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Зміст 3. The determination of the optical constants. 

The Determination of the Optical Constants of the Plant Leaves with Polarized Emission Using. A.G. Ushenko, A.V. Motrich, S.G. Guminietski Chernivtsi National University, 2 Kotsyubinsky Str., 58012 Chernivtsi, Ukraine. motrich@gmail.com Abstract. There have been given the results of the investigating the reflection coefficients of some plants’ leaves with very smooth surfaces in the polarized light for those wave lengths for which the inner composition in the reflected emission does not practically exist. The optical constants for the mentioned objects of investigation have been defined, also Fresnel formulas as well as the method of the determination of the optical constants based on the measuring the reflection coefficients of the polarized emission for the definite fixed angles of incidence. Keywords: polarized emission, reflection coefficient, index of refraction, index of absorption. PACS: 42.25.Ja, 42.62.Be 1. Introduction Determination of optical constants of matter is one of actual tasks at its researches optical and spectral, for example [1, 2]. The methods of the determination of the optical constants of different substances are widely presented in the scientific research letters and mainly with the polarized emission using in the matter of the semiconductor crystals in operation [3]. As far as the biological objects are concerned, for instance the plants’ leaves, then the index of refraction n is evaluated for them as being equal approximately to 1,4 while the index of absorption k (extinction) is considered to be far less to compare to it [4, 5]. Herewith the radiated wave length λ has not being concretized. That’s why the purpose of writing this paper was in the determining the exact values of n and k of the mentioned objects for the definite wave lengths. It becomes even more complicated even because of that as it is known from [6, 7], in the reflected emission from the plant leaf there two components: a surface component (at the expense of the reflection boundary airleaf) and the inner one (at the expense of the scattering in the backward direction on the optical heterogeneities of the leaf structure). As in methods based on the Fresnel formulas the reflection coefficient has being used only for the boundary of two media (in our case it is an airleaf surface), then they could be used only for those wave lengths λ, for which the inner component is absent or I far less to compare to a surface one. It is known that [4, 7], that is in the field of λ < 420 nm and about 660 670 nm. 2. Methods and the objects of investigation The reflection of the polarized emission from a smooth surface boundary between two media with the indexes of refraction n_{1} and n_{2} are described by Fresnel coefficients and – are Fresnel coefficients for the wave which is correspondingly polarized perpendicularly and in parallels towards the incidence area. The measured indexes of reflection and are equal to the squares and and are the functions of three variables, i.e. (n, k, φ) end (n, k, φ). Thus, – is a relative index of refraction; k  is an index of extinction; φ – the angle of incidence. For the experimental determination of n and k it is necessary to have two equations, which could be obtained by the measuring of the indexes of reflection and for two fixed angles of incidence φ_{1} and φ_{2}, then: (1) they are solved by numerical, graphical or computerized methods. The plants’ leaves with very smooth surfaces served as the objects of investigation such as ones of Ficus australis, Philodendron selloun, Tetrastigma voneriana, Opuntia compressa. and measurements for two angles of incidence φ_{1} = 40° , φ_{2} = 65° and at the normal incidence were done on the spherical photometer and by the methods described in [8, 9] with Nicol’s prism being used as a polarizer at its inlet port. Relative error of measuring and does not exceed 0.2 – 0.4 %. The investigation at the normal incidence gave a possibility to choose those wave lengths for which it is not essential to insert into the value of the index of reflection the diffusing scattered component out of the inner leaf parts particularly for each of the objects. However, using the given works [4] was possible to take into account the size of the internal diffusely dissipated constituent in the values and at the calculations of n and k for Ficusa australis. The choice of the indicated angles of incidence is in a certain measure arbitrary, but he must provide a sufficient difference between the values and, that is instrumental in more exact establishment of functions and . The results of investigation for the chosen λ are given in tabl.1 tabl.4. Table 1 The reflection index value for Ficus australis from the consideration internal diffusely dissipated constituent.
Table 2 The reflection index value for Philodendron selloun
It is known, that at the normal incidence , (2) because as if , then with the decrease of R_{o} the value of n also has to be decreased; if R_{0} with the wave length is not practically being changed then the value of n would be invariable. Table3 The reflection index value for Opuntia compressa
Table 4 The reflection index value for Tetrastigma voneriana
As the plant leaf is a dielectric then this circumstance becomes a condition of making the choice of those λ for which the inner component at the reflection is yet not enough significant to compare to a surface one and what is shown in tabl.1 tabl.4. It’s worthwhile to underline that the investigations at λ < 280 nm were not carried out as nicol does not practically transmit in this region, though a spherical photometer gives such an opportunity to arrange the measurement of the reflection indexes starting from λ = 220 nm [8,9]. Using measuring of value andand size of relative errors here, on the basis of formulas (1) it is possible to expect absolute errors in determination of functions and . They are found within the limits of 0.0016 – 0.0028 for and 0.001 – 0.0017 for . ^ Using computer potentialities n and k could be found in the following way. Let’s there were experimentally found the relations /at the angles of incidence and that are equal correspondingly to and . It is apparent, that the functions are equal to zero at the correct values of n and k. (3) Fig. 1. Values of  curve 1 and  curve 2 for Ficus australis at nm; n = 1.26. In order to find these values it is necessary to construct the relationships of and in dependence from k, changing values in wide limits and assuming the reasonable values of n, as shown in fig.1 and fig. 2, for the wave length nm in case of Ficus australis leaf. At the selected value of n the points of crosssection of both curves with the axis (marked by the circles in fig. 1) suit the equations (3) at various values of k. Changing values of n (fig.2), it might be possible to reduce two points of the crosssection into one. Fig.2. Attached to the methods of the determination of the optical constants n and k for Ficus australis at nm. This condition meets the correct values of n and k. We act in an analogous way towards all other λ and the objects of investigation. The determined in that way values of n and k are given in table 5. For Ficus australis as the graphs on fig.3. Table 5 Values of the optical constants of plants’ leaves.
The values of absorption factor , sm^{1}, shown on it are obtained using equation . Similarly one can obtain the error values of and using functions and . Thay are in range of 0.02 – 0.05 for k and 0.01 – 0.03 for n . Fig.3. Spectrum: 1 of refraction index n; 2 – absorption factor α. Taken into account this fact, one can see that the absorption factor k for investigated leafs of plants in investigated spectral range is practically equal, although in range of nm and nm one can observe maxima. At the same time, on the curve for leafs of Ficus australis one can see two anomalous regions: in range nm and near nm (fig.3). 4. Conclusion The determined values of the optical constants of the plants’ leaves could be used while studying the phenomena of absorption and the emission scattering inside the plant medium with using the methods of optics and spectroscopy of the scattered objects [10, 11].
