ISSN 0030-400X, Optics and Spectroscopy, 2006, Vol. 100, No. 2, pp. 291–299. © Pleiades Publishing, Inc., 2006. Original Russian Text © L.A. Ageev, E.D. Makovetskiœ, V.K. Miloslavskiœ, 2006, published in Optika i Spektroskopiya, 2006, Vol. 100, No. 2, pp. 328–336.

PHYSICAL OPTICS

Spontaneous S– Gratings and Specific Features of Their Formation and Evolution in Photosensitive AgCl–Ag Films
L. A. Ageev, E. D. Makovetskiœ, and V. K. Miloslavskiœ
Kharkov National University, Kharkov, 61077 Ukraine e-mail: evgeny.d.makovetsky@univer.kharkov.ua
Received February 22, 2005

Abstract—The formation and evolution of spontaneous S– gratings in photosensitive waveguide AgCl–Ag ﬁlms under the action of a laser beam are investigated experimentally and theoretically. It is found that in the cases of s and p polarizations of laser radiation these gratings differ signiﬁcantly not only in the period but also in the structure and spatial and temporal stability. The different character of the gratings is explained in detail by the competition between the S– gratings and other gratings evolving simultaneously in the ﬁlm and the change in the scattering indicatrix in the case of p polarization as a result of intense evolution of degenerate C gratings. The regularity of S– gratings on íå0 modes in the case of p polarization and large angles of incidence makes it possible to increase the accuracy of determining the refractive indices of substrates using AgCl ﬁlms corresponding to the cutoff thicknesses of TM0 waveguide modes. PACS numbers: 42.62 DOI: 10.1134/S0030400X06020214

INTRODUCTION As is known [1, 2], spontaneous gratings are formed on surfaces of solids irradiated with a high-power laser beam. The reasons for this phenomenon are the interference of an incident wave (with a wavelength λ) with the surface TM modes excited as a result of the light scattering and the mass transfer in the interference ﬁeld. At oblique incidence of an inducing beam, the type of a spontaneous grating depends on the beam polarization. In the case of linear p polarization, in a wide range of angles of incidence ϕ, so-called [3] S– and S+ gratings evolve with the following azimuthal angles α of the grating vectors: α = 0° and 180° (α is counted from the plane of incidence) and periods d S–, S+ = λ / ( 1 − + sin ϕ ) . (1)

in the nonlinear evolution stage were predicted theoretically in [6]. However, as far as we know, such oscillations have not been revealed experimentally. Spontaneous gratings have also been found in photosensitive ﬁlms with waveguide properties: Ag-doped AgCl ﬁlms (AgCl–Ag) [7]; AgBr–Ag, AgI, and As2S3 ﬁlms [8]; and photopolymer ﬁlms [9]. Here, we will restrict our consideration to AgCl–Ag ﬁlms since similar regularities in the evolution of spontaneous S– and S+ gratings have been observed for the ﬁlms of other materials. A thin AgCl ﬁlm deposited on a transparent substrate (glass) is an asymmetric waveguide, in which both TEm and TMm modes can propagate [10]. However, these ﬁlms are insensitive to visible light. To increase the ﬁlm sensitivity, silver is incorporated into the ﬁlm to precipitate as very small grains and form an absorption band peaked at 500 nm in the AgCl spectrum. Exposure of a ﬁlm to a low-power gas laser beam leads to the excitation of waveguide modes in the ﬁlm due to the light scattering by Ag grains. In the interference ﬁeld formed by the incident wave and a scattered mode, silver mass transfer to the interference minima occurs, which ﬁnally leads to the formation of a quasiperiodic structure composed of microgratings (spontaneous grating domains). The mechanism of silver mass transfer was described in [7, 8]. In contrast to spontaneous gratings on surfaces of solids, evolution of spontaneous gratings on scattered TE and TM modes is possible in ﬁlms. These gratings have different propagation constants depending on the photosensitive layer thick-

These spontaneous gratings are dominant because, in the case of p polarization, scattered TM modes have the largest amplitude at α = 0° and 180°. The dominance of spontaneous S– and S+ gratings is conﬁrmed by the calculation of the azimuthal dependence of the increment in their evolution due to positive feedback [4, 5]. However, the diffraction reﬂections in the given geometry are crescent-shaped rather than pointlike, which indicates the formation of S– like gratings evolving on the modes with α ≠ 0°. It was ascertained in [3] that, irrespective of the type of the irradiated material, at ϕ տ 35°, spontaneous S– and S+ gratings are attenuated and disappear due to the evolution of regular degenerate C gratings. Periodic oscillations of the amplitudes of spontaneous S– and S+ gratings due to their competition

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Fig. 1. Schematic diagram of the setup for generation and investigation of spontaneous gratings: (1) single-mode He–Ne laser (λ = 632.8 nm, P = 8 mW), (2) λ/2 plate on a vertical goniometer (for rotation of the plane of polarization), (3) collecting lens (F = 9.5 cm), (4) screen with an aperture for the laser beam, and (5) a sample on a horizontal goniometer.

ness; i.e., spontaneous gratings in photosensitive planar waveguides are characterized by a larger variety. Spontaneous S– and S+ gratings were ﬁrst found in thin (h < h TM0 , where h TM0 is the cutoff thickness of the TM0 mode) AgCl–Ag films exposed to He–Ne [7] and He–Cd [11] laser radiation. Spontaneous gratings arose in the case of an s-polarized laser beam, exciting scattered TE0 modes with periods similar to (1) in the ﬁlm: d S–, S+ = λ / ( n eff − + sin ϕ ) , (2)

EXPERIMENTAL TECHNIQUE We studied spontaneous S –, TE0 and S –, TM0 gratings in AgCl–Ag ﬁlms in the thickness range h TM0 ≤ h < h TE1 , where h TM0 , and h TE1 are the cutoff thicknesses of the corresponding modes: λ 2 ( ns – 1 ) ------------------------- , h TM0 = -------------------------------arctan n 1/2 2 2 2 2 1/2 ( n – ns ) 2 π ( n – ns ) h TE1 ( ns – 1 ) λ - arctan ------------------------= -------------------------------2 1/2 2 1/2 2 2 ( n – ns ) 2 π ( n – ns ) λ + -----------------------------, 2 1/2 2 2 ( n – ns ) where λ = 632.8 nm is the He–Ne laser wavelength. The cutoff thicknesses h TM0 = 94 nm and h TE1 = 273 nm were calculated using the tabular values of n = nAgCl = 2.06 and ns = 1.515 (glass). The photosensitive layer thickness was determined from the mass thickness calculated for the given geometry of material evaporation using a planar evaporator [16]. In addition, the photosensitive layer thickness was measured from the lines of equal monochromatic order by the Tolansky method [17]. AgCl–Ag ﬁlms were deposited in vacuum on cold substrates by successive evaporation of AgCl and Ag. The mass thickness of Ag was about 10 nm, which determined the ﬁlling number for Ag ﬁlms: q ≤ 0.1. A schematic diagram of sample exposure is shown in Fig. 1. A narrow He–Ne laser beam (P = 8 mW, the mode waist width at the output mirror w0 = 3 × 10−2 mm) passed through a quartz λ/2 plate mounted on a vertical goniometer. Then the beam passed through a collecting lens (F = 9.5 cm) and an aperture in a screen oriented perpendicularly to the beam direction, after which it fell on a sample mounted on a horizontal goniometer. At a distance of 60 mm between the laser and the lens, the width of the focused beam waist on the sample was wF = 62 µm. The illuminated spot at ϕ ≠ 0° has an elliptical form with an area SF(ϕ) = SF(0)sec ϕ, where SF(0) = 3 × 103 µm2. The diffraction from sponOPTICS AND SPECTROSCOPY Vol. 100 No. 2 2006
2 1/2 2 1/2

where neff is the effective mode index (here, neff = n TE0 ). Then, the unusual dynamics of S– and S+ gratings was revealed [12]: spontaneous S+ gratings, evolving in the initial stages of exposure, disappear with increasing exposure time and are replaced by S– gratings. The competition of spontaneous S– and S+ gratings becomes especially pronounced when the laser beam is focused [13]. The pattern of anisotropic light scattering observed in the direction opposite to the incident beam direction exhibits a system of randomly arising, moving, and disappearing spots, which indicate the evolution of some microgratings and the disappearance of other microgratings (nonlinear optical turbulence). It was found that this turbulence is related to the evolution of spontaneous S– gratings due to the suppression and elimination of S+ gratings. The competition observed is somewhat like that predicted in [6]; however, instead of periodic oscillations (according to [6]), random oscillations of spots related to leaky modes with an average frequency decreasing with increasing exposure time were observed in [13]. We observed S– gratings when a p-polarized beam was introduced into AgCl–Ag ﬁlms with h > h TM0 either through a prism [8, 14] or from air (at large angles of incidence) [15]. Optical microscopy measurements [15] indicate a signiﬁcant difference in the structure of S –, TE0 and S –, TM0 gratings. In this study, we performed a more detailed investigation of the formation and evolution of spontaneous S– gratings in ﬁlms exposed to s- and p-polarized laser beams to explain the difference noted.

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4 1 y x (c) 5 6 4 1 1 E0 kx (d) 5 1 kx E0

Fig. 2. Light scattering patterns on the screen (Fig. 1, 4) during the exposure of an AgCl–Ag ﬁlm: (a) s polarization, angle of incidence ϕ = 35°; (b) p polarization, ϕ = 25°; (c) p polarization, ϕ = 35°; and (d) p polarization, ϕ = 50°. Designations: (1) laser beam, (2) arc of diffraction of the laser beam to the order –1 by S –, TE -like gratings, (3) anisotropic scattering ﬂame related to the diffrac0

tion of S + − , TE modes by S ±, TE gratings, (4) anisotropic scattering arcs related to the dominance of C TE gratings, (5) arc of dif0 0 0

fraction of the laser beam to the order –1 by S –, TM -like gratings, and (6) ﬂame related to the existence of P TE -like gratings.
0 0

taneous S– gratings to the order m = –1 at ϕ > ϕ*, where ϕ* = arcsin [( n eff – 1)/2] (see [13]), was observed on the screen, which made it possible to investigate the spatial and temporal evolution of spontaneous S– gratings during the exposure. Since neff changes from ns to n, the least value ϕ* = 14°55′ is obtained at h = h TM0 and neff = n TM0 = ns. The periods of S– gratings at different h and ϕ were measured by autocollimation. The required (s or p) polarization of the inducing beam was obtained by rotating the λ/2 plate. At ϕ > 50°, the periods of S– gratings d S– > λ, which makes it possible to observe their spatial structure in an optical microscope (MII-4, observation in reﬂected light). EXPERIMENTAL RESULTS The patterns observed on the screen (Fig. 2; h = 100 nm; the exposure time t = 5 min; ϕ = 25°, 35°, and
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50°) demonstrate a significant difference in the S –, TE0 and S –, TM0 gratings. In the case of s polarization, the diffraction reﬂection 2 from the S –, TE0 grating (the order m = –1, Fig. 2a) is crescent-shaped and has a signiﬁcant angular spread in the transverse (horizontal) direction. In addition, a small-angle anisotropic scattering reﬂection 3 (a so-called ﬂame), intersecting the center of the laser beam 1, is observed along the y axis, which is perpendicular to the plane of incidence. The patterns obtained at ϕ = 25° and 50° are not shown here since they do not differ qualitatively. They also demonstrate a ﬂame and a crescent-shaped reﬂection corresponding to the diffraction to the order m = –1. The position of the reﬂection 2 depends on ϕ: at ϕ < 33°, it is located on the left of the beam 1. With an increase in ϕ, the reflection 2 shifts to kx (here, kx = i(2π/ λ)sin ϕ is the component of the wave vector of the laser beam that is parallel to the photosensitive layer); crosses the center of the beam 1; and, at ϕ > 33°, is located on the right

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400 2 380

1

360

80

120

160

200 h, nm

Fig. 3. Dependences of d0 on the thickness h of AgCl ﬁlms (d0 = λ/neff). Filled circles and the black curve show, respectively, the experimental values and the results of calculation by the dispersion equation for íå0 gratings (neff = 2.12). Open circles and the gray curve show the same for TE0 gratings (neff = 1.94).

of the beam (Fig. 2a). The intensity of the reﬂection 2 does not change. In the case of p polarization, the patterns obtained at different ϕ are signiﬁcantly different. At ϕ < 25°, the vertical diffraction reﬂection is absent; i.e., S –, TM0 gratings are not formed (Fig. 2b). However, one can see two anisotropic scattering arcs 4, passing through the centers of the initial and reﬂected beams. It will be shown below that the appearance of the arcs 4 is related to the formation of dominant C TE0 gratings. At ϕ = 35° (Fig. 2c), the reﬂection 5 corresponding to the diffraction to the order m = –1 can be clearly seen simultaneously with the attenuation of the arcs 4 from the C gratings. At the points of intersection of the arcs 4 and 5 (Figs. 2b, 2c), a signiﬁcant increase in brightness is observed, which corresponds to the enhancement of certain S–-like gratings. The reasons for this enhancement were discussed in [18]. At ϕ = 50° (Fig. 2d), the arcs 4 disappear and a weak scattering reﬂection 6 extended along the kx axis becomes pronounced. This scattering reﬂection exists at all values of ϕ; however, it is weak against the background of the bright arcs 4. At the same time, the scattering reﬂection extended along the y axis is absent at all values of ϕ. Measurement of the periods of S– gratings in the two polarizations gives ds = 560 nm and dp = 660 nm at ϕ = 35°. These results, according to (2) and the calculation of neff by the dispersion equations [10], indicate the formation of gratings in the cases of s and p polarizations on the TE0 and íå0 modes, respectively.

The time evolution of different S– gratings is also different. With an increase in the exposure time, the diffraction reﬂection 2 from TE0 gratings shifts signiﬁcantly in the kx direction, which indicates an increase in the period of the spontaneous íÖ0 gratings with time. In this case, the angular width of the reﬂection narrows from several degrees to 30′. In the initial stage of exposure, the ﬂame 3 demonstrates a strong optical turbulence. With increasing exposure time, the frequency of random oscillations of spots in the pattern decreases and the small-angle scattering reﬂection 3 in the counterpropagating beam decreases to complete disappearance. These results are similar to those obtained in [13], where this character of evolution was explained. The character of the evolution of the reﬂection 5 from S –, TM0 gratings is different: long-term exposure does not lead to a signiﬁcant shift or change in the halfwidth of this reﬂection; only its length slightly decreases. The dependences of the periods d on the ﬁlm thickness h for the S –, TE0 and S –, TM0 gratings were measured at ϕ = 40° in the range of h from 70 to 180 nm, which corresponds to the existence of TE0 and TM0 modes. Since the period of S –, TE0 gratings depends on the exposure time, we measured the limiting values of d obtained at long exposure times. Figure 3 shows the values d0 = λ/neff, where d0 = d[1 + (d/ λ)sin ϕ]–1. A systematic decrease in d0 with increasing h is observed. Figure 3 also shows the dependences d0(h) calculated by the dispersion equations. The calculation method is considered below. The micrographs in Fig. 4 were obtained at ϕ = 70° since the periods of spontaneous S– gratings are much larger than λ at this angle of incidence (the observation and photographing were performed in white light). The illuminated spot had the form of an ellipse extended in the kx direction, with axes of 60 and 170 µm (Fig. 4c). A portion of the spot with an area of 80 × 60 µm2 near the spot center was photographed. One can see a significant difference in the spontaneous S –, TE0 and S –, TM0 gratings not only in periods but also in shape. On the whole, the quasi-periodic structure of spontaneous S –, TE0 gratings (Fig. 4a) consists of individual small microgratings extended in the kx direction, with an average number of lines of about 10–15. Some microgratings are up to 10 µm in length and 3 µm in width. The vectors K of the microgratings have a spread with respect to the dominant direction, perpendicular to E0; they are separated by fairly wide bright areas, where their evolution is signiﬁcantly reduced. The microgratings are poorly matched with each other in phase. The quasi-periodicity of the structure of S –, TM0 gratings (Fig. 4b) is more pronounced. In the middle of the
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E0 (c) 80 × 60 µm

E0

kx

Fig. 4. Micrographs of spontaneous gratings obtained in an MII-4 optical microscope in reﬂected white light. The gratings were formed at an angle of incidence ϕ = 70° in (a) s polarization ( S –, TE -like gratings can be seen) and (b) p polarization ( S –, TM -like
0 0

gratings can be seen); (c) the position of the portion of the irradiated spot that was photographed.

illuminated spot, some microgratings have the form of narrow, often lancet-shaped, bands, which are signiﬁcantly extended in the kx direction. Neighboring microgratings are separated by narrow bright bands of about 1 µm in size. The number of lines of the microgratings varies from 20 to 30; their vectors K are oriented predominantly parallel to E0. The length of some lancetshaped microgratings reaches 30 µm and their width in the middle region is 3–5 µm. At the edges of the photograph, close to the transverse boundaries of the illuminated spot, the micrograting axes deviate from the plane of incidence by 10°–20°, with increasing tilt angle as the microgratings approach the boundaries. In addition, in contrast to the central region of the spot, the grating lines on the periphery are not perpendicular to the grating axes. Also, the period of spontaneous gratings at the spot periphery decreases on average to 0.9 with respect to the period in the central area of the spot. DISCUSSION To establish the conditions for the evolution of spontaneous S– gratings on scattered TE0 and TM0 modes, one has to know the indicatrices of scattered radiation. Generally, these indicatrices in a waveguide ﬁlm are rather complicated. They change during the formation and evolution of spontaneous gratings due to the redistribution of Ag grains, which are the main scattering centers in AgCl–Ag ﬁlms. Here, we will restrict ourselves to a simple case: light is scattered by small spherical Ag grains with a diameter much smaller than λ (about 10 nm). At a small filling number of the initial
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ﬁlm (q ≤ 0.1), we can neglect the multiple scattering and consider one-particle scattering. In this case, the indicatrices can be calculated using the conventional theory of scattering by small particles [19], with the 4 × 4 Müller matrix. Since we are interested in the angular distribution of the scattered light intensity, we do not take into account its radial distribution in the waveguide layer. Determination of the radial distribution is an independent problem [20, 21]. For an s-polarized incident beam (the polarization vector E0 || y lies in the plane (x, y) of the photosensitive layer), the angular intensity distribution of the light scattered to the TE and TM modes in the far zone is I s, TE ∝ cos α ,
2

I s, TM ∝ sin α cos θ ,

2

2

(4)

where α and θ are, respectively, the azimuthal and meridional angles determining the scattering direction: α = ∠(b, x), θ = ∠(ks, z) (the z axis is perpendicular to the photosensitive layer); ks is the wave vector of the scattered wave; and b is the wave vector of the excited mode (equal to the tangential component ks). It follows from (4) that the scattering to the TE mode depends only on α and is maximum at α = 0, π. The intensity of light scattered to the TM mode is highest at α = ±π/2 and depends on θ. In our case, the angle θ is ﬁxed and related to the propagation constant of the TM mode by the expression [10] β TM = k 0 n TM = k 0 n sin θ , where k0 = 2π/ λ.

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For a p-polarized incident beam (E0 is in the plane of incidence (x, z)), the angular distribution of scattered light is set by the relations I p, TE ∝ sin α cos ψ , I p, TM ∝ ( cos α cos θ cos ψ + sin θ sin ψ ) ,
2 2 2

(5)

where ψ is the angle of refraction of the beam in the ﬁlm. In contrast to the case of s polarization, the intensity of scattered light depends on the angle of incidence since n0 sin ϕ = n sin ψ, where n0 is the refractive index of the medium surrounding the photosensitive layer on the glass plate. Indicatrices (4) and (5) determine the dominance of particular spontaneous gratings in the initial stage of their formation. In the case of s polarization, S–, TE and S+, TE gratings are the dominant ones (α = 0 or π). In the case of p polarization, so-called parquet (P) gratings [11], evolving on scattered TE modes at α = ±π/2, and S–, TM and S+, TM gratings (α = 0°, θ > 0 and α = 180°, π/2 < θ < π, respectively) can be considered to be dominant. In this case, the dependences of Ip, TE and Ip, TM on the angle of incidence can be used to determine the result of the competition of the S–, + and P gratings. At ϕ ≠ 0°, the dominance of gratings is determined not only by the indicatrices (4) and (5). Let us consider the diffraction from plane spontaneous gratings forming during illumination of a sample by a laser beam. In this case, the tangential component kd of a diffracted wave is kd = k x + m K ( α ) = [ k x + m ( β cos α – k x ) ] i + m β sin α j , (6)

where K(α) is the vector of a plane grating formed on the waveguide mode scattered at an azimuthal angle α; m = ±1, ±2 … is the diffraction order; and i and j are the unit vectors of the x and y axes, respectively. For the order m = 1, we have kd = b; i.e., the grating introduces a mode whose interference with the incident beam causes the evolution of this mode. Thus, all spontaneous gratings evolve as a result of positive feedback. However, if m = –1 in (6), kdx = 2kx – β cos α and, generally, kd ≠ b. At m = –1, the diffraction may excite leaky modes into air (if kd < k0), leaky modes into the substrate (if k0 < kd < k0ns), and evanescent modes (if kd > k0ns). The only exception is the azimuthal angles α * = ± arccos ( k x / β ) . (7) At α = α*, diffraction to the orders m = ±1 introduces waveguide modes on which degenerate C gratings [3, 2 5] with antiparallel vectors K = ±(β2 – k x )1/2j are formed. Since C gratings evolve on double Wood anomalies (i.e., waves of two diffraction orders propagate along the grating), the increment in their evolution

is much larger than the increment of spontaneous gratings evolving at azimuthal angles close to α* [5, 22]. This circumstance determines the regularity of C gratings. Figure 5 shows the dependence of the relative intensities of scattered light on the angle of incidence of a laser beam (formulas (4) and (5)) for the azimuthal angles α = 0, π/2, and α*, corresponding to the formation of dominant S–, P, and C gratings. The calculation was performed at the same values nAgCl = 2.06 and ns = 1.515 at which the cutoff thicknesses of the waveguide modes (3) were determined above and at the values n TE0 = 1.63 and n TM0 = 1.52, corresponding to the experimental conditions. It can be seen from Fig. 5a that, in the case of s polarization, S –, TE0 and S +, TE0 gratings should be dominant at all values of ϕ, which is consistent with the experimental data (Fig. 2a). In the case of p polarization, the situation is different (Fig. 5b). At ϕ < 40°, the scattering to the TE0 mode at α = ±π/2 or α = α* is dominant. However, with increasing ϕ, the intensity of scattering to the íå0 mode at α = 0 increases, whereas the intensity of scattering to the P TE0 modes at α = ±π/2 or α* decreases. In addition, according to (7), α* decreases with increasing ϕ, which leads to a decrease in the intensities of the modes of C gratings even when the scattering indicatrix does not vary with ϕ. Thus, at ϕ < 40°, the C and P gratings on TE0 modes should prevail, while, at ϕ > 40°, S –, TM0 gratings should be dominant. This analytical result is in qualitative agreement with the experiment: the diffraction reﬂection 5 with m = –1 from S –, TM0 gratings at ϕ = 25° (Fig. 2b) can hardly be seen, whereas, at ϕ = 35° (Fig. 2c), its intensity is high and increases with increasing ϕ. Due to the short period of C and P gratings, diffraction from these gratings to the order –1 is not observed. However, their existence is conﬁrmed by the anisotropic scattering patterns on the screen. Anisotropic (small-angle) scattering, as was shown in [8], is the result of the diffraction of waveguide modes excited on the dominant gratings with the vector K0 and on the neighboring gratings with the vector K only slightly differing from K0. In this case, leaky modes arise (leading to the occurrence of ﬂames on the screen) with the tangential (in the layer plane) components kr = b0 + m K ( α ) = [ β cos α 0 + m ( β cos α – k x ) ] i + ( β sin α 0 + m β sin α ) j , where b0 is the wave vector of the dominant mode, α0 is its azimuthal angle, α is the azimuthal angle of the waveguide mode forming a neighboring grating, and m = ±1. According to (8), for the values of the parameters m = 1; α0 = 0, π; and α = π, 0 (diffraction of a mode
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SPONTANEOUS S– GRATINGS AND SPECIFIC FEATURES Is 1.0 (a) S±, TE0 CTE0 0.8 0.8 Ip 1.0 (b) PTE0

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0.6 PTM0 0.4

0.6 S±, TM0 0.4 CTM0 0.2 CTE0 CTM0 60 80 ϕ, deg 0 20 40 60 80 ϕ, deg

0.2

0

20

40

Fig. 5. Dependences of the relative intensities of scattered TE0 and TM0 modes on the angle of incidence of a laser beam: (a) spolarized beam, formulas (4), and (b) p-polarized beam, formulas (5). For the ﬁlm with nAgCl = 2.06 on a substrate with ns = 1.515 in air, n TE = 1.63, n TM = 1.52, and θ = 48° for TM0 modes. The curves for the S± gratings (azimuthal angles of the modes α =
0 0

π, 0) and P gratings (α = π/2) evolving on ordinary Wood anomalies and for the C TE and C TM gratings azimuthal angles of the modes α* depend on ϕ, see formula (7)) are shown.
0 0

from an S– grating on an S+ grating, and vice versa), kr = –kxi; i.e., a leaky mode arises in the direction opposite to the laser beam direction. At α0 = 0, π and α only slightly differing from either 0 or π, the spots from leaky modes appearing on the screen form anisotropic scattering arcs extended along the y axis. The tangents to these arcs at the point on the laser beam path corresponding to kr = –kxi are parallel to the y axis. Thus, in the case of s polarization, the scattering related to the penetration of S − + , TE 0 modes to neighboring S ±, TE 0 gratings is observed on the screen. This scattering is accompanied by turbulence and gradual disappearance of S+ gratings (see [13] for details). In the case of p polarization, the anisotropic scattering arcs 4 (Figs. 2b, 2c) are related to the dominant 2 C TE0 gratings, for which b0 = kxi ± (β2 – k x )1/2j. At m = 1, the tangents to the arcs at the point kr = –kxi are tilted 2 to the plane of incidence at an angle of − + sin ϕ ( n TE – sin2ϕ)–1/2; i.e., the convergence of the arcs 4 increases with increasing ϕ. In addition, these arcs weaken with increasing ϕ and disappear completely at ϕ > 45°, which indicates indirectly the disappearance of C TE0 gratings. However, at ϕ = 50°, the anisotropic scattering at small angles to the plane of incidence increases sigOPTICS AND SPECTROSCOPY Vol. 100 No. 2 2006
0

niﬁcantly (Fig. 2d, the ﬂame 6). We assign this scattering to the dominant P gratings formed on TE0 modes with b0 ≠ ±βj. In this case, the mean tangent to the arcs (at the point where they intersect the laser beam) lies in the plane of incidence. The anisotropic scattering spread is related to the irregularity of the P gratings. Thus, the results of the analysis of the patterns on the screen are implicitly in agreement with the calculation of the intensity of the radiation scattered to the modes generating S –, TM0 , P TE0 , and C TE0 gratings. At the same time, the absence of the vertical ﬂame characteristic of the s polarization of the laser beam at all ϕ indicates the absence of S +, TM0 gratings at all exposure times. The absence of these gratings may be caused by the existence of C TE0 or P TE0 gratings at large ϕ. As the electron micrographs show [11], the lines from regular C TE0 gratings are formed by individual silver grains extended in the kx direction. In the case of incidence of p-polarized light, along with TE0 modes, which facilitate the evolution of C TE0 gratings, light scattering from individual lines occurs. In the ﬁrst-order approximation, a chain of extended grains forming a line can be approximated by a cylinder with a permittivity different

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from that of AgCl, oriented parallel to the x axis. When the vector of the electric ﬁeld of the incident beam lies in the plane of incidence (p polarization), the front of the wave scattered from the cylinder has the form of a cone moving along the line axis in the far zone [19]. The wave vector of scattered light (normal to the plane that is tangent to the cone at a given point) always has a component ksx = kxi at any azimuthal angle α. Thus, at α = 0, scattering to the TM0 mode along the kx direction occurs. This scattering facilitates the evolution of spontaneous S –, TM0 gratings and impedes the formation of spontaneous S +, TM0 gratings. In the case of s polarization, C gratings are not formed, which leads to equiprobable scattering of TE0 modes at azimuthal angles α = 0 and π. This circumstance leads to competition between the spontaneous S –, TE0 and S +, TE0 gratings. At ϕ > 45°, in the case of p polarization, C TE0 gratings vanish; however, irregular P gratings arise, whose lines are inclined at angles of 20°–30° and –20° to –30° to the plane of incidence. The relatively small angle of inclination of these gratings also facilitates scattering of TM0 modes at azimuthal angles α close to 0, i.e., the preferred evolution of spontaneous S –, TM0 gratings (Fig. 2d). It is now clear why the structures in the photographs of portions of spontaneous S –, TE0 and S –, TM0 gratings in Figs. 4a and 4b differ radically. The small length of S –, TE0 gratings and large gaps between them are the result of their competition during evolution with S +, TE0 gratings, which ﬁll these gaps and cannot be resolved in an optical microscope because of their very short periods. At the same time, the absence of S +, TE0 gratings in the case of p polarization facilitates a freer evolution of S –, TM0 microgratings in the kx direction. The microgratings have a large length in the kx direction, which is limited only by their competition with neighboring microgratings evolving in the same direction but formed at other scattering centers. As was mentioned above, the diffraction reﬂection from spontaneous S –, TE0 gratings shifts in the kx direction with increasing exposure time due to an increase in the period of S –, TE0 gratings. This effect was revealed for S –, TE0 gratings in [13] and studied in detail in [23] for AgCl–Ag ﬁlms at normal incidence of the inducing beam with λ = 633 nm. It was ascertained in [23] that, before the formation of spontaneous gratings, the refractive index of a composite AgCl–Ag ﬁlm, due to the presence of a strong colloidal absorption band of Ag at 500 nm, exceeds the value of nAgCl and becomes as high as 2.4 at λ = 633 nm and q Ӎ 0.3. With an increase in the exposure time, the period increases by a factor of 1.25 due to a decrease in n. The reason for this phenomenon is as

follows: during the formation of spontaneous gratings, silver precipitates in minima of the interference ﬁeld, and the effective exponent of the TE0 mode is determined primarily by the refractive index in the interference maxima, which tend to be free of silver. The values of d0 = λ/neff shown in Fig. 3 were obtained at long exposure times (1 h or more). The value of neff found by the dispersion equation was used to calculate the dependence d0(h), which is in good agreement with experiment at n = 1.94. The smaller value of n in comparison with nAgCl = 2.06 is related to the porosity of AgCl ﬁlms in interference maxima, caused by the dissolution of Ag grains in the lines of spontaneous S +, TE0 gratings and their precipitation in the lines of spontaneous S –, TE0 gratings. At the same time, similar calculation of d0(h) for spontaneous S –, TM0 gratings gives the best agreement with experiment at n = 2.12. In our case, in contrast to [23], the ﬁlling number for a ﬁlm containing silver is q Ӎ 0.1, which yields (at λ = 633 nm) a refractive index of the initial ﬁlm equal to 2.18. The small difference in n before and after the exposure indicates the conservation of Ag grains in the maxima of interference of TM0 modes with the incident beam and explains the absence of a signiﬁcant shift of diffraction reﬂections from spontaneous S –, TM0 gratings during the exposure. Simultaneous existence of C TE0 and P TE0 gratings is the reason for the small variation in n. At a ﬁxed exposure time, the period of spontaneous gratings depends on the laser beam intensity. This fact was established in [23] in measuring the radial dependence of the period of spontaneous gratings upon illumination by an extended Gaussian beam. A signiﬁcant decrease in the period of íÖ0 gratings and, accordingly, an increase in n with increasing distance from the beam center were revealed. The increase in n at the periphery of the waist of a focused Gaussian beam was explained by the inclination of the axes of S –, TM0 microgratings and the decrease in their period (Fig. 4b). Indeed, on the slope of the Gaussian beam intensity, where the derivatives |dI/dy | and dn/dy have the largest values, a TM0 mode propagating along peripheral regions will shift to the beam periphery due to refractive index gradient. CONCLUSIONS An analysis of the formation and evolution of spontaneous S– gratings under the action of s- and p-polarized laser beams revealed a signiﬁcant difference between the gratings not only in the period but also in the structure and the spatial and temporal stability. The evolution of S –, TE0 gratings in the case of s polarization is controlled by their competition with S +, TE0 gratings, which determines their instability during the ﬁlm expoOPTICS AND SPECTROSCOPY Vol. 100 No. 2 2006

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sure, whereas the evolution of S –, TM0 gratings in the case of p polarization depends on the existence of C TE0 and P TE0 gratings. At all angles of incidence of a p-polarized beam, S +, TM0 gratings are absent, which leads to a better stability and structure of S –, TM0 gratings in comparison with S –, TE0 gratings. As was shown in [24], the use of S– gratings formed at the cutoff thicknesses of TE0 modes makes it possible to determine with a high accuracy the refractive indices of substrates in a wide range (up to 2.5). The better structure of S –, TM0 gratings, which manifests itself in narrower diffraction reﬂections and better stability, makes it possible to increase the accuracy in determining the refractive index n of substrates using AgCl ﬁlms formed at the cutoff thicknesses of waveguide TM0 modes. REFERENCES
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Translated by Yu. Sin’kov

OPTICS AND SPECTROSCOPY

Vol. 100

No. 2

2006