# L A Ageev - Spontaneous s_ gratings and specific features of their formation and evolution in photosensitive agcl_ag films - страница 2

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 micro-gratings are separated by narrow bright bands of about 1 |im 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 lancet-shaped microgratings reaches 30 | m and their width in the middle region is 3-5 |im. 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 X (about 10 nm). At a small filling number of the initial

OPTICS AND SPECTROSCOPY Vol. 100 No. 2 2006

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 x 4 Miiller 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

2 2 2

h, TE x cos a, Is TM °= sin a cos 6, (4)

where a and 6 are, respectively, the azimuthal and meridional angles determining the scattering direction: a = Z(P, x), 6 = Z(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 a and is maximum at a = 0, n. The intensity of light scattered to the TM mode is highest at a = ±n/2 and depends on 6. In our case, the angle 6 is fixed and related to the propagation constant of the TM mode by the expression [10]

Ptm = ^0 «tm = M sin 6, where k0 = 2п/ X.

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

Ip TE °= sin a cos \|/,

2

Ip, TM 00 ( cos a cos 6 cos у + sin 6 sin\|/) ,

(5)

where | 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 n0sin cp = n sin \|/, 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 (a = 0 or n). In the case of p polarization, so-called parquet (P) gratings [11], evolving on scattered TE modes at a = ±n/2, and S_ TM and S+ TM gratings (a = 0°, 6 > 0 and a = 180°, n/2 < 6 < n, respectively) can be considered to be dominant. In this case, the dependences of IA TE and I

p, TM

on

the angle of incidence can be used to determine the result of the competition of the S_ + and P gratings.

At cp Ф 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 (a) [kx + m(P cos a - kx)]i + mP sinaj,

(6)

where K(a) is the vector of a plane grating formed on the waveguide mode scattered at an azimuthal angle a; 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 - P cos a and, generally, kd Ф 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 a* = ± arccos (kx/P).

(7)

At a = a*, diffraction to the orders m = ±1 introduces waveguide modes on which degenerate C gratings [3,

5] with antiparallel vectors K = ±(P2 - kx2 )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 a* [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 a = 0, n/2, and a*, 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 nTE(0 = 1.63 and nTM0 = 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 c , which is consistent with the experimental data (Fig. 2a). In the case of p polarization, the situation is different (Fig. 5b). At c < 40°, the scattering to the TE0 mode at a = ±n/2 or a = a* is dominant. However, with increasing c , the intensity of scattering to the TM0 mode at a = 0 increases, whereas

the intensity of scattering to the PTE(0 modes at a = ±n/2

or a* decreases. In addition, according to (7), a* decreases with increasing c , which leads to a decrease in the intensities of the modes of C gratings even when the scattering indicatrix does not vary with c . Thus, at cp < 40°, the C and P gratings on TE0 modes should prevail, while, at c > 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 c = 25° (Fig. 2b) can

hardly be seen, whereas, at cp = 35° (Fig. 2c), its intensity is high and increases with increasing c .

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 anisotro-pic 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 = bo + mK (a) [P cos a0 + m(P cos a - kx)]i + (P sin a0 + m P sin a) j,

(8)

where b0 is the wave vector of the dominant mode, a0 is its azimuthal angle, a 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; a0 = 0, n; and a = n, 0 (diffraction of a mode

Fig. 5. Dependences of the relative intensities of scattered TE0 and TM0 modes on the angle of incidence of a laser beam: (a) s-polarized beam, formulas (4), and (b) p-polarized beam, formulas (5). For the film with nAgC[ = 2.06 on a substrate with ns = 1.515 in air, = 1.63, n-jM = 1.52, and 6 = 48° for TM0 modes. The curves for the S± gratings (azimuthal angles of the modes a =

я, 0) and P gratings (a = я/2) evolving on ordinary Wood anomalies and for the Cje and Cjm gratings azimuthal angles of the modes a* depend on cp, see formula (7)) are shown.

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 a0 = 0, я and a only slightly differing from either 0 or n, 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- TEo modes to neighboring S±_ TEo 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

Сте0 gratings, for which bo = kxi ± (P2 - kx2 )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 cp(nTEo -

sin2cp)-1/2; i.e., the convergence of the arcs 4 increases with increasing cp. In addition, these arcs weaken with increasing cp and disappear completely at cp > 45°, which indicates indirectly the disappearance of CTE0

gratings. However, at cp = 50°, the anisotropic scattering at small angles to the plane of incidence increases significantly (Fig. 2d, the flame 6). We assign this scattering to the dominant P gratings formed on TE0 modes with b0 Ф ±Pj. 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 tm 0 , Рте0 , and Ce0 gratings. At the

same time, the absence of the vertical flame characteristic of the s polarization of the laser beam at all c indicates the absence of S+_ TM0 gratings at all exposure times.

The absence of these gratings may be caused by the existence of CTE0 or PTE0 gratings at large c . As the electron micrographs show [11], the lines from regular CTE0 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 CTE0 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

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 a. Thus, at a = 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 a = 0 and я. This circumstance leads to competition between the spontaneous S_ TE(0 and S+_ TE0 gratings. At cp > 45°, in the case of p polarization, CTE0 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 a 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_ tMq 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 micro-

gratings 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 X = 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 X = 633 nm and q = 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 = X/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_, те0 gratings.

At the same time, similar calculation of d0(h) for spontaneous S_ TM0 gratings gives the best agreement