NDT.net • July 2004 • Vol. 9 No.07 
Optical Nondestructive Method for Measuring Inhomogeneously Distributed Parameters in Semitransparent SamplesA.E.MartirosyanInstitute for Physical Research, Armenian Academy of Sciences, Ashtarak2, 378410, Armenia, email: armar@ipr.sci.am PACS cods: 07.60.j; 61.72.y; 61.72.Dd; 61.80.Fe; 78.20.a; 78.55.m. Corresponding Author Contact: Email: armar@ipr.sci.am AbstractIn the present paper, the diagnostic technique for volumetric analysis of inhomogeneous materials, as well as for investigating samples subjected to any kinds of affections is described. The set of equations which characterize spatially propagations of counter laser beams through an arbitrary semitransparent sample allows one to obtain the distributions of doped or impurity ions, scattering at 900 and extinction coefficients. The experimental study of the LiF crystal with Fcenters produced by electron beams is described.Key words: inhomogeneous, semitransparent, local, density, scattering, extinction. Optical methods are widely used for nondestructive investigations of various materials [1]. If a sample is homogeneous, an exact study of its internal regions is easy to perform, because the light attenuation obeys an exact exponential dependence on the passed distance, and, therefore, the testing beam intensity in an arbitrary internal point is known. But if a sample is subjected to affections (by laser or electron beams, thermal, chemical, xray, etc.) [2,3], its internal regions can loss a spatial homogeneity. The volumetric investigation of inhomogeneous semitransparent samples faces some difficulties of principles, because their internal light attenuation factors are unknown. Methods of counterpropagating beams allow to obtain exact quantities of inhomogeneously distributed parameters of the atmosphere [4,5], and optical materials [6]. On the basis of this technique, the optical device has been designed. In [6], because of small optical densities of presented crystals along their transverse directions, attenuation factors of fluorescence and scattered beams along these directions have been neglected. This paper presents the solution of the set of equations describing a propagation of counter laser beams through a sample for the general case, i.e. for arbitrary semitransparent materials (crystals, glasses, minerals, liquids, etc.) with arbitrary external forms, optical densities and with arbitrary distributions of internal parameters. Obtained formulas can be used for volumetric investigations of a) semitransparent materials, and b) samples which have been subjected to any kinds of affections At first, let us obtain the extinction coefficient within two arbitrary points A and B in a semitransparent sample with an any external form and with fluorescing ions or colored centers (Fig.1a,b).
Let the laser beam with a wavelength l and power I_{l} propagates along MABN and excites ions. The observation of ionic fluorescence (at the wavelength L ) from local regions centered at A and B is performed at small aperture angles along arbitrary directions AA_{1} and BB_{1}. Then, if the spectral bandwidth of the ionic excitation line is much wider than the line width of the laser radiation, the fluorescence powers from these regions are described by following Eqs. [1]: F_{L} ( AA_{1})= s r I_{l} N(A)d(AA_{1})[ a^{2}( AA_{1})/16][1R(M)][1R (A_{1})]T_{l} (MA) T_{L} ( AA_{1}) (1) F_{L} ( BB_{1})= s r I_{l} N(B)d(BB_{1})[ a^{2} (BB_{1})/16][1R(M)][1R (B_{1})]T_{l} (MA) T_{l} (AB) T_{L} ( BB_{1}) (2) Here R(M), R (A1) and R (B_{1}) are the light reflection coefficients from the sample's surface at M, A_{1}, B_{1}, respectively, T_{l} (MA), T_{l} (AB), T_{L} (AA_{1}) and T_{L} (BB_{1}) are the light transmission factors at the wavelength or within corresponding points, N(A) and N(B) are the ionic densities in local regions, d(AA_{1}) and d(BB_{1}) are the lengths of fluorescence intervals at A and B, which are observed instantaneously by a recording system along AA_{1} and BB_{1}, a (AA_{1}) and a (BB_{1}) are the observation aperture angles by recording system (if a is small, a 2/16 is the spatial portion of fluorescence radiated in the observing direction), a is the absorption cross section of ions at l , r is the ratio of the photon appearance probability at L and the total probability of the excited level decay of ions. Now let the counter laser beam with the wavelength and power I * propagates through a sample along the opposite direction NBAM. Then, the fluorescence powers at emitted from local regions centered at A and B are given by: F_{L} *( AA_{1})= s r I_{l} *N(A)d(AA_{1})[ a^{2} ( AA_{1})/16][1R(N)][1R (A_{1})]T_{l} (NB) T_{l} (AB) T_{L} ( AA_{1}) (3) F_{L} *( BB_{1})= s r I_{l} *N(B)d(BB_{1})[ a^{2} (BB_{1})/16][1R(N)][1R (B_{1})] T_{l} (NB) T_{L} ( BB_{1}) (4) Where R(N) is the light reflection coefficient from the sample's surface at N, T_{ l} (NB) is the light transmission factor at l within N and B. After multiplication of Eq. (2) by (3) and division by (1) and (4), we obtain the extinction coefficient k_{l} (AB)=ln[T_{l} (AB)/l(AB) between local regions centered at A and B ( l(AB) is the length of AB): (5) Therefore, for calculation of absolute values of the extinction coefficient between two internal points by using Eq.(5), only the distance between these points and relative fluorescence powers should be measured. If the observation of fluorescence is performed along AA_{2} (A_{2} is a point where the straight line AA_{1} is crossed with the sample's surface on the opposite side, see Fig.1) when A is illuminated by a counter laser beam, then a( AA_{1}), d(AA_{1}), R (A_{1}) and T_{L} (AA_{1}) in Eq.(3) should be replaced by a (AA_{2}), d(AA_{2}), R (A_{2}) and T_{L} (AA_{2}), respectively. In this case, after multiplication of Eq.(1) and revised Eq.(3), we find: (6) where m(A)= s r a ( AA_{1}) ( AA_{2})[d(AA_{1}) d(AA_{2}) I_{L} I_{l} *]^{1/2} /16, T_{L}'(A_{1}A_{2})=T_{L} (A_{1}A2)[1R (A_{1})][1 R(A_{2})]=T_{L} (AA_{1})T_{L} (AA_{2})[1R(A_{1})][1R(A2)], and T_{l}'(MN)=T_{l} (MN)[1R(M)][1R(N)]=T_{l} (MA) T (AB) T_{l} (BN)[1R(M)][1R(N)] are the light total transmission factors between AA_{1} and MN which include the light attenuation due to reflections on sample's surfaces: these factors are equal to ratios of passed (through a sample) and incident laser beams powers. It is obvious, that all values in the right part of Eq. (6) can be measured or calculated. Therefore, we can obtain the absolute ionic density at an arbitrary point of the sample. Let us note, that for obtaining the distribution of relative local densities of ions, the measurement of some parameters is not required. Certainly T_{l}'(MN)=const, when we register a fluorescence from any points on MN. If fluorescence beams are observed at conditions =const, and d=const, then m(A) in Eq.(6) is also constant. Therefore, by measuring only relative values of F_{L} (AA_{1}), F_{L} *(AA_{1}) and T_{L}'(A_{1}A_{2}), one can find a relative ionic distribution along MN. It is also easy to derive a formula similar to (6) for local values of the scattering coefficient at an angle of 90°  P_{l} (A, p/2), when beams which are scattered by inhomogeneties and crystal defects at the right angle with respect to MN are registered. Let the observation of local regions centered at A and B is performed along AA_{1}' (see Fig.1a ) and BB_{1}' which form right angles respect to MN. In this case, F_{L} (AA_{1}), F_{L} *(AA_{1}) F_{L} (BB_{1}) and F_{L} *(BB_{1}) in Eqs. (14) should be obviously replaced by F_{l} (AA_{1}'), F_{l} *(AA_{1}') F_{l} (BB_{1}') and F_{l} *(BB_{1}'), respectively, i.e. by powers of scattered radiation at the right angle. Besides that, s r N a ^{2}/16 should be replaced by P _{ l}p a ^{2}/4 [7] (where p a ^{2}/4 is the observing solid angle, if aperture angles are small). After transformation of obtained equations, it is easy to find k_{ l} (AB) and P_{ l} (A, /2): (7) (8) Here n(A)= p a ( AA_{1}') a (AA_{2}')[d(AA_{1}') d(AA_{2}') I_{l} I_{l} *]^{1/2}/4, and A_{2}' is the point where the straight line AA_{1}' is crossed with a sample's surface on the opposite side (see Fig.1a), T'(A_{1}'A_{2}') is the light total transmission factors between A_{1}' and A_{2}'. P_{l} (A, p /2) is actually one of the parameters characterizing the optical quality of sample in local regions. As an example, let us present investigation results of a LiF cubic crystal with Fcenters. The wide emission spectrum of colored LiF crystals is used for laser generation [8,9] and for amplification of femtosecond laser pulses [10]. Fcenters in the crystal have been produced by irradiation of electron beams with the energy of 7 Mev. The examination of this crystal is performed by using a modified optical device with the micrometric spatial resolution [6]. The crystal was irradiated by a second harmonic (SH) beam (l =532 nm) of cw Nd:YAG laser with a power of 50 mW. The observation of scattering at 532 nm and fluorescence from Fcenters at 633 nm is performed at the right angle respect to laser beams (Fig.2). After being reflected by mirrors M1 and M2, the fluorescence and scattered beams from the local region LR fall on the scanning lenses L1 and L2. These lenses moving along the direction of laser beams collect the scattering and fluorescing light from various local sample's regions on micrometric slits and light guides, along which the light enters a recording system [6]. During the scanning time, two channels of a recording system collect 4 data sets: a) beams passed through the lens L1 when we irradiate the local region by laser first beam (F1); b) beams passed through the lens L1 when we irradiate the local region by counter laser beam (C1); c) beams passed through the lens L2 when we irradiate the local region by laser first beam (F1); d) beams passed through the lens L2 when we irradiate the local region by counter laser beam (C1). These data sets are used for calculation of local parameters of the crystal by Eqs.(58). Experimental data were processed with a computer. During the scanning time, irradiated local regions of the crystal should be located on the object plane of lenses L1 and L2. For calculation of the relative distribution of Fcenters and scattering coefficient at the right angle by Eqs. (6) and (8), two channels should be used (data sets F1 and C2, or C1 and F2). Because m(A) and n(A) are constant along a fixed direction of counterpropagating laser beams, the relative distributions of N(A) and P_{l} (A, p/2) along this direction can be calculated by measuring only relative powers of fluorescence or scattered beams and light total transmission factors in transverse directions (see Fig.2). Total transmission factors along these directions is obtained by using SH Nd:YAG (l =532nm) and HeNe (L =633nm) lasers.
The Fig.3 illustrates parameters distributions along the direction of counter laser beams. Because the irradiation of the crystal by electron beams have been performed from one of its surface, regions with high Fcenter densities is located no far from this surface. The absolute error of the extinction coefficient is 0.1 cm^{1}. The relative error of fluorescence and scattering measurements is equal to 1%.
In summary, the technique for an exact study of spatial inhomogeneties of semitransparent materials is described. Eqs (58) allow to obtain internal parameters of samples with unlimited optical densities, but one should take into account, that the sharp reduction of intensities of incident laser beams in optically dense materials can leads to dramatically decreasing the fluorescing and scattering pulses. In this case, more intense laser beams should be used. References

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