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DOI 10.1007/s11182-018-1303-z Russian Physics Journal, Vol. 60, No. 11, March, 2018 (Russian Original No. 11, November, 2017) INVESTIGATION OF BACKWARD REFLECTION FROM AQUEOUS MEDIA AT ARBITRARY GRAZING ANGLES Yu. L. Lomukhin and V. P. Butukhanov UDC 519.2 A model of backward reflection is presented which is distinguished by the fact that it takes account of excitation of counterpropagating waves in bounding media. For the case of a homogeneous aqueous medium, it is shown that the proposed model describes distinguishing features of the angular dependence of the backward reflection coefficients for all angles of incidence of the waves on the air–water interface. Characteristic regions of the angular dependence, associated with different mechanisms of formation of the backward reflection, are elucidated. Keywords: backward reflection, grazing angle, radar, polarization, Brewster angle, counterpropagating waves. It is possible to consider an aqueous medium (a body of water) to be homogeneous and, moreover, its surface can be considered calm (flat) or rough. Numerous experimental data show that backward reflection in the air–water system exists in any case. However, so far there does not exist a theoretical model adequately describing the backward reflection at arbitrary angles of irradiation of the air–water boundary. In the given work we construct an electrodynamic model of backward reflection from an aqueous medium for arbitrary angles of irradiation, using the effect of excitation of counterpropagating return (backward reflected) waves in bounding media. It was established in [1] that upon irradiation of the interface of two media, in space together with the reflected and refracted waves, waves with negative angle of refraction and waves traveling back toward the source are excited. These latter are the counterpropagating return waves. The physical cause of the formation of counterpropagating waves in bounding media is emission by secondary sources of waves not only in the direction of the primary wave exciting these field sources, but also in the exactly reverse direction. Indeed, from a classical point of view, the process of interaction of electromagnetic waves and media consists in the fact that a primary wave passing through matter along its entire path causes oscillations of charges in atoms. In the case of water, a reorientation of the dipole moments takes place, which is equivalent to motion of charges. Both the free and the bound oscillating electrons excite secondary waves propagating in all directions. Since the secondary waves are coherent, they cancel each other out as a result of interference in all directions except the direction coinciding with the primary wave and the exactly opposite direction – the direction counter to the primary wave [2]. The existence of a counterpropagating wave in the given case also follows from the condition of the impossibility of compensation of coherent waves in the given direction. Thus, in homogeneous absorbing media, the propagating radiation gives rise to counterpropagating waves. The presence of counterpropagating waves in media also follows of necessity from Maxwell’s equations [3]. Indeed, as is well known, the complete solution of the homogeneous wave equation ΔE 2 E 1 0, c , 2 с t 00 Institute of Physical Material Science of the Siberian Branch of the Russian Academy of Sciences, Ulan-Ude, Russia, e-mail: Lomukhin_yuriy@mail.ru; vbut1951@gmail.com. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 61–66, November, 2017. Original article submitted July 14, 2017. 1064-8887/18/6011-1917 2018 Springer Science+Business Media, LLC 1917 Fig. 1. Counterpropagating waves in bounding media. the derivation of which takes account of secondary emission by secondary sources, is a superposition of two counterpropagating waves: E E1 r exp i t kr E 2 r exp i t kr , k (1) , ' i'' , ' '' . c The solution in the form given by Eq. (1), in which one wave is primary and the second wave is excited by a secondary source, in the case of absorbing media is unique and satisfies all of the radiation conditions although it also assumes motion of energy both to the interface and from it in the case of bounding media. The presence of counterpropagating waves does not rule out the possibility of explaining both rectilinear propagation of light and deviations from rectilinear propagation with the help of the Huygens method, all the more so as the secondary sources introduced by Huygens radiate waves both in the direction of the exciting wave and in the reverse direction. The need to exclude backward waves from the Huygens–Fresnel principle is caused by the fact that counterpropagating backward reflected radiation in the optical range is very weak due to the inertia of the charges when acted upon by high-frequency fields; therefore, the idea that counterpropagating radiation was simply absent arose. Taking a phenomenological approach to the description of wave propagation, we will assume in the given work that secondary radiators are excited in media under the action of the field of an external source, and we model these secondary radiators by dipoles [4]. It is well known that the given sources radiate waves in the backward direction, i.e., in the direction counter to the exciting waves [5]. We will show that radar backward reflection is the resulting radiation of waves by the secondary sources. '' '' ' '' Let two semi-infinite isotropic linear contiguous media with 1 1 i1 , 1 1 and 2 2 () i 2 () , 1 2 1 be separated by a flat boundary (Fig. 1). A plane wave is incident from the first medium (free space) at an angle 1 upon the interface: E1p , s (1 ) E0p , s exp i ( k1r ) t . The indices p and s denote respectively ТМ and ТЕ modes. The line АА is a trace of the plane perpendicular to the vector k1 , i.e., to the wave front. In the second medium, the given wave excites secondary sources in the plane of the wave front – dipoles radiating waves only in the directions along and opposite to the normal to the front [5]. These waves, in turn, interact with the field of the incident 1918 wave. In accordance with the extinction theorem, a refracted mode arises in the second medium and the incident wave is extinguished [4]. The given mode propagates along the normal to the front BB with wave vector k2 : E2p , s (1 ) E1p , s (1 )T12p , s (1 )exp i (k2 r ) t . p,s Here T12 1 is the transmission coefficient of the wave from the first into the second medium, k2 k1 2 () , and k1 / c . In the expression 2 () that branch is chosen for which E 2p , s (1 ) does not grow. Simultaneously, as has already been stated, a wave is excited in the direction opposite to the normal of the front BB . Since its front is also BB , it propagates exactly counter to E 2p , s (1 ) , i.e., its wave vector is k2 . The counterpropagating wave (Fig. 1) is incident upon the interface from below. Thus, according to the laws of reflection and refraction, a wave propagates in the first medium precisely in the direction k1 . This is the backward wave: E 4p , s (1 ) p , s E 2p , s (1 )T21p , s (2 ) exp i k1 r t p, s E0p , s exp 2i k1r1 k2 r2 t 1 V12p, s 1 2 . Here it has been taken into account that V12 (1 ) V21 (2 ) ; therefore, T12p , s (1 )T21p , s (2 ) 1 V12p , s 1 p,s p,s 2 , 1 V p , s 2 is the coefficient taking into account the intensity of radiation arising in the medium excited by the 12 1 p,s p,s penetrating field, and V12 (1 ) and V21 (2 ) are reflection coefficients. The factor p,s was introduced to take account of the proportionality of the amplitudes of the primary wave and the wave re-radiated by the secondary sources. p ,s At a certain depth h2 h20 cos(1 ) the power of the wave E 2 (1 ) is comparable to the power of the thermal radiation; thus, the backward reflection, configured by the spatially-coherent structure of the secondary sources, from depths greater than h2 is indistinguishable from the background noise. Thus, it can be assumed that the counterpropagating backward waves are radiated only by a layer of thickness h20 located in free space. From the condition that the powers be equal, and also from an analysis of the available experimental data, it is possible to obtain an estimate of the interval of possible values of h20 : 1 h20 Im k2 3 , with the most probable (mean) value being h20 1/ Im(k2 ) [6]. As can be seen, h20 is close to the value of the skin-layer thickness, which does not contradict the physical sense of the given parameter as the maximum penetration depth from which a coherent backward wave is still observed. Obviously, in the case of normal incidence of waves onto the interface, the reflected field will be a superposition of waves re-radiated by the above-mentioned layer of thickness h20 , and waves reflected from the interface, i.e., E Equating E p,s 0 p, s (0) p,s and E4 E0p , s p,s V12 (0) 1 exp(2ik2 h20 ) p, s V12 (0) . exp(2ik1r1 ) 2 p,s 1 V12 (0) exp(2ik2 h20 ) 0 , we find p,s . Thus, the expression for the return wave takes the form 1919 Fig. 2. Diagram of radar sensing. 2 E4p , s 1 p , s E0p , s 1 V12p , s 1 exp(2ik2 h2 ) , p,s (2) V12p , s (0) 1 exp(2ik2 h20 ) , 1 2 2 1 V p , s 0 exp(2ik h ) 1 V p , s (0) exp(2ik h ) 2 20 12 2 20 12 where h2 h20 cos 2 . To compare the theoretical and experimental data, it is necessary to generalize formula (1) to the case of a point source with directed radiation (Fig. 2). Representing the field of a point source in the form of an expansion over plane return waves (Eq. (2)), and also taking into account the presence of an antenna, we obtain an equation for the return wave in the case of irradiation of the interface by radar: E p , s (1 ) E0p , s exp 2ik1r1 2H W p , s (1 ) . (3) Here 2 W p , s (1 ) 1 V12p , s 1 exp(2ik2 h2 )cos(1 ) F (1 ) , and H r1 cos(1 ) is the altitude of the radar above the surface (air–water interface). The function F (1 ) 1 (1 p,s 2 ) exp 1 G models the directivity diagram, is its width, and G is the amplification coefficient (gain) of the antenna, simultaneously taking into account various mechanisms of backward reflection for 0 1 and 1 1920 . From Eq. (3) we determine the backward reflection intensity coefficients: 2 Fig. 3. Angular dependence of the backward reflection coefficients σ of aqueous media: curves 1 and 2 are theoretical curves of σVV and σ НН for ∆ = 0.02 and 0.002 m, respectively; ▲, ♦, ■, and ● are experimental mean values of σVV for wind velocities Uw = 8.4, 5.6, 3.88, and 2.5 m/s, respectively [8]; and + and are experimental values of σVV and σ НН for Uw = 2.57 m/s [9]. The dependence of the standard deviations of the 2.5 [8]. roughness on the wind speed is represented in the form 4.52 103 U w VV 1 W p 1 W p 1 , HH 1 W s 1 W s 1 , (4) VH (1 ) HV (1 ) VV (1 ) HH (1 ) P , where P is the probability of displacement of charged particles in a direction perpendicular to the p- or s-direction of the primary waves. In this work we take P to be equal to 1/4: P = 1/4. Interfaces between media are often uneven. Therefore, to account for the possible roughness of the interface, we multiply the Fresnel coefficients in Eqs. (2) and (3) 2 2 2 by the Rayleigh factor: exp(2k1 Δ cos (1 )) , where Δ is the standard deviation of the roughness distributed according to a normal law [7]. Figure 3 plots the angular dependence – as calculated by formula (4) and experimental values taken from [8, 9] – of the backward reflection coefficients of an aqueous medium in the L-band with central frequency f = 1.4 GHz for vertical and horizontal polarization of microwaves. Here and below, by angular dependence we understand the dependence of the backward reflection on the grazing angle ψ 90 1 , where 1 is the incident angle. The frequency dispersion of the aqueous medium was taken into account in the calculations according to the Debye formula. The experimental data from [8] were obtained by processing and averaging measurements over a multiyear period. It can be seen from Fig. 3 that the values calculated according to formula (4) and the experimental data from [8] are in good qualitative and quantitative agreement. The data from [9] are in qualitative agreement with the calculations based 1921 on formula (4). Thus, the model of backward reflection proposed in this work based on an account of excitation of counterpropagating waves describes the real mechanism of backward reflection. It follows from Fig. 3 and formula (3) that three characteristic regions stand out in the angular dependence of the backward reflection. In the region near normal irradiation 80 ψ 90 , the background reflection from the surface prevails. In the region of volume reflection ψ Br ψ 80 , a weak dependence of reflection on the grazing angle is observed. Here ψBr is the Brewster angle. In the region of backward reflection at small grazing angles 0 ψ ψ Br , the backward reflection falls abruptly to zero. A polarization feature of the angular dependence should also be noted: for 45 ψ 90 there is practically no difference in the values of σVV and σ HH , whereas for 0 ψ 45 a substantial difference between the values of σVV and σ HH is observed, with σVV being greater than σ HH : σVV > σ HH . Here, with increase of both σVV and σ HH increase, but σ HH grows faster, approaching σVV without exceeding it. These peculiarities were noted in [10]. In the case of normal incidence σVV (90 ) σ HH (90 ) σ(90 ) , σ decreases with increase of the standard deviation of the boundary roughness. The calculated and measured values of σ(ψ) show that for the grazing angle equal to the Brewster angle ( ψ Br 6.3 ), there takes place an equalization of the backward reflection coefficients σVV for different roughnesses of the water surface. The mechanism of formation of backward reflection proposed in this work is analogous to mechanisms of wavefront reversal [11] and formation of a radar return signal [12]. In the given case, the role of the reversing medium is played by the volume of the homogeneous aqueous medium, and the presence of roughness on the boundary is equivalent to a scattering plate placed in front of the reversing medium [11]. Thus, the model of backward reflection presented here and constructed with excitation of counterpropagating waves taken into account adequately describes the angular dependence of the backward reflection coefficient of an aqueous medium for all grazing angles. Three regions of grazing angles with characteristic behavior of the backward reflection have been distinguished. In the example of an aqueous medium, the mechanisms of wavefront reversal, generation of a radar return signal, and backward reflection have been shown to be equivalent. The influence of the Brewster effect on backward reflection for oblique irradiation of an air–water interface has been demonstrated. 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