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Application of the optical method for determining of the RMS roughness of porous glass surfaces
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Journal of Non-Crystalline Solids 354 (2008) 3241–3245 www.elsevier.com/locate/jnoncrysol
Application of the optical method for determining of the RMS roughness of porous glass surfaces E. Dobierzewska-Mozrzymas *, E. Rysiakiewicz-Pasek, P. Biegan´ski, J. Polan´ska, E. Pieciul Institute of Physics, Wrocław University of Technology, Wybrze_ze Wyspian´skiego 27, 50-370 Wrocław, Poland Received 3 August 2007; received in revised form 13 February 2008 Available online 8 April 2008
Abstract The application of an optical method for characterizing surface roughness is presented. This method was used for an examination of porous glass surfaces. The expressions relating the root mean square rms (r) of a surface to its specular reflectance at normal incidence are used for r k, (k – wavelength). For light of sufficiently long wavelength the decrease in the measured specular reflectance due to the surface roughness depends only on the root mean square (rms) height of the surface irregularities. On the basis of reflectance spectra, one can determine r for the porous glass surfaces after technological processes. The measured reflectance spectra were compared with calculated ones for which the scattered component of light was taken into account. The parameters rms determined from the optical method are comparable to those obtained from atomic force microscopy examinations. Ó 2008 Elsevier B.V. All rights reserved. PACS: 78.68.+m Keywords: Glasses; Optical spectroscopy; Atomic force and scanning tunneling microscopy; Porosity; Reflectivity
1. Introduction Surface microstructure of porous glasses strongly depends on the local composition of glass and the technology of its preparation. The surfaces usually exhibit roughness. The heights and diameters of the surface heterogeneities may be described by means of different statistical distributions, i.e. they are more or less inhomogeneous. The reflectance of an electromagnetic wave depends on a state of the surface, on its roughness characterized by the root mean square roughness r or other parameters. Several examinations of the relation between the roughness of metal surfaces or ground glass surfaces and the specular or diffuse reflectance have been reported [1–5]. Bennett and Porteus pointed out that the root mean square of the surface roughness could be determined on *
Corresponding author. E-mail address: [email protected] (E. Dobierzewska-Mozrzymas). 0022-3093/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2008.02.013
the basis of specular reflectance measurements [6]. This method can be applied when r k, k denotes the wavelength. Therefore, a degree of the suitable smoothness is required. The reflectance is more sensitive to the surface roughness than the transmittance. Previously this method was used to examine glass surfaces polished by different methods [6]. The relation between the specular reflectance and the root mean square of roughness is obtained from the treatment of the electromagnetic wave reflection from a rough surface derived by Davis [7]. In this model of a surface, the following conditions have been satisfied: 1. the root mean square roughness r, defined as a root mean square deviation of the surface from the mean surface level is small compared with the wavelength k, 2. the surface is perfectly conducting and hence would have a specular reflectance of unity if it is perfectly smooth, 3. the distribution of heights of the surface irregularities is Gaussian about the mean,
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4. the autocovariance function of the surface irregularities is also Gaussian with the standard deviation ra. The conditions 3 and 4 mean that the surface irregularities in the plane of the surface and in the direction perpendicular to the surface are homogeneous. If the surface is illuminated with a parallel beam of the monochromatic light, the reflectance is divided into two components; one from the specular reflectance and the other connected with scattering. Davis obtained the following expression for the specular reflectance of the rough surface at the normal incidence [7]: " 2 # 4pr Rs ¼ R0 exp ; ð1Þ k where R0 denotes the reflectance of a perfectly smooth surface of the some conducting material. If the reflectance at the normal incidence is measured within an instrumental acceptance angle DH, the complete expression for the measured reflectance R takes the form [7]: " 2 # 4pr 25 p4 r4 2 R ¼ R0 exp ðDHÞ ; ð2Þ þ R0 2 k k m where m is the root mean square slope of the surface profile. The following correlation between the autocovariance length a and m is satisfied [7,8] pffiffiffi r ð3Þ a¼ 2 : m The value of a may be determined by atomic force microscopy (AFM) examination or adopted as a fitting parameter. The second term in Eq. (2) is connected with the scattered reflectance measurement. For a sufficiently long wavelength this term may be neglected. In this case, r is calculated from the measured reflectance, making use of Eq. (1), which takes the form R0 2 1 ln ð4Þ ¼ ð4prÞ 2 : Rs k If ln (R0/Rs) versus 1/k2 is a straight line, its slope makes it possible to determine r. This is a method for calculation of the root mean square roughness from the measured values of the reflectance at normal incidence (method I). There is another graphic method for determining of the value r. For this purpose one can present the plot r versus k, making use of Eq. (2). For the appropriate wavelength range, r is independent of k. The root mean square roughness may be also determined from this interval, (method II). The different methods are used for the surface morphology examination. The nanoscale roughnesses of glass induced by fast electron irradiation or oxide glass are studied by AFM and controlled by the surface tension of the melt at the glass transition temperature, respectively [9,10]. Micro-roughnesses arising on the glass surface, for example, during the interaction of femto- and nanosecond laser pulses with glass surface, are examined by scanning
electron and optical microscopes [11]. In order to monitor corrosion mechanisms on the glass surface the Fourier transform infrared reflectance spectroscopy is used [12]. In the presented paper a simple and precise method for determination of the rms for the rough surface, when r k, is described (nanoscale roughnesses). This method is applied to an examination of porous glass surfaces. The root mean square roughness r determined from the optical measurements is compared with that obtained from AFM. 2. Experimental A silica porous layer was formed at one side of the regular wafer of sodium borosilicate glass. The glass surface was polished before the layer fabrication. The porous layer was obtained by etching off sodium borate phase from the two-phase sodium borosilicate glass [13,14]. The phase separation temperature was 490 °C. During the leaching process the soluble sodium and boron oxides of a chemically less durable phase were extracted. The obtained samples were heated at 393 K for 0.5 h. The previous examinations were performed for the samples with geometrical sizes of 10 10 0.5 mm3, the thickness of porous layer was about 250 lm. Application of the presented optical method for determination of structural parameter r, requires the surface of porous glass to be flat. However, the surfaces after leaching process usually exhibit deformations. In order to verify the surface after the technological process, the equal thickness fringes obtained from the porous glass surface covered with Al film were observed. Fig. 1(a) presents these fringes from deformed surface. In this case the optical method cannot be used. The fringes are straight and parallel lines when the surface is flat, which is shown in Fig. 1(b). An application of the described optical method is possible for such a surface. It has been found that the samples approximately thick retain flat surface after the technological process. The optimal thickness of the sodium borosilicate glass plate equals 2 mm. The glass surface image is shown in Fig. 2(a) for the sample with rAFM = 12.4 nm, the distribution of the heights of the surface irregularities is presented in Fig. 2(b). It is seen that the surface microstructure is homogeneous and the distribution of the heights is close to Gaussian one (conditions 3 and 4). In our examinations 0.01 6 r/k 6 0.05 (condition 1). The surfaces of the porous layers were overcoated with an opaque Al films, evaporated under appropriate conditions, making possible to reproduce the surfaces examined [15] (condition 2). The conditions required in the Davis model were satisfied. The reflectance at normal incidence was measured as a function of the wavelength in the range from 200 to 2500 nm with the Jasco spectrophotometer. 3. Results The reflectance spectra for the various surfaces of the porous glasses are shown in Figs. 3(a)–(c). The experimental
E. Dobierzewska-Mozrzymas et al. / Journal of Non-Crystalline Solids 354 (2008) 3241–3245
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Fig. 1. Equal thickness fringes obtained: (a) from the deformed glass surface and b) from the flat glass surface.
Fig. 2. (a) AFM image of a rough porous glass surface, rAFM = 12.4 and (b) statistical distribution of heights of the surface irregularities for this sample.
values are presented, the solid curves were computed from Eq. (2) taking into account a contribution from both specular and scattered reflectance. In order to estimate the root mean square r, Eq. (1) was used for longer wavelengths, when the specularly reflected light is taken into account only. If the relation ln (R0/Rs) versus 1/k2 (Eq. (4)) is a straight line, its slope makes it possible to determine r. The example of this procedure is shown in Fig. 4(a). However, due to the measurement accuracy and some other errors, such as the effect of detector change in the spectrophotometer and surrounding humidity, the presented plot may not be strictly linear. There is another method for determination of the value of r. For this purpose the relation of r versus wavelength k is presented on the basis of Eq. (2). The example of this plot is shown in Fig. 4(b). For the sample examined, r is independent of wavelength, the parameter r is determined from this range of k. The values of the r calculated for the same sample, by both the methods, are the same or comparable (Fig. 4(a) and (b), Table 1). In order to calculate the reflectance on basis of Eq. (2), the instrumental acceptance angle DH for light at the normal incidence was measured, DH 0:1 for spectrophotometer used, the root mean square slope of the profile of the surface m may be assumed as a fitting parameter. If the structural examinations are conducted independently by means of AFM, the parameter m may be calculated making use of Eq. (3), for the determined autocovariance length
a. This method was used in our investigations, the values of m are presented in Table 1. 4. Discussion The theoretical wavelength dependences of the reflectance are in a good agreement with the experiment in the visible and near infrared. The root mean square in the range 5–20 nm describes well the reflectance at normal incidence for the surfaces examined. Various values of r may be connected with a different state of the surfaces after the polish process and also with surface layer composition differences of porous glass. As a result of the technological process [13,14], the different silica surface layers are formed. In the ultraviolet, the decrease of the measured reflectance is more rapid than the calculated one. It means that there is an appreciable contribution from the scattered reflectance near the normal one. Moreover, the requirement r k is violated, and the theory would not be expected to hold. It is seen from Table 1 that the parameters r calculated obtained by the optical methods are comparable with those determined by means of AFM. This procedure makes possible to estimate the root mean square for rough surfaces and porous glass surfaces on the basis of the optical measurements of reflectance at normal incidence. It is
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Fig. 4. Illustration of the methods for determination of r value: (a) from the slope of the plot ln (R0/Rs) l versus 1/k2, and (b) from the dependence r versus k.
Table 1 The structural parameters determined with different methods Sample
Root mean square roughness ropt ± 1 (nm)
1 2 3 4 5 6
Fig. 3. Measured and calculated reflectance spectra for the porous glass surfaces with different values of the r: (a) r = 6.5 nm, (b) r = 9.7 nm and (c) r = 17.5 nm.
Method I (nm)
Method II (nm)
6.5 7.5 9.7 17.0 18.2 23.5
– 8.0 7.5 17.5 16.5 23.5
rAFM (nm)
– 8.7 12.5 – 15.5 20.0
Root mean square slope of surface profile m
0.072 0.074 0.088 0.130 0.131 0.161
interesting that the reflectance versus r for different wavelengths is satisfied for regular dependences shown in Fig. 5. For k = 1500 nm R is practically independent of
E. Dobierzewska-Mozrzymas et al. / Journal of Non-Crystalline Solids 354 (2008) 3241–3245
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rough surfaces and porous glass surfaces. The surface state depends on the local composition of the porous glass surface layer and technological process used on the surface. For a light of a sufficiently long wavelength, the decrease in measured specular reflectance depends only on the root mean square of height of the surface irregularities. The measured specular reflectance spectra provide a simple and sensitive method for determination of the rms roughness. This method requires the surface to be flat and the root mean square surface roughness r k. The results obtained, with this method, are comparable with those from atomic force microscopy examinations. Acknowledgement This work has been done under the contract for Wrocław University of Technology 2006. Fig. 5. Dependences of the reflectance as function of the root mean square roughness for different wavelengths.
r, so in this range the scattered reflectance can be neglected. In the visible, for k = 500nnm the dependence R on r is more distinct, the reflectance decreases from 0.92 to 0.70 with the increasing r from 5 to 23 nm. The strongest decrease of the reflectance is seen in the ultraviolet (k = 300 nm). In the presented range of r, R decreases from 0.92 to 0.40. It should be noted that the regular dependences, presented in Fig. 5, are satisfied only for the calculated points (Eq. (2)) that are in an agreement with the measured ones. These curves may be used to estimate the r on the basis of the appropriate optical measurements. The root mean square in the range of 5–20 nm describes well the reflectance at normal incidence for the surfaces examined in the considered range of wavelengths.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
5. Conclusions The optical method was used to determine the structural parameter, the root mean square roughness, characterizing
[14] [15]
Lord Rayleigh, Nature 64 (1901) 385. R.S. Hunter, J. Opt. Soc. Am. 36 (1946) 178. J.M. Bennett, Appl. Opt. 15 (1976) 2705. B. van Ginneken, M. Stavridi, J.J. Koenderink, Appl. Opt. 37 (1998) 130. L.B. Wolff, J. Opt. Soc. Am. 11 (1994) 2956. H.E. Bennett, J.O. Porteus, J. Opt. Soc. Am. 51 (1961) 123. H. Davies, Proc. Inst. Electr. Eng. 101 (1954) 209. G. Rasigni, F. Varnier, M. Rasigni, J.P. Palmari, Phys. Rev. B 25 (1982) 2315. O. Gedeon, K. Jurek, I. Drbohlav, J. Non-Cryst. Solids 353 (2007) 1946. P.K. Gupta, D. Inniss, Ch.R. Kurkjian, Q. Zhong, J. Non-Cryst. Solids 262 (2000) 200. M.R. Kasaai, V. Kacham, F. Theberge, See Leang Chin, J. NonCryst. Solids 319 (2003) 129. M.E. Lynch, D.C. Folz, D.E. Clark, J. Non-Cryst. Solids 353 (2007) 2667. O.V. Mazurin, G.P. Roskova, V.I. Avierianow, T.V. Antropowa, Dwukfaznyje stiokla structura, svojstva, primienienie, Nauka, Laningrad, 1991 (in Russian). Ya.O. Roizin, S.A. Gevelyuk, L.P. Prokopovich, D.P. Savin, E. Rysiakiewicz-Pasek, K. Marczuk, J. Porous Mater. 4 (1997) 151. G. Hass, J. Waylonis, J. Opt. Soc. Am. 51 (1961) 719.
Journal of Non-Crystalline Solids 354 (2008) 3241–3245 www.elsevier.com/locate/jnoncrysol
Application of the optical method for determining of the RMS roughness of porous glass surfaces E. Dobierzewska-Mozrzymas *, E. Rysiakiewicz-Pasek, P. Biegan´ski, J. Polan´ska, E. Pieciul Institute of Physics, Wrocław University of Technology, Wybrze_ze Wyspian´skiego 27, 50-370 Wrocław, Poland Received 3 August 2007; received in revised form 13 February 2008 Available online 8 April 2008
Abstract The application of an optical method for characterizing surface roughness is presented. This method was used for an examination of porous glass surfaces. The expressions relating the root mean square rms (r) of a surface to its specular reflectance at normal incidence are used for r k, (k – wavelength). For light of sufficiently long wavelength the decrease in the measured specular reflectance due to the surface roughness depends only on the root mean square (rms) height of the surface irregularities. On the basis of reflectance spectra, one can determine r for the porous glass surfaces after technological processes. The measured reflectance spectra were compared with calculated ones for which the scattered component of light was taken into account. The parameters rms determined from the optical method are comparable to those obtained from atomic force microscopy examinations. Ó 2008 Elsevier B.V. All rights reserved. PACS: 78.68.+m Keywords: Glasses; Optical spectroscopy; Atomic force and scanning tunneling microscopy; Porosity; Reflectivity
1. Introduction Surface microstructure of porous glasses strongly depends on the local composition of glass and the technology of its preparation. The surfaces usually exhibit roughness. The heights and diameters of the surface heterogeneities may be described by means of different statistical distributions, i.e. they are more or less inhomogeneous. The reflectance of an electromagnetic wave depends on a state of the surface, on its roughness characterized by the root mean square roughness r or other parameters. Several examinations of the relation between the roughness of metal surfaces or ground glass surfaces and the specular or diffuse reflectance have been reported [1–5]. Bennett and Porteus pointed out that the root mean square of the surface roughness could be determined on *
Corresponding author. E-mail address: [email protected] (E. Dobierzewska-Mozrzymas). 0022-3093/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2008.02.013
the basis of specular reflectance measurements [6]. This method can be applied when r k, k denotes the wavelength. Therefore, a degree of the suitable smoothness is required. The reflectance is more sensitive to the surface roughness than the transmittance. Previously this method was used to examine glass surfaces polished by different methods [6]. The relation between the specular reflectance and the root mean square of roughness is obtained from the treatment of the electromagnetic wave reflection from a rough surface derived by Davis [7]. In this model of a surface, the following conditions have been satisfied: 1. the root mean square roughness r, defined as a root mean square deviation of the surface from the mean surface level is small compared with the wavelength k, 2. the surface is perfectly conducting and hence would have a specular reflectance of unity if it is perfectly smooth, 3. the distribution of heights of the surface irregularities is Gaussian about the mean,
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E. Dobierzewska-Mozrzymas et al. / Journal of Non-Crystalline Solids 354 (2008) 3241–3245
4. the autocovariance function of the surface irregularities is also Gaussian with the standard deviation ra. The conditions 3 and 4 mean that the surface irregularities in the plane of the surface and in the direction perpendicular to the surface are homogeneous. If the surface is illuminated with a parallel beam of the monochromatic light, the reflectance is divided into two components; one from the specular reflectance and the other connected with scattering. Davis obtained the following expression for the specular reflectance of the rough surface at the normal incidence [7]: " 2 # 4pr Rs ¼ R0 exp ; ð1Þ k where R0 denotes the reflectance of a perfectly smooth surface of the some conducting material. If the reflectance at the normal incidence is measured within an instrumental acceptance angle DH, the complete expression for the measured reflectance R takes the form [7]: " 2 # 4pr 25 p4 r4 2 R ¼ R0 exp ðDHÞ ; ð2Þ þ R0 2 k k m where m is the root mean square slope of the surface profile. The following correlation between the autocovariance length a and m is satisfied [7,8] pffiffiffi r ð3Þ a¼ 2 : m The value of a may be determined by atomic force microscopy (AFM) examination or adopted as a fitting parameter. The second term in Eq. (2) is connected with the scattered reflectance measurement. For a sufficiently long wavelength this term may be neglected. In this case, r is calculated from the measured reflectance, making use of Eq. (1), which takes the form R0 2 1 ln ð4Þ ¼ ð4prÞ 2 : Rs k If ln (R0/Rs) versus 1/k2 is a straight line, its slope makes it possible to determine r. This is a method for calculation of the root mean square roughness from the measured values of the reflectance at normal incidence (method I). There is another graphic method for determining of the value r. For this purpose one can present the plot r versus k, making use of Eq. (2). For the appropriate wavelength range, r is independent of k. The root mean square roughness may be also determined from this interval, (method II). The different methods are used for the surface morphology examination. The nanoscale roughnesses of glass induced by fast electron irradiation or oxide glass are studied by AFM and controlled by the surface tension of the melt at the glass transition temperature, respectively [9,10]. Micro-roughnesses arising on the glass surface, for example, during the interaction of femto- and nanosecond laser pulses with glass surface, are examined by scanning
electron and optical microscopes [11]. In order to monitor corrosion mechanisms on the glass surface the Fourier transform infrared reflectance spectroscopy is used [12]. In the presented paper a simple and precise method for determination of the rms for the rough surface, when r k, is described (nanoscale roughnesses). This method is applied to an examination of porous glass surfaces. The root mean square roughness r determined from the optical measurements is compared with that obtained from AFM. 2. Experimental A silica porous layer was formed at one side of the regular wafer of sodium borosilicate glass. The glass surface was polished before the layer fabrication. The porous layer was obtained by etching off sodium borate phase from the two-phase sodium borosilicate glass [13,14]. The phase separation temperature was 490 °C. During the leaching process the soluble sodium and boron oxides of a chemically less durable phase were extracted. The obtained samples were heated at 393 K for 0.5 h. The previous examinations were performed for the samples with geometrical sizes of 10 10 0.5 mm3, the thickness of porous layer was about 250 lm. Application of the presented optical method for determination of structural parameter r, requires the surface of porous glass to be flat. However, the surfaces after leaching process usually exhibit deformations. In order to verify the surface after the technological process, the equal thickness fringes obtained from the porous glass surface covered with Al film were observed. Fig. 1(a) presents these fringes from deformed surface. In this case the optical method cannot be used. The fringes are straight and parallel lines when the surface is flat, which is shown in Fig. 1(b). An application of the described optical method is possible for such a surface. It has been found that the samples approximately thick retain flat surface after the technological process. The optimal thickness of the sodium borosilicate glass plate equals 2 mm. The glass surface image is shown in Fig. 2(a) for the sample with rAFM = 12.4 nm, the distribution of the heights of the surface irregularities is presented in Fig. 2(b). It is seen that the surface microstructure is homogeneous and the distribution of the heights is close to Gaussian one (conditions 3 and 4). In our examinations 0.01 6 r/k 6 0.05 (condition 1). The surfaces of the porous layers were overcoated with an opaque Al films, evaporated under appropriate conditions, making possible to reproduce the surfaces examined [15] (condition 2). The conditions required in the Davis model were satisfied. The reflectance at normal incidence was measured as a function of the wavelength in the range from 200 to 2500 nm with the Jasco spectrophotometer. 3. Results The reflectance spectra for the various surfaces of the porous glasses are shown in Figs. 3(a)–(c). The experimental
E. Dobierzewska-Mozrzymas et al. / Journal of Non-Crystalline Solids 354 (2008) 3241–3245
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Fig. 1. Equal thickness fringes obtained: (a) from the deformed glass surface and b) from the flat glass surface.
Fig. 2. (a) AFM image of a rough porous glass surface, rAFM = 12.4 and (b) statistical distribution of heights of the surface irregularities for this sample.
values are presented, the solid curves were computed from Eq. (2) taking into account a contribution from both specular and scattered reflectance. In order to estimate the root mean square r, Eq. (1) was used for longer wavelengths, when the specularly reflected light is taken into account only. If the relation ln (R0/Rs) versus 1/k2 (Eq. (4)) is a straight line, its slope makes it possible to determine r. The example of this procedure is shown in Fig. 4(a). However, due to the measurement accuracy and some other errors, such as the effect of detector change in the spectrophotometer and surrounding humidity, the presented plot may not be strictly linear. There is another method for determination of the value of r. For this purpose the relation of r versus wavelength k is presented on the basis of Eq. (2). The example of this plot is shown in Fig. 4(b). For the sample examined, r is independent of wavelength, the parameter r is determined from this range of k. The values of the r calculated for the same sample, by both the methods, are the same or comparable (Fig. 4(a) and (b), Table 1). In order to calculate the reflectance on basis of Eq. (2), the instrumental acceptance angle DH for light at the normal incidence was measured, DH 0:1 for spectrophotometer used, the root mean square slope of the profile of the surface m may be assumed as a fitting parameter. If the structural examinations are conducted independently by means of AFM, the parameter m may be calculated making use of Eq. (3), for the determined autocovariance length
a. This method was used in our investigations, the values of m are presented in Table 1. 4. Discussion The theoretical wavelength dependences of the reflectance are in a good agreement with the experiment in the visible and near infrared. The root mean square in the range 5–20 nm describes well the reflectance at normal incidence for the surfaces examined. Various values of r may be connected with a different state of the surfaces after the polish process and also with surface layer composition differences of porous glass. As a result of the technological process [13,14], the different silica surface layers are formed. In the ultraviolet, the decrease of the measured reflectance is more rapid than the calculated one. It means that there is an appreciable contribution from the scattered reflectance near the normal one. Moreover, the requirement r k is violated, and the theory would not be expected to hold. It is seen from Table 1 that the parameters r calculated obtained by the optical methods are comparable with those determined by means of AFM. This procedure makes possible to estimate the root mean square for rough surfaces and porous glass surfaces on the basis of the optical measurements of reflectance at normal incidence. It is
3244
E. Dobierzewska-Mozrzymas et al. / Journal of Non-Crystalline Solids 354 (2008) 3241–3245
Fig. 4. Illustration of the methods for determination of r value: (a) from the slope of the plot ln (R0/Rs) l versus 1/k2, and (b) from the dependence r versus k.
Table 1 The structural parameters determined with different methods Sample
Root mean square roughness ropt ± 1 (nm)
1 2 3 4 5 6
Fig. 3. Measured and calculated reflectance spectra for the porous glass surfaces with different values of the r: (a) r = 6.5 nm, (b) r = 9.7 nm and (c) r = 17.5 nm.
Method I (nm)
Method II (nm)
6.5 7.5 9.7 17.0 18.2 23.5
– 8.0 7.5 17.5 16.5 23.5
rAFM (nm)
– 8.7 12.5 – 15.5 20.0
Root mean square slope of surface profile m
0.072 0.074 0.088 0.130 0.131 0.161
interesting that the reflectance versus r for different wavelengths is satisfied for regular dependences shown in Fig. 5. For k = 1500 nm R is practically independent of
E. Dobierzewska-Mozrzymas et al. / Journal of Non-Crystalline Solids 354 (2008) 3241–3245
3245
rough surfaces and porous glass surfaces. The surface state depends on the local composition of the porous glass surface layer and technological process used on the surface. For a light of a sufficiently long wavelength, the decrease in measured specular reflectance depends only on the root mean square of height of the surface irregularities. The measured specular reflectance spectra provide a simple and sensitive method for determination of the rms roughness. This method requires the surface to be flat and the root mean square surface roughness r k. The results obtained, with this method, are comparable with those from atomic force microscopy examinations. Acknowledgement This work has been done under the contract for Wrocław University of Technology 2006. Fig. 5. Dependences of the reflectance as function of the root mean square roughness for different wavelengths.
r, so in this range the scattered reflectance can be neglected. In the visible, for k = 500nnm the dependence R on r is more distinct, the reflectance decreases from 0.92 to 0.70 with the increasing r from 5 to 23 nm. The strongest decrease of the reflectance is seen in the ultraviolet (k = 300 nm). In the presented range of r, R decreases from 0.92 to 0.40. It should be noted that the regular dependences, presented in Fig. 5, are satisfied only for the calculated points (Eq. (2)) that are in an agreement with the measured ones. These curves may be used to estimate the r on the basis of the appropriate optical measurements. The root mean square in the range of 5–20 nm describes well the reflectance at normal incidence for the surfaces examined in the considered range of wavelengths.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
5. Conclusions The optical method was used to determine the structural parameter, the root mean square roughness, characterizing
[14] [15]
Lord Rayleigh, Nature 64 (1901) 385. R.S. Hunter, J. Opt. Soc. Am. 36 (1946) 178. J.M. Bennett, Appl. Opt. 15 (1976) 2705. B. van Ginneken, M. Stavridi, J.J. Koenderink, Appl. Opt. 37 (1998) 130. L.B. Wolff, J. Opt. Soc. Am. 11 (1994) 2956. H.E. Bennett, J.O. Porteus, J. Opt. Soc. Am. 51 (1961) 123. H. Davies, Proc. Inst. Electr. Eng. 101 (1954) 209. G. Rasigni, F. Varnier, M. Rasigni, J.P. Palmari, Phys. Rev. B 25 (1982) 2315. O. Gedeon, K. Jurek, I. Drbohlav, J. Non-Cryst. Solids 353 (2007) 1946. P.K. Gupta, D. Inniss, Ch.R. Kurkjian, Q. Zhong, J. Non-Cryst. Solids 262 (2000) 200. M.R. Kasaai, V. Kacham, F. Theberge, See Leang Chin, J. NonCryst. Solids 319 (2003) 129. M.E. Lynch, D.C. Folz, D.E. Clark, J. Non-Cryst. Solids 353 (2007) 2667. O.V. Mazurin, G.P. Roskova, V.I. Avierianow, T.V. Antropowa, Dwukfaznyje stiokla structura, svojstva, primienienie, Nauka, Laningrad, 1991 (in Russian). Ya.O. Roizin, S.A. Gevelyuk, L.P. Prokopovich, D.P. Savin, E. Rysiakiewicz-Pasek, K. Marczuk, J. Porous Mater. 4 (1997) 151. G. Hass, J. Waylonis, J. Opt. Soc. Am. 51 (1961) 719.