ISSN 0030-400X, Optics and Spectroscopy, 2007, Vol. 102, No. 3, pp. 442–448. © Pleiades Publishing, Ltd., 2007. Original Russian Text © L.A. Ageev, V.K. Miloslavskiœ, E.D. Makovetskiœ, 2007, published in Optika i Spektroskopiya, 2007, Vol. 102, No. 3, pp. 489–495.

CONDENSED-MATTER SPECTROSCOPY

Coloring of Silica Glass with Silver Nanoparticles
L. A. Ageev, V. K. Miloslavskiœ, and E. D. Makovetskiœ
Kharkov National University, Kharkov, 61077 Ukraine e-mail: Leonid.A.Ageev@univer.kharkov.ua
Received February 20, 2006; in ﬁnal form, August 18, 2006

Abstract—The possibility of doping of surface layers of optical fused silica with silver nanogranules upon heating of a thin Ag ﬁlm on the silica surface by a CO2 laser beam (P ≈ 30 W, λ = 10.6 µm) is established. The effect of exposure time on the doped layer structure has been investigated. The absorption band of the colloidal solution of Ag in silica has been studied. It is shown that this band is homogeneously broadened and its peak (at 420 nm) corresponds to the small volume ﬁlling factor (q < 0.1) at an average granule radius of about 3 nm. Based on the measurements of the radial distributions of the reﬂectance and refractive index over the doped region at λ = 633 nm, it is revealed that the doped layer is an area-irregular asymmetric step waveguide. PACS numbers: 78.20.Ci DOI: 10.1134/S0030400X07030198

INTRODUCTION The technique of coloring dielectrics by introducing very small metal particles into them has been known for a very long time. However, the interest in this problem remains in view of the development of nonlinear optics, waveguide optics, and optics of nanostructures [1, 2]. The studies published approximately before 1995 and devoted to optical properties of metal clusters were reviewed in monograph [2]. In particular, it was noted in [2] that implantation of clusters into fused silica is a fairly difﬁcult problem. In this review, the data from different studies were summarized, which indicate the possibility of implanting Ag particles in fused silica by combined vacuum deposition of metal and quartz [3] and no other doping techniques were mentioned. In [1], ion implantation was used to dope silicate glass with silver; however, according to the noted value of the refractive index (n = 1.5), this glass is not optical fused silica. Recently, much attention has been given to study of the clusters that are chemically implanted (with subsequent annealing) into specially prepared porous silica with pores about 10 nm in diameter [4]. In this study, we show that the surface layer of fused silica can be doped using irradiation by a vacuum-deposited thin Ag ﬁlm on silica by a CO2 laser beam. EXPERIMENTAL We used polished plane-parallel KU silica plates [5, 6] with a thickness of 2 mm and area of 10 cm2. A thin (≈10 nm) Ag ﬁlm was deposited in vacuum (5 × 10−5 Torr) on the surface of a cold plate. A film of such thickness was obtained upon complete evaporation of 8 mg pure Ag from a planar molybdenum evaporator located at a distance of 15 cm from the evaporator to the

substrate center. Then, the ﬁlm thickness was measured by the interference method of lines of equal chromatic order [7]. Electron microscopy analysis shows that ﬁlms of such thickness are not continuous but have an island structure [8]. The plate with an Ag ﬁlm was then irradiated in air from the ﬁlm side by a Gaussian beam of an ILGN-704 cw CO2 laser (λ = 10.6 µm, P ≈ 30 W, the effective beam diameter about 3–4 mm). Preliminary investigation of the thermal action of such a beam on absorbing media showed that it can heat surfaces of different samples to 900–1200°C. The temperature of a heated surface can be estimated, ﬁrst of all, from the luminescence brightness of the irradiated area and by measuring the luminous temperature with a pyrometer. The absorption coefﬁcient α of fused silica at λ = 10.6 µm is ≈400 cm–1 [9]; the beam penetration depth δ = α–1 ≈ 25 µm is much smaller than the plate thickness and the beam is completely absorbed in it. However, the plate heating due to the beam absorption does not lead to visible luminescence of the irradiated region since the emissive power of silica is low in the visible spectral range. The temperature of the rear plate surface (T ≈ 600°C) was measured by a Chromel–Alumel thermocouple. One might predict that the front surface is heated to T ≈ 1000°C. Then, assuming that T linearly decreases along the beam axis z and the plate thickness ∆z = 2 mm, we ﬁnd that the temperature gradient is fairly high: dT/dz ≈ 200 K/mm. In the ﬁrst stage of irradiation of the plate with an Ag ﬁlm, signiﬁcant changes in the ﬁlm are observed. It can be seen in transmitted white light that, in the ring zone surrounding the beam, the ﬁlm is rapidly colored yellow; coloring propagates in the radial direction far from

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the beam center. This coloring is related to the transformations in the ﬁlm structure caused by thermal diffusion processes. Upon heating, the ﬁlm on the surface acquires a structure of a two-dimensional colloid and absorbs light through the known mechanism of plasma resonance [10]. After about a minute, luminescence arises in the irradiated region, which rapidly acquires an almost white color. Pyrometer measurements showed that the color temperature is about 960°C in this case. Since the true temperature is somewhat higher than the color temperature, the ﬁlm is heated above the melting temperature of bulk silver (Tm ≈ 960.8°C). It has been known for a long time from measurements in the visible spectral range that island Ag ﬁlms with a thickness of about 10 nm have large absorptance (30–35%, [10]). Apparently, this is the reason for the bright thermal luminescence of a ﬁlm heated to high temperatures. Further exposure leads to gradual disappearance of luminescence in the central irradiation zone, and the luminous zone turns into a slowly expanding ring. At a long exposure, the luminescence in the central zone completely disappears, and the surrounding luminous ring reaches the maximum (dependent on the exposure) diameter. For further investigations, we used the samples irradiated to the stage of appearance of luminescence in the central zone. After irradiation, a sample was immersed in a sulfuric acid solution to remove Ag from the silica surface. The silver that penetrated into the quartz is protected from the action of the acid and its state remains invariable for an arbitrarily long time (under normal conditions). The doped region is about 3–4 mm in diameter and has a yellow color upon observation in transmitted white light. In reﬂected light, interference coloring, similar to that of thin ﬁlms, can be seen. Different colors indicate nonuniform distribution of Ag over the area and depth of the doped region. Figure 1 shows a reﬂected light photograph of the (a) doped region in the central irradiation zone and (b) the doped ring zone, which is obtained at a long exposure. A series of experiments showed that, despite the Gaussian intensity distribution in the CO2 laser beam, the coloring of the doped region is not concentric. The centers of rings of the same color do not coincide and rings decompose into arcs. In black and white photographs, nonuniform coloring manifests itself as nonuniform blackening. A possible reason for nonuniform coloring is ﬂuctuations in the azimuthal dependence of the transverse gradient ∇T(x, y). All further investigations were performed with the sample shown in Fig. 1a. The doped surface was ﬁrst studied by transmission electron microscopy. To this end, we prepared a replica of the surface after its shadowing with an ultrathin gold ﬁlm deposited in vacuum at an angle of 10° to the surface. The image obtained (Fig. 2) shows that there is a very ﬁne chaotic relief on the doped surface. Spherical protrusions have an averOPTICS AND SPECTROSCOPY Vol. 102 No. 3 2007

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Fig. 1. Photographs of the doped regions in reﬂected white light: (a) irradiation by a CO2 laser beam was interrupted in the white luminescence stage (nonuniform black and white contrast is related to the difference in the interference coloring of this doped region) and (b) long laser beam exposure with the formation of a ring. Shadow from the doped regions can be seen on the rear side of the silica plate.

age height of about 5 nm. The average distance between the relief elements is ≈100 nm. Then, we investigated the optical properties of the doped region: the radial distributions of the reﬂectance and the refractive index and the absorption spectrum. The reﬂectance R under normal incidence was measured according to the scheme shown in Fig. 3 with the use of a focused Gaussian He–Ne laser beam (λ = 633 nm) with a waist in the focus of about 30 µm. The sample on a movable holder was displaced along the x axis passing through the center of the doped region. Values of R(x) were measured with a step ∆x = 0.2 mm. Due to the slight tapering of silica plate, the narrow laser beam was split into two beams after reﬂection from two boundaries. This circumstance made it possible to measure R(x) only for the doped surface layer. The values of R(x) were calculated by the formula R ( x ) = 0.035 i ( x ) / i 0 , (1)

where i(x) is a signal of a photoelectron multiplier from the doped region, i0 is the signal from a pure surface, and 0.035 is the reﬂectance of a pure surface at λ = 633 nm. The dependence R(x) is shown by a dashed line in Fig. 4. The beam from the doped region is brighter but is does not have any spatial distortions, except for a very weak scattering halo. This pattern indicates that there are no signiﬁcant violations of the polished surface within the doped region. The refractive index n was measured according to the technique described in [11] (Fig. 3). A sample was prepared as follows: a thin (≈30 nm) AgCl ﬁlm and an Ag ﬁlm with a thickness of ≈7 nm were subsequently

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100 nm

(b)

100 nm

Fig. 2. Electron microscopy images of the silica surface: (a) undoped and (b) doped regions.

deposited on the sample surface in vacuum; then the sample was installed on a goniometer and irradiated with a focused s-polarized beam. As a result of scattering, the beam excites a waveguide TE0 mode in the photosensitive AgCl–Ag ﬁlm and, as will be shown below, in the doped layer. This mode induces a diffraction grating of poor quality in the photolayer. Diffraction from the grating into the order m = –1 results in a narrow

fringe on a screen installed parallel to the polarization vector E of the beam. At the thickness of the AgCl–Ag ﬁlm smaller than the cutoff thickness of the TE0 mode (45 nm), measurement of the grating period yields the refractive index of the substrate [11]. In measurements, the angle of incidence ϕ of the beam is varied until the autocollimation condition is satisﬁed. In this case, n = 3 sin ϕ . (2) At the error ∆ϕ = ±0.1°, the error in n is ±0.005. Measurements of n(x) were performed in almost the same direction x as the measurements of R(x); the results are shown in Fig. 4 (curve 1). Finally, the absorption spectrum was measured in the wavelength range λ = 270–620 nm (Fig. 5). The measuring beam of the spectrophotometer was limited by a diaphragm so as to make the beam diameter smaller than the doped region diameter. Measurements were performed with respect to a pure silica plate. RESULTS AND DISCUSSION

6

7

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Fig. 3. Schematic diagram of the setup for measuring the optical characteristics of doped samples: (1) He–Ne laser beam (λ = 633 nm); (E) direction of linear beam polarization, perpendicular to the plane of the ﬁgure, (2) focusing lens (F = 9.4 cm), (3) screen with a diaphragm for beam limiting, (4) beam-splitting glass plate (installed to measure R(x)), (5) sample on a movable holder and goniometer; (6, 7) optical fiber and photoelectron multiplier (used to measure R(x)).

Let us consider the doping mechanism. Apparently, when the surface temperature reaches the value of Tm, emission of electrons into silica and their trapping occur. Molten particles are positively charged, and an electrostatic ﬁeld is induced between them and traps, providing detachment of Ag+ ions and their transport to traps. This process repeats many times and, as a result, leads to precipitation of Ag granules in the surface layer of silica. Note that the granule precipitation is faciliOPTICS AND SPECTROSCOPY Vol. 102 No. 3 2007

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tated by the porosity of fused silica: the ratio of its density to the density of crystalline SiO2 is 0.83. The expected doping process occurs in the initial stage, when the temperature in the central zone reaches the melting point. Furthermore, luminescence at the center of the irradiated region disappears, the dark region gradually expands, and the luminescing region acquires the shape of a ring whose diameter increases with a simultaneous decrease in its width. Apparently, such transformations are related to the disappearance of Ag from the silica surface in the central zone and gradual increase in T on the surface in the directions transverse to beam direction. This large-scale doping inhomogeneity, dependent on the exposure time, corresponds to a Gaussian beam; however, thermal processes lead also to additional smaller inhomogeneities (Figs. 1a, 4). The surface microrelief (Fig. 2) revealed by electron microscopy is due to doping. Formation of Ag particles should increase the volume and induce internal stress in the silica. If a particle is located directly under the surface, a convexity should arise on the surface, whose size and shape are related to the size and shape of the particle. Since some microrelief elements have a spherical shape, we can suggest that they are formed by spherical granules with a radius approximately equal to the height of the relief element (≈5 nm). Optical measurements can yield information on the doped layer structure. The absorption spectrum (Fig. 5) is shown in the energy range from 2 to 4.6 eV, which contains not only the resonant band of Ag particles but also the intrinsic absorption edge of silver. It can be seen that the absorption band peaks at បωm = 2.95 eV (λm = 420 nm) and has the half-width γ = 0.47 eV. The band is almost symmetric and the proﬁle is close to Lorentzian: γ -, D = D m ----------------------------------------------------2 2 2 4 [ ប ( ω – ω m ) + γ /4 ]
2

n 1.50 2 1.48

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1.46 0 1 2 3 x, mm

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Fig. 4. Dependences (1) n(x), (2) R(x), and (3) q(x).

600 1.5

λ, nm 500 450 400 350

300

270

1.0 2 1 D

0.10 D2 0.05
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0.5

0

2

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បω, eV

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where D = αh is the optical density, α is the absorption coefﬁcient, h is the thickness of the doped layer, and Dm = 1.49 is the maximum optical density. The Lorentzian proﬁle indicates homogeneous broadening of the absorption band. Obviously, inhomogeneous broadening also contributes to the bandwidth. However, this contribution can be estimated from the ﬁlling number q. Calculation by the Maxwell–Garnet formula shows that, at a change in the ﬁlling number in the range 0 ≤ q ≤ 0.15, the band peak shift is only 0.05 eV; i.e., the inhomogeneous broadening is an order of magnitude smaller than the homogeneous broadening (0.47 eV). A homogeneously broadened band can be due to very small isolated granules of almost identical size. In this case, the frequency ωm should obey the Fröhlich formula [12]; i.e., ωm = ωF, where ωF = ω p / ( ε M + 2 ε0 ) .
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Fig. 5. (1) Absorption spectrum and (2) the dependence D2(បω). Circles and solid lines show the experimental data. The dot-and-dash curve is the Lorentzian proﬁle. The longand short dashed curves show the proﬁles calculated by the Maxwell–Garnet formula at q = 0.05 and 0.1, respectively.

The exact value of the plasmon frequency ωp for Ag is unknown. According to the data of different researchers, the plasmon energy lies in the range បωp = 8.6– 9.1 eV [13]. To calculate បωF, we will take the average value បωp = 8.85 eV. The dielectric constant εM of the metal is related to interband transitions in Ag. According to the data of [13] εM = 4.5 for the energy បωm. The permittivity of silica for the same energy is ε0 = n 0 = 2.15. Calculation according to (4) for these data yields បωF = 2.98 eV, which is in good agreement with the value of 2.95 eV for the experimentally determined band peak position.

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The threshold energy is known for silver beginning with which absorption of light is due to direct interband transitions. The measurements we performed cover the interband absorption edge. Direct transitions suggest a linear dependence D2(បω), which is actually observed (Fig. 5). Intersection of the straight line with the energy axis gives the known value បωth ≈ 3.9 eV [2]. The coincidence of the experimental value of បωth with the threshold energy for bulk silver indicates the absence of quantum size effects despite the small size of silver particles. The granules precipitated in the silica should form a three-dimensional colloid. Therefore, to describe the band shape, it is also expedient to apply the well-known Maxwell–Garnet formula. This formula relates the effective permittivity εef of the bulk colloid with the complex permittivity ε of the metal in granules [2, 10]: ( ε ef – ε 0 ) / ( ε ef + 2 ε 0 ) = q ( ε – ε 0 ) / ( ε + 2 ε 0 ) , (5)

where q = NV is the average volume ﬁlling factor of the dielectric with granules and N and V are, respectively, the concentration and average volume of granules. The absorption band is almost entirely beyond the interband absorption region (Fig. 5). Therefore, the frequency dependence ε = ε1 – iε2 can be described by the Drude– Lorentz formula: ε1 = ε M ( ω ) – ω p / ( ω + ν ) , ε2 = ω p ν / ω ( ω + ν ) .
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To calculate ε1 and ε2, we used the above-mentioned value of បωp and took into account the dispersion of εM(ω) since, in the measured region, the value of εM(ω) ranges from 4.0 at បω = 2.00 eV to 5.3 at បω = 3.70 eV [13]. Expression (6) contains the electron scattering frequency ν, which signiﬁcantly differs from the corresponding bulk value in very small granules [12]. Taking into account that electron scattering occurs mainly at the boundaries of a granule of radius a, ν can be written as ν = ν0 + κ v F / a , (7)

Formulas (5) and (6) were used to derive expressions for ε1ef and ε2ef and calculate the optical constants nef and κef and the absorption coefficient αef = (2ω/c)κef, which determines the optical density D = αef h. The values of αef were calculated for q = 0.05 and 0.1. In both cases, maxima of αef (ω) were obtained at បω = 2.98 eV, which coincides with the Fröhlich energy calculated above and (within the calculation and measurement errors) the experimental value 2.95 eV. In comparison of the calculated and experimental data, it was assumed that Dmax = αmax, 1h1 = αmax, 2h2. Here, the subscripts 1 and 2 refer to q = 0.05 and 0.1, respectively, and h1 = 96.5 nm and h2 = h1/2 are the thicknesses of the surface doped layers for these values of q. The assumption that Dmax is an invariant independent of q is justiﬁed by the fact that the diffusion ﬂux of electrons and ions at the center of the irradiated region is directed mainly along the normal to the sample surface. It can be seen in Fig. 5 that the calculated dependences D1(ω) and D2(ω) almost merge, which is indicative of small contribution of the inhomogeneous broadening related to the dispersion of the ﬁlling number q at q ≤ 0.1. The calculated values of nef at ω < ωm are larger than n0 of silica. This fact indicates that, in the transparency region (បω ≤ 2 eV), the surface layer containing Ag granules is a planar asymmetric waveguide, irregular in thickness and mode indices. At ﬁrst glance, with allowance for the diffusion of electrons and Ag+ ions, the waveguide should be a gradient with a change in nef from the maximum value at z = 0 to n0 at large z (the z axis is directed along the normal into the silica bulk). However, the presence of interference colors on the irradiated area contradicts this suggestion. In addition, it is inconsistent with the data on the radial reﬂectance distribution (Fig. 4). For a gradient waveguide of this type in the transparency region, R is determined as [16] n ef ( z = 0 ) – 1 2 - . R = r 1 = ------------------------------n ef ( z = 0 ) + 1 (8)

where κ is a coefﬁcient that depends on the granule shape and is close to unity, vF is the velocity of electrons on the Fermi surface, and ν0 is the scattering frequency in silver bulk (according to the data of [14], vF = 2.3 × 108 cm s–1 and ν0 ~ 1013 s–1). Since ν determines the broadening of the colloid absorption band, we can assume that ν = γ/ប = 7.14 × 1014 s–1. At these values and κ = 1, we derive from (7) the average granule radius a ≈ 3 nm. Note that good agreement between formula (5) and the reﬂection and absorption spectra of spherical granules with a ≤ 10 nm was found in [15]; i.e., in the case under consideration, application of (5) is quite justiﬁed.

Calculation of r1 for λ = 633 nm at q = 0.1 gives r1 = 0.08, whereas, on the experimental dependence of R (Fig. 4), R > 0.1 for a number of points. For this reason, we suggested that the waveguide is not a gradient but has a step form with a narrow boundary between the doped layer and pure silica. Note that this form of a waveguide is known for the case of photodiffusion doping of As2S3 ﬁlms with silver [17]. At a step change in nef at the rear boundary (nef (z = h) = nef (z = 0)), the reflectance should obey the Airy formula. On the assumption of weak absorption (nef ӷ κef), this formula can be written as r 1 + r 2 η – 2 r 1 r 2 η cos 2 δ -, R = -----------------------------------------------------------------2 1 + r 1 r 2 η – 2 r 1 r 2 η cos 2 δ
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where r2 = (nef – n0)2/(nef + n0)2, δ = (ω/c)nef h, and η = exp(−αef h). Large changes in R(x) (Fig. 4) indicate signiﬁcant changes in nef, q, and h within the doped region. Since the values of R were measured using a focused beam with a small waist (≈30 µm), we can assume that each value of x corresponds to a plane-parallel layer of very small area, which satisﬁes formula (9). This formula was used to solve the inverse problem: the values of q(x) and h(x) were found by matching the values of R calculated by formula (9) with the measured values. To this end, according to (5) and (6), we calculated n for λ = 633 nm and q in the range from 0.03 to 0.1. Applying the previously used relation κef h = const at h = 96.5 nm, which was previously found at q = 0.05 from the value of Dmax at λ = 420 nm, we determined the radial distribution q(x) (Fig. 4). It can be seen that, within the doped region, the factor q changes from 0.04 to 0.085 with respect to the average value q ≈ 0.06. This range is in agreement with the previous estimate q = 0.05–0.1 made in the analysis of the resonant absorption band. Measurements of the radial distribution of the refractive index n(x) (Fig. 4) by the method proposed in [11] showed that n irregularly changes in the range 1.473–1.503 within the doped region and coincides with the value n = 1.457 for silica beyond this region. The values of n(x) are much smaller than the calculated values of nef: with a change in q from 0.03 to 0.1, n changes from 1.56 to 1.8. Therefore, we assumed that, in the case under consideration, the method proposed in [11] gives, not the value of nef, but the effective refractive index nTE for the waveguide TE0 mode. To verify this suggestion, we used the model of a two-layer step waveguide consisting of an AgCl–Ag ﬁlm with n1 = 2.06 (λ = 633 nm) and h1 = 30 nm and a doped layer silica with n2 ≡ nef and the thickness h2 located on a silica substrate. The dispersion equation for this case [18] can be written as tan ( kh 2 p 2 )
2 p 1 p 2 ( p 0 + p 3 ) + p 2 ( p 3 p 0 – p 1 ) tan ( kh 1 p 1 ) (10) -----------------------------------------------------------------------------------------------------, = 2 2 2 p 1 ( p 2 – p 3 p 0 ) + ( p 3 p 1 + p 2 p 0 ) tan ( kh 1 p 1 )

This example is an additional proof that the doped region is a step waveguide. CONCLUSIONS We found the possibility of doping thin (≈50– 100 nm) surface layers of optical silica glass with colloidal Ag (granule size 3 nm) by heating a thin Ag ﬁlm on the silica surface by a CO2 laser beam (λ = 10.6 µm, P ≈ 30 W). It is shown that the doped layer is an areairregular step waveguide for red and near-infrared spectral ranges. Note that the method considered here is valid specifically for fused silica since this material has a high melting temperature (1773°C) and a high absorption coefﬁcient (at λ = 10.6 µm); is heat-resistant upon local heating by a laser beam; and, ﬁnally, has a porous structure allowing formation of small Ag granules in such a matrix. With all that, the following conditions are very important: the initial Ag ﬁlm on the surface must be thin, have an island structure, and be heated by a beam to the melting temperature. Doped layers can be used for stable labeling and coloring of fused silica particles, fabricating waveguide paths, imprinting into silica holographic and another structures formed previously in a thin silver ﬁlm [19] on its surface, etc. ACKNOWLEDGMENTS We are grateful to A.P. Kryshtal’ for electron microscopy analysis of the doped samples. REFERENCES
1. R. A. Ganeev, A. I. Ryasnyanskiœ, A. L. Stepanov, et al., Opt. Spektrosk. 95 (6), 1034 (2003) [Opt. Spectrosc. 95, 967 (2003)]. 2. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, Berlin, 1995). 3. H. Hovel, S. Fritz, A. Hilder, et al., Phys. Rev. B: Condens. Matter Mater. Phys. 48, 18178 (1993). 4. W. Cai, H. Hofmeister, T. Rainer, and W. Chen, J. Nanopart. Res. 3, 443 (2001). 5. Encyclopedia of Inorganic Materials, I. M. Fe-dorchenko, Ed. (Glav. Red. Ukr. Sov. Éntsiklop., Kiev, 1977) [in Russian]. 6. M. Ya. Kruger, V. A. Panov, V. V. Kulagin, et al., Handbook for a Designer of Opticomechanical Instruments (Mashgiz, Moscow, 1963) [in Russian]. 7. Physics of Thin Films, G. Hass and R. E. Thun, Eds. (Acad. Press, London, 1967; Mir, Moscow, 1970), Vol. 4. 8. L. A. Ageev and Kh. I. Él’-Ashkhab, Zh. Prikl. Spektrosk. 60 (1–2), 152 (1994). 9. E. M. Voronkova, B. N. Grechushnikov, G. I. Distler, and I. P. Petrov, Optical Materials for IR Applications (Nauka, Moscow, 1965) [in Russian].

where k = 2π/ λ, p0 = ( n TE – 1)1/2, p1 = ( n 1 – n TE )1/2, p2 = ( n 2 – n TE )1/2, p3 = ( n TE – n 3 )1/2, and n3 ≡ n0 = 1.457 (n0 = 1.457 is the silica refractive index). For example, using (10), we calculated nTE for the average value q = 0.06 using the previously calculated values n2 ≡ nef = 1.673 and h2 = 80 nm, as well as the values of h1, n1, and n3. The calculation gave nTE = 1.472. For the central part of the doped region, this value, in comparison with nef, is in much better agreement with the average experimental value n = 1.487 at q = 0.06 (Fig. 4).
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AGEEV et al. 15. W. T. Doyle, Phys. Rev. B: Condens. Matter Mater. Phys. 39 (14), 9852 (1989). 16. Physics of Thin Films, G. Hass, M. Frankombe, and R. Hoffman, Eds. (Acad. Press, London, 1975; Mir, Moscow, 1978), Vol. 8. 17. V. A. Dan’ko, I. Z. Indutnyœ, A. A. Kudryavtsev, et al., Ukr. Fiz. Zh. 36 (6), 937 (1991). 18. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981; Moscow, Mir, 1984). 19. L. A. Ageev and V. K. Miloslavsky, Opt. Eng. 34 (4), 960 (1995).

10. G. V. Rozenberg, Optics of Thin-Layer Coatings (Glav. Izd. Fiz.-Mat. Lit., Moscow, 1958) [in Russian]. 11. L. A. Ageev, V. K. Miloslavskiœ, O. V. Tyutyunnik, and Kh. I. Él’-Ashkhab, Zh. Prikl. Spektrosk. 68 (2), 270 (2001). 12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983; Mir, Moscow, 1986). 13. V. K. Miloslavskiœ and R. G. Yarovaya, Opt. Spektrosk. 21, 708 (1966). 14. V. G. Padalka and I. N. Shklyarevskiœ, Opt. Spektrosk. 11 (1961).

Translated by Yu. Sin’kov

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