the object features is very high, which is particularly noticeable in the minima angular positions observed in the backscattering patterns (hi varying from normal to grazing) when illuminating and detecting with a given polarization. Furthermore, the ¯exibility of the MDIM allows the introduction of simple modi®cations which help explain what is happening on the surface. 2.2. Description of the experiment The experimental setup is similar to others used before, and is described in detail in Refs. [17,18]. In this case the object is a 1.1 lm diameter ®ber upon a substrate, and both coated with gold. A beam splitter cube is used in order to collect the light scattered in the illuminating (backscatter) direction (see Fig. 2). An additional cylindrical lens is introduced close to the object, focusing the beam on a narrow stripe centered at a given point of the cylinder. This target includes a ®ber length of approximately 60 lm. It is selected by a longitudinal
Fig. 2. (a) Scheme of the experimental setup used for the backscattering measurements. BS: beam splitter, CL: cylindrical lens. The target can be displaced along its axis and turns both clockwise and counter-clockwise. (b) Pro®le of the incident beam.
movement of the ®ber along its axis. The scattering is produced in all directions ± though most of it is contained in the plane of incidence ± and the focusing lens collects and collimates the backscattered light. Finally, another lens focuses the beam onto the PIN detector. In order to increase the signal-to-noise ratio (SNR) the signal is driven to a lock-in ampli®er, synchronized with the modulated source (a chopped He±Ne laser operating at 633 nm). The polarization selected is perpendicular to the plane of incidence, (i.e. parallel to the cylinder axis) which is called s polarization. In Fig. 3 the experimental result obtained from the majority of points along the ®ber is plotted together with that obtained with the model for a radius of R 0:55 lm. 3. Anomalous results and irregularities. MDIM tests When the ®ber is moved, in 40 lm steps, along its axis, the backscattering pattern is almost repeated for each position, because the scattering object is nearly identical along the scan process. However, a point may be reached where the pattern
36
J.M. Saiz et al. / Optics Communications 196 (2001) 33±39
Fig. 3. Experimental s-polarized backscattering pattern obtained from an ordinary point of the ®ber (circles). Continuous line corresponds to the result obtained with the model for a radius of R 0:55 lm. Backscattering cross-section is expressed in arbitrary units (a.u.).
is severely altered. In these cases we immediately know that something has changed in the basic scattering object, which is no longer a simple cylinder lying on a ¯at surface. This is shown in the following examples, together with the results given by some small changes introduced in the model, trying to account for the variations in the scattering patterns.
Fig. 4. Experimental backscattering patterns obtained at different points of the ®ber. Circles: the pattern at that point shows an extra minimum; asterisks: pattern corresponding to the most of the points, like that shown in Fig. 3.
θo
go
h
3.1. A side defect Fig. 4 shows the pattern obtained at a certain point of the ®ber (circles). When compared with the experimental pattern obtained in most of points (asterisks), like that of Fig. 3, an extra minimum appears in 50°, and a distortion of the pattern in the approximate angular interval [40°,60°] is observed. However, the rest of the pattern is very much the same, suggesting that, whatever takes part in the scattering, it is in a very particular angular range. It is easy to notice that every backscattering angle is associated to a surface point in the substrate (see Fig. 5). A local defect, like a neighboring bump or dip in the surface not far from the cylinder, would presumably produce such a localized alteration of the pattern. Therefore, a simple, though only approximate, modi®cation may be introduced in the model, consisting in de®ning a Gaussian distributed height for each distance z to the base of the cylinder, thus
wo Fig. 5. Scheme showing how MDIM implements the presence of a bump (go < 0 corresponds to a dip).
producing a curved substrate, with dierent value h0 for each position z (or angle hi ) (such a surface defect is represented in Fig. 5): ! 2 2 h tan h tan h i o h0 z ' h go exp 6 w2o where the parameter go is the maximum height, or depth, of the bump go > 0 or dip go < 0 in the vicinity of the cylinder, and wo its width. In Eq. (6) tan ho
zo h
7
zo being the central point of the bump (or dip) measured from the base of the cylinder. ho is the center of the aected angular interval.
37
B.S. Cross-Section (a.u.)
B.S. Cross-Section (a.u.)
J.M. Saiz et al. / Optics Communications 196 (2001) 33±39
0
15
30
45
60
75
90
0
θi Fig. 6. Agreement between the experimental backscattering pattern obtained at a point of the ®ber (circles) and the result obtained with the model for a ®ber radius of R 0:55 lm and a neighboring bump characterized by wo ' 0:2 lm and go 0:4 lm (zo h tan ho ' 1:0 lm). Both show the same departure from the ordinary lobbed pattern. 0
The variable h , when taken to the double interaction model, produces a change in the phase shift associated to the optical path of each component given by d h0 2kh0 cos hi
8
Fig. 6 shows the pattern produced for wo ' 0:2 lm and go 0:4 lm (zo h tan ho ' 1:0 lm). This corresponds to a neighboring bump, something that agrees with the fact that a sputtering gold layer may lift easily over the lower substrate (this was observed before from electron-microscope images). The presence of an extra minimum could be explained in this way. 3.2. Other possibilities Fig. 7 shows two experimental results corresponding to another position of the illuminating spot. Both clearly dier from the ordinary result of Fig. 3 because of the sudden presence of a higher number of lobes. Furthermore, they correspond to the backscattering produced at each side of the ®ber, obtained when the measurement is performed by turning the sample clockwise or counter-clockwise. The results are similar but not identical, giving
15
30
45 θi
60
75
90
Fig. 7. Experimental backscattering patterns obtained in an irregular zone of the ®ber. Circles: sample turning clockwise; asterisks: sample turning counter-clockwise.
evidence that the change in the scattering object may be nearly, but not totally, symmetric. In principle, a higher number of lobes immediately suggests the idea of having a bigger cylinder in this point [19]. Though this assumption can be valid, the high regularity of the ®ber size, a property derived from the fabrication process itself (i.e. a pulled silica ®ber), leads to eliminate this possibility and to check with the help of the model a more likely possibility: a local lifting of the ®ber (see Fig. 8a), due to longitudinal thermal stress, unavoidable due to the geometry of the sample. In this case, if h is increased from its initial value (h R, corresponding to the cylinder resting on the substrate) the number of lobes increases, even if the size is kept the same. For h 1:7k (i.e. h ' 2:3R) the pattern obtained is shown in Fig. 9. The small lack of symmetry may be produced by the gold coating of the substrate, broken in a dierent way at each side of the cylinder. It is necessary to admit that other possibilities, like a small contaminating particle on or near the ®ber (see Fig. 8b), may produce scattering patterns similar to those found in our measurements. The presence of several objects associated to a given scattering pattern is something we can expect when dealing with an inverse problem like this. However, it is realistic to consider that the knowledge about the scattering object may always help us in
38
J.M. Saiz et al. / Optics Communications 196 (2001) 33±39
›
ε1
›
ε1
›
ε2
›
ε2
ho
›
ε1
B.S. Cross-Section (a.u.)
Fig. 8. Two possibilities explored with MDIM: (top) the ®ber is lifted above the substrate; (bottom) a small particle is resting on the ®ber.
the surfaces we have used to check the possibilities of this model, the basic 1D scatterer is a cylinder on a ¯at substrate, both of which are metallic. The changes explored consisted of defects on the substrate and changes in the basic object geometry. However, other cases may be subjected to this kind of analysis. For instance, other con®gurations given by 1D objects of arbitrary section. In this case the MDIM model would need the scattering pattern for the isolated object for any dierent angle of incidence transverse to the main axis. This may be provided by other models [20] whose exact solutions would be used as data®les, allowing again fast calculation. Also the presence of a local change in the optical constants, or a misalignment might be detected. Concerning the measurement of the backscattering pattern, it is interesting to remark the high sensitivity of this measurement to the presence of local defects and the ability to measure small asymmetries in the scattering objects.
Acknowledgements
0
15
30
45 θi
60
75
90
Fig. 9. Agreement between the experimental backscattering pattern obtained at the point shown in Fig. 7 (circles) and the result obtained with the model for a ®ber radius of R 0:55 lm showing a local lifting of h ' 2:3R.
selecting some of them, once we have reached reliable results from a suciently versatile model. In this case, for instance, the choice of a lifted ®ber is determined by a combination of theoretical ®t and plausibility. 4. Conclusions A fast, reliable and versatile model for light scattering, like the MDIM, has shown to be a very useful tool in detecting and recognizing anomalies produced in regularly structured 1D surfaces. In
The authors wish to thank the Direcci on General de Ense~ nanza Superior for its support through project PB97-0345. J.L. de la Pe~ na acknowledges the FPI grant conceded by the Ministerio de Educaci on y Ciencia of Spain.
References [1] P.H. McMurry, A.A. Naqwi (Eds.), Proceedings of the 5th International Congress on Optical Particle Sizing, Minneapolis MN, USA, 1998. [2] J.C. Stover, Optical Scattering: Measurements and Applications, SPIE Press, Bellingham, Washington, 1995. [3] E.D. Hirleman, C.F. Bohren (Eds.), Optical Particle Sizing, Appl. Opt. 30 (1991) 4685. [4] G. Gouesbet, G. Grehan (Eds.), Optical Particle Sizing. Theory and Practice, Plenum Press, New York, 1988. [5] R. Schmehl, B.M. Nebeker, E.D. Hirleman, Discretedipole approximation for scattering by features on surfaces by means of a two-dimensional fast Fourier transform technique, J. Opt. Soc. Am. A 14 (11) (1997) 3026. [6] J. Ripoll, A. Madrazo, M. Nieto-Vesperinas, Scattering of electromagnetic waves from a body over a random rough surface, Opt. Commun. 142 (1997) 173.
J.M. Saiz et al. / Optics Communications 196 (2001) 33±39 [7] A. Madrazo, M. Nieto-Vesperinas, Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface, J. Opt. Soc. Am. A 14 (8) (1997) 1859. [8] M.L. Liswith, E.J. Bawolek, E.D. Hirleman, Modeling of light scattering by submicrometer spherical particles on silicon and oxidized silicon surfaces, Opt. Eng. 35 (3) (1996) 858. [9] E. Yablonovitch, G.D. Cody, Intensity enhancement in textured optical sheets for solar cells, IEEE Trans. Electron. Dev. ED-29 (1982) 300. [10] J.M. Saiz, J.L. de la Pe~ na, P.J. Valle, F. Gonzalez, F. Moreno, Light scattering by regular particles on ¯at substrates, in: F. Moreno, F. Gonzalez (Eds.), Light Scattering from Microstructures, 2000, Springer, Berlin, pp. 285±299. [11] J.L. de la Pe~ na, J.M. Saiz, F. Gonzalez, P.J. Valle, F. Moreno, Sizing particles on substrates. A general method for oblique incidence, J. Appl. Phys. 85 (1) (1999) 432. [12] J.L. de la Pe~ na, J.M. Saiz, P.J. Valle, F. Gonzalez, F. Moreno, Tracking scattering minima to size metallic particles on ¯at substrates, Part. & Part. Syst. Charact. 16 (1999) 113.
39
[13] K.B. Nahm, W.L. Wolfe, Light-scattering models for spheres on a conducting plane: comparison with experiment, Appl. Opt. 26 (15) (1987) 2995. [14] H.C. van de Hulst, Light Scattering by Small Particles, Dover, New York, 1981. [15] J.L. de la Pe~ na, J.M. Saiz, F. Gonzalez, Pro®le of a ®ber from backscattering measurements, Opt. Lett. 25 (2000) 1699. [16] J.L. de la Pe~ na, J.M. Saiz, G. Videen, F. Gonzalez, P.J. Valle, F. Moreno, Scattering from particulate surfaces: visibility factor and polydispersity, Opt. Lett. 24 (1999) 1451. [17] J.M. Saiz, P.J. Valle, F. Gonzalez, F. Moreno, D.J. Jordan, Backscattering from particulate surfaces: experiments and theorical modeling, Opt. Eng. 33 (4) (1994) 1261. [18] J.M. Saiz, F. Gonzalez, F. Moreno, P.J. Valle, Application of ray-tracing model to the study of back-scattering from surfaces with particles, J. Phys. D: Appl. Phys. 28 (1995) 1040. [19] F. Moreno, J.M. Saiz, P.J. Valle, F. Gonzalez, Metallic particle sizing on ¯at substrates: application to conducting substrates, Appl. Phys. Lett. 68 (22) (1996) 3087. [20] F. Moreno, F. Gonzalez (Eds.), Light Scattering from Microstructures, Springer, Berlin, 2000.
Optics Communications 196 (2001) 33±39
www.elsevier.com/locate/optcom
Detection and recognition of local defects in 1D structures ~a, F. Gonz Jose M. Saiz *, J.L. de la Pen alez, F. Moreno Grupo de Optica, Departamento de Fõsica Aplicada, Universidad de Cantabria, 39005 Santander, Spain Received 26 March 2001; accepted 5 June 2001
Abstract The use of a scattering model, based on the double interaction model, to explore microdefects in 1D microstructures on ¯at surfaces is proposed. Due to their high sensitivity, backscattering experiments have been performed for the case of a cylindrical ®ber resting on a ¯at substrate. They show that some local defects may be identi®ed by comparing the predictions of the model with the experimental results. The possibilities of this procedure for quality control in semiconductor industry are envisaged. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Light scattering; Backscattering; Surface defects
1. Introduction Eorts devoted to the scattering inverse problem for surfaces (knowledge of the properties of the scattering surface from the scattered light) are constant in the optics literature [1±4]. There has been a number of approaches to this problem, some general and some more speci®c. They always involve some kind of approximation and, in general, some assumption about the scattering surface is necessary to reach particular solutions. Some models are clearly addressed to a restricted kind of objects, wavelengths, etc., if not directly to a particular application [5]. In the case of surfaces composed of scattering objects on or above regular substrates there are two situations of special interest: (i) the case of a known object above an unknown surface, which
*
Corresponding author. Fax: +34-942-201402. E-mail address: [email protected] (J.M. Saiz).
has proved itself useful to model the surface exploration with a small tip in near ®eld microscopy [6,7] and (ii) the case of an unknown object on a well known surface. Here, the interest lies either in the recognition of the object features (shape, size, composition) or in the mere detection of such object changing the scattering pattern of the surface alone (contamination of electronic circuits [2,8], or optical surfaces, performance of solar cells [9]). In addition, the recognition of the scattering pattern produced by such surfaces (particles on substrates) helps in detecting changes produced on it, which may be useful in the case that some microstructure might be repetitive. Therefore, in order to carry out such analyses, it is needed a certain knowledge of the parameters of the problem (i.e. to know the surface characteristics) and a good theoretical model, able to reproduce the scattering patterns from these surfaces. The meaning of good in the former sentence is not clear, as it depends on the nature of the problem and on our requirements. Good may be exact, not so exact but
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 3 6 7 - 0
34
J.M. Saiz et al. / Optics Communications 196 (2001) 33±39
θs θi R εp
‹
h
εs
‹
fast, or not so exact but very versatile. The model that we propose in this work, the modi®ed double interaction model (MDIM), belongs to this last kind, in the sense that it is transparent (the components of the ®eld in a particular direction are easy to identify), plastic (many changes in the model ± accounting for dierent scattering situations ± are easy to implement), and fast. These properties confer this model the ability of suggesting interesting surface solutions to some particular scattering pattern observed. Once a solution is proposed, and if more accuracy is required, it may be checked afterwards with any other more exact method. However, for our purposes, MDIM has proved itself very reliable for surfaces composed of particles on substrates. In this research we apply this model to a particular kind of surface, consisting on a regular 1D metallic structure (a cylinder of approximately 1 lm diameter), whose basic backscattering pattern, when illuminating small lengths, is very sensitive, repetitive, and easy to reproduce with many models. However, the experimental pattern may be altered either by geometrical changes in the ®ber or by the presence of neighboring defects, possibilities that will be examined by means of the MDIM. 1D geometry surfaces are found for instance in wafers used in the semiconductor industry, whose level of contamination and defects must be explored as part of the quality control process. This paper is organized as follows: In Section 2 the model and the experimental setup are described brie¯y, and some results will be shown for the basic surface. In Section 3 various anomalous results are presented, together with some others obtained by means of the model, suggesting the possible origin of such anomalies. Finally, the main conclusions are summarized.
Fig. 1. Scatter geometry considered in the double interaction model. R: radius of the particle; h: height; ^es , ^ep : dielectric constants of substrate and particle; hi , hs : incidence and scattering angle; k: wavelength; A: incident amplitude.
the double interaction model [13], four ®elds contribute to the total ®eld, each calculated from the Mie solution for the isolated cylinder [14] and weighted to account for the dierent shadowing of each component produced in the illumination and observation of the image particle. Fresnel coecients are applied at the re¯ection on the ¯at surface and phase shifts corresponding to the dierent optical paths are introduced. The total scattered ®eld in a given direction may be written as: ET
4 X j1
Es j
1
with b Es 1 A0 A p b i Es A0 A h 2
b i Es 3 A0 A h b Es A0 A p 4
1
hs
hi
hs br hs 1 hs br hi 1
1=2
exp fid hs g
1=2
exp fid hi g
S hs S hi
hs br hi br hs 1
hi
1=2
S hi
S hs
1=2
exp fid hs g exp fid hi g 2
2. Cylinder-surface geometry 2.1. The modi®ed double interaction model For this geometry, whose pro®le is depicted in Fig. 1, and for a metallic scatter system, a model has been developed and described in previous works [10±12]. Brie¯y, since this model is based in
where hs is the scattering angle, measured from the normal to the substrate and positive towards the specular direction, hi is the angle of incidence, A0 b is the incident amplitude, A h is the complex scattered amplitude in direction h given by Mie calculation for the isolated cylinder, br h is the Fresnel complex re¯ection coecient for incidence h and the considered polarization, d h the phase
J.M. Saiz et al. / Optics Communications 196 (2001) 33±39
35
due to the additional path of each component, given by d h
2p 2R cos h k
3
where R is the radius of the cylinder and k is the incident wavelength. In Eq. (2), shadowing eects are characterized by the factor S h, which for the cylindrical geometry is given by [10,11] S h 1
sin jhj
4
In the pure backscattering direction (hs Eq. (2) takes the following form: b Es 1 A0 A p b i br hi 1 Es A0 A 2h 2
Es 3 E s 2 b Es A0 A pb r 2 hi 1 4
1=2
S hi
hi ,
exp fid hi g
S hi exp fi2d hi g 5
The ®t of the theory to the experiment has allowed us to ®nd, among other things, an estimate of the mean size [11,12,15] and size polydispersity [16] of the ®ber. When observing the pure backscattering (hs hi ), its sensitivity to the object features is very high, which is particularly noticeable in the minima angular positions observed in the backscattering patterns (hi varying from normal to grazing) when illuminating and detecting with a given polarization. Furthermore, the ¯exibility of the MDIM allows the introduction of simple modi®cations which help explain what is happening on the surface. 2.2. Description of the experiment The experimental setup is similar to others used before, and is described in detail in Refs. [17,18]. In this case the object is a 1.1 lm diameter ®ber upon a substrate, and both coated with gold. A beam splitter cube is used in order to collect the light scattered in the illuminating (backscatter) direction (see Fig. 2). An additional cylindrical lens is introduced close to the object, focusing the beam on a narrow stripe centered at a given point of the cylinder. This target includes a ®ber length of approximately 60 lm. It is selected by a longitudinal
Fig. 2. (a) Scheme of the experimental setup used for the backscattering measurements. BS: beam splitter, CL: cylindrical lens. The target can be displaced along its axis and turns both clockwise and counter-clockwise. (b) Pro®le of the incident beam.
movement of the ®ber along its axis. The scattering is produced in all directions ± though most of it is contained in the plane of incidence ± and the focusing lens collects and collimates the backscattered light. Finally, another lens focuses the beam onto the PIN detector. In order to increase the signal-to-noise ratio (SNR) the signal is driven to a lock-in ampli®er, synchronized with the modulated source (a chopped He±Ne laser operating at 633 nm). The polarization selected is perpendicular to the plane of incidence, (i.e. parallel to the cylinder axis) which is called s polarization. In Fig. 3 the experimental result obtained from the majority of points along the ®ber is plotted together with that obtained with the model for a radius of R 0:55 lm. 3. Anomalous results and irregularities. MDIM tests When the ®ber is moved, in 40 lm steps, along its axis, the backscattering pattern is almost repeated for each position, because the scattering object is nearly identical along the scan process. However, a point may be reached where the pattern
36
J.M. Saiz et al. / Optics Communications 196 (2001) 33±39
Fig. 3. Experimental s-polarized backscattering pattern obtained from an ordinary point of the ®ber (circles). Continuous line corresponds to the result obtained with the model for a radius of R 0:55 lm. Backscattering cross-section is expressed in arbitrary units (a.u.).
is severely altered. In these cases we immediately know that something has changed in the basic scattering object, which is no longer a simple cylinder lying on a ¯at surface. This is shown in the following examples, together with the results given by some small changes introduced in the model, trying to account for the variations in the scattering patterns.
Fig. 4. Experimental backscattering patterns obtained at different points of the ®ber. Circles: the pattern at that point shows an extra minimum; asterisks: pattern corresponding to the most of the points, like that shown in Fig. 3.
θo
go
h
3.1. A side defect Fig. 4 shows the pattern obtained at a certain point of the ®ber (circles). When compared with the experimental pattern obtained in most of points (asterisks), like that of Fig. 3, an extra minimum appears in 50°, and a distortion of the pattern in the approximate angular interval [40°,60°] is observed. However, the rest of the pattern is very much the same, suggesting that, whatever takes part in the scattering, it is in a very particular angular range. It is easy to notice that every backscattering angle is associated to a surface point in the substrate (see Fig. 5). A local defect, like a neighboring bump or dip in the surface not far from the cylinder, would presumably produce such a localized alteration of the pattern. Therefore, a simple, though only approximate, modi®cation may be introduced in the model, consisting in de®ning a Gaussian distributed height for each distance z to the base of the cylinder, thus
wo Fig. 5. Scheme showing how MDIM implements the presence of a bump (go < 0 corresponds to a dip).
producing a curved substrate, with dierent value h0 for each position z (or angle hi ) (such a surface defect is represented in Fig. 5): ! 2 2 h tan h tan h i o h0 z ' h go exp 6 w2o where the parameter go is the maximum height, or depth, of the bump go > 0 or dip go < 0 in the vicinity of the cylinder, and wo its width. In Eq. (6) tan ho
zo h
7
zo being the central point of the bump (or dip) measured from the base of the cylinder. ho is the center of the aected angular interval.
37
B.S. Cross-Section (a.u.)
B.S. Cross-Section (a.u.)
J.M. Saiz et al. / Optics Communications 196 (2001) 33±39
0
15
30
45
60
75
90
0
θi Fig. 6. Agreement between the experimental backscattering pattern obtained at a point of the ®ber (circles) and the result obtained with the model for a ®ber radius of R 0:55 lm and a neighboring bump characterized by wo ' 0:2 lm and go 0:4 lm (zo h tan ho ' 1:0 lm). Both show the same departure from the ordinary lobbed pattern. 0
The variable h , when taken to the double interaction model, produces a change in the phase shift associated to the optical path of each component given by d h0 2kh0 cos hi
8
Fig. 6 shows the pattern produced for wo ' 0:2 lm and go 0:4 lm (zo h tan ho ' 1:0 lm). This corresponds to a neighboring bump, something that agrees with the fact that a sputtering gold layer may lift easily over the lower substrate (this was observed before from electron-microscope images). The presence of an extra minimum could be explained in this way. 3.2. Other possibilities Fig. 7 shows two experimental results corresponding to another position of the illuminating spot. Both clearly dier from the ordinary result of Fig. 3 because of the sudden presence of a higher number of lobes. Furthermore, they correspond to the backscattering produced at each side of the ®ber, obtained when the measurement is performed by turning the sample clockwise or counter-clockwise. The results are similar but not identical, giving
15
30
45 θi
60
75
90
Fig. 7. Experimental backscattering patterns obtained in an irregular zone of the ®ber. Circles: sample turning clockwise; asterisks: sample turning counter-clockwise.
evidence that the change in the scattering object may be nearly, but not totally, symmetric. In principle, a higher number of lobes immediately suggests the idea of having a bigger cylinder in this point [19]. Though this assumption can be valid, the high regularity of the ®ber size, a property derived from the fabrication process itself (i.e. a pulled silica ®ber), leads to eliminate this possibility and to check with the help of the model a more likely possibility: a local lifting of the ®ber (see Fig. 8a), due to longitudinal thermal stress, unavoidable due to the geometry of the sample. In this case, if h is increased from its initial value (h R, corresponding to the cylinder resting on the substrate) the number of lobes increases, even if the size is kept the same. For h 1:7k (i.e. h ' 2:3R) the pattern obtained is shown in Fig. 9. The small lack of symmetry may be produced by the gold coating of the substrate, broken in a dierent way at each side of the cylinder. It is necessary to admit that other possibilities, like a small contaminating particle on or near the ®ber (see Fig. 8b), may produce scattering patterns similar to those found in our measurements. The presence of several objects associated to a given scattering pattern is something we can expect when dealing with an inverse problem like this. However, it is realistic to consider that the knowledge about the scattering object may always help us in
38
J.M. Saiz et al. / Optics Communications 196 (2001) 33±39
›
ε1
›
ε1
›
ε2
›
ε2
ho
›
ε1
B.S. Cross-Section (a.u.)
Fig. 8. Two possibilities explored with MDIM: (top) the ®ber is lifted above the substrate; (bottom) a small particle is resting on the ®ber.
the surfaces we have used to check the possibilities of this model, the basic 1D scatterer is a cylinder on a ¯at substrate, both of which are metallic. The changes explored consisted of defects on the substrate and changes in the basic object geometry. However, other cases may be subjected to this kind of analysis. For instance, other con®gurations given by 1D objects of arbitrary section. In this case the MDIM model would need the scattering pattern for the isolated object for any dierent angle of incidence transverse to the main axis. This may be provided by other models [20] whose exact solutions would be used as data®les, allowing again fast calculation. Also the presence of a local change in the optical constants, or a misalignment might be detected. Concerning the measurement of the backscattering pattern, it is interesting to remark the high sensitivity of this measurement to the presence of local defects and the ability to measure small asymmetries in the scattering objects.
Acknowledgements
0
15
30
45 θi
60
75
90
Fig. 9. Agreement between the experimental backscattering pattern obtained at the point shown in Fig. 7 (circles) and the result obtained with the model for a ®ber radius of R 0:55 lm showing a local lifting of h ' 2:3R.
selecting some of them, once we have reached reliable results from a suciently versatile model. In this case, for instance, the choice of a lifted ®ber is determined by a combination of theoretical ®t and plausibility. 4. Conclusions A fast, reliable and versatile model for light scattering, like the MDIM, has shown to be a very useful tool in detecting and recognizing anomalies produced in regularly structured 1D surfaces. In
The authors wish to thank the Direcci on General de Ense~ nanza Superior for its support through project PB97-0345. J.L. de la Pe~ na acknowledges the FPI grant conceded by the Ministerio de Educaci on y Ciencia of Spain.
References [1] P.H. McMurry, A.A. Naqwi (Eds.), Proceedings of the 5th International Congress on Optical Particle Sizing, Minneapolis MN, USA, 1998. [2] J.C. Stover, Optical Scattering: Measurements and Applications, SPIE Press, Bellingham, Washington, 1995. [3] E.D. Hirleman, C.F. Bohren (Eds.), Optical Particle Sizing, Appl. Opt. 30 (1991) 4685. [4] G. Gouesbet, G. Grehan (Eds.), Optical Particle Sizing. Theory and Practice, Plenum Press, New York, 1988. [5] R. Schmehl, B.M. Nebeker, E.D. Hirleman, Discretedipole approximation for scattering by features on surfaces by means of a two-dimensional fast Fourier transform technique, J. Opt. Soc. Am. A 14 (11) (1997) 3026. [6] J. Ripoll, A. Madrazo, M. Nieto-Vesperinas, Scattering of electromagnetic waves from a body over a random rough surface, Opt. Commun. 142 (1997) 173.
J.M. Saiz et al. / Optics Communications 196 (2001) 33±39 [7] A. Madrazo, M. Nieto-Vesperinas, Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface, J. Opt. Soc. Am. A 14 (8) (1997) 1859. [8] M.L. Liswith, E.J. Bawolek, E.D. Hirleman, Modeling of light scattering by submicrometer spherical particles on silicon and oxidized silicon surfaces, Opt. Eng. 35 (3) (1996) 858. [9] E. Yablonovitch, G.D. Cody, Intensity enhancement in textured optical sheets for solar cells, IEEE Trans. Electron. Dev. ED-29 (1982) 300. [10] J.M. Saiz, J.L. de la Pe~ na, P.J. Valle, F. Gonzalez, F. Moreno, Light scattering by regular particles on ¯at substrates, in: F. Moreno, F. Gonzalez (Eds.), Light Scattering from Microstructures, 2000, Springer, Berlin, pp. 285±299. [11] J.L. de la Pe~ na, J.M. Saiz, F. Gonzalez, P.J. Valle, F. Moreno, Sizing particles on substrates. A general method for oblique incidence, J. Appl. Phys. 85 (1) (1999) 432. [12] J.L. de la Pe~ na, J.M. Saiz, P.J. Valle, F. Gonzalez, F. Moreno, Tracking scattering minima to size metallic particles on ¯at substrates, Part. & Part. Syst. Charact. 16 (1999) 113.
39
[13] K.B. Nahm, W.L. Wolfe, Light-scattering models for spheres on a conducting plane: comparison with experiment, Appl. Opt. 26 (15) (1987) 2995. [14] H.C. van de Hulst, Light Scattering by Small Particles, Dover, New York, 1981. [15] J.L. de la Pe~ na, J.M. Saiz, F. Gonzalez, Pro®le of a ®ber from backscattering measurements, Opt. Lett. 25 (2000) 1699. [16] J.L. de la Pe~ na, J.M. Saiz, G. Videen, F. Gonzalez, P.J. Valle, F. Moreno, Scattering from particulate surfaces: visibility factor and polydispersity, Opt. Lett. 24 (1999) 1451. [17] J.M. Saiz, P.J. Valle, F. Gonzalez, F. Moreno, D.J. Jordan, Backscattering from particulate surfaces: experiments and theorical modeling, Opt. Eng. 33 (4) (1994) 1261. [18] J.M. Saiz, F. Gonzalez, F. Moreno, P.J. Valle, Application of ray-tracing model to the study of back-scattering from surfaces with particles, J. Phys. D: Appl. Phys. 28 (1995) 1040. [19] F. Moreno, J.M. Saiz, P.J. Valle, F. Gonzalez, Metallic particle sizing on ¯at substrates: application to conducting substrates, Appl. Phys. Lett. 68 (22) (1996) 3087. [20] F. Moreno, F. Gonzalez (Eds.), Light Scattering from Microstructures, Springer, Berlin, 2000.