DOI 10.1007/s1596-013-0156-7 RESEARCH ARTICLE Analysis of reflected intensities of linearly polarized electromagnetic plane waves on parabolic boundary surfaces with different focal lengths Hossein Arbab & Mehdi Rezagholizadeh Received: 19 November 01 /Accepted: October 013 # Optical Society of India 013 Abstract The aim of the study is to investigate the reflection of linearly polarized electromagnetic plane wave from common parabolic boundary surface between two different mediums. For this purpose, a discrete model is developed for the analysis of reflected power. Results of the study show that reflection of incident wave depends on the optical properties of two mediums besides the boundary surface geometry. Furthermore, results imply that when the opening area of parabola keeps constant and the focal length is changing, the contribution of S and P components of incident wave differs with respect to changing the focal length. Keywords Parabolic solar dish. Electromagnetic wave. Polarization. Flat square elements. Focal length. Fresnel coefficients H. Arbab (*) Department of Physics, Faculty of Science, University of Kashan, Kashan, I. R., Iran e-mail: arbabpen@kashanu.ac.ir Introduction In the near future, the energy supplement of humankind will be of vital importance. Dependence of human to the fossil fuel and nuclear energy which are nonrenewable sources and make pollution, will pose the future life to the real danger. In the present time, the significant issue is to harnessing renewable energies such as solar radiation, wind, wave power, and hydroelectricity as much as possible. Total solar irradiance describes the radiant energy emitted by the sun over all wavelengths that falls each second on1m outside the Earth atmosphere. Solar energy flux on the outer limit of the Earth atmosphere is measured about 1,367 W per square meter [1]. There are several known ways to exploit solar energy for sustainable energy provision purposes. A solar dish concentrator converts energy from the sun and can be used for hydrogen production at high temperatures [, 3] or surgery [4]. Theoretical analysis for thermodynamic limit of concentration [5, 6] and cavity optimization [7] is an important problem has been investigated. In the literature, several methods of optimizing performance of solar dishes have been proposed from different point of views such as certain optical properties of the substrate material [8], dependence of optical efficiency to the incident angle [9], and increasing the concentration ratio and obtaining the uniform illumination distribution for PV cells [10]. Solar dishes are constructed in different shapes and structures. A typical dish concentrator which is made in the University of Kashan by Arbab et al, as shown in M. Rezagholizadeh Department of Electrical and Computer Engineering, Faculty of Engineering, McGill University, Montreal, Quebec, Canada H. Arbab Energy Research Institute of University of Kashan, Kashan, I. R., Iran
Fig. 1, made out of plane squared mirrors set on the coaxial rings using the laser spot image processing. The plane mirror elements could be chosen from any conductor or dielectric materials. This concentrator has the advantages of simplicity and accuracy and is capable of tracking the sun using intelligent controller based on the image processing of bar shadow [11]. In this work, we investigate electrodynamical effects of changing the focal length of a parabolic boundary surface of two semi-infinite neighbor dielectric mediums on the reflected energy distribution. The proposed analysis in this study can be further developed for real cases such as the one that isshowninfig.1 where the parabolic dish concentrator is covered with mirrors. Before entering the main discussion of our work, it is necessary to introduce an explicit explanation of the dish structure. The base of dish concentrator is covered with same size plane square elements set on coaxial rings of different radii.thenumberofsquareelementsofeachring is the function of its radius. The number of rings that cover the base surface is the function of parabola s focal length. Since the base is of parabolic shape, square elements cannot cover the whole surface and there will be some uncovered area. In our work, the diameter of opening of the parabolic dish is constant and the effects of changing its focal length on the total reflected power and the ratio of olarization part to the P polarization part of a linearly polarized electromagnetic plane wave is investigated. This work is organized as follows. The introduction was presented in Section 1. In the second section, mathematical formulation of covering the parabolic boundary surface with squared elements is proposed. In Numerical simulation of the reflected energy distribution section, the reflected intensities of linearly polarized electromagnetic plane wave by parabolic boundary surfaces of different focal lengths will be analyzed using a numerical simulation framework. The last section contains discussion and conclusion. Preliminaries: covering the parabolic boundary surface with squared elements From a general point of view the geometry of the parabolic dish surface can be chosen smooth and continuous or discrete as mentioned in the preceding section. When the dimensions of discrete elements are small enough, the discrete model will be a good approximation of the continuous form. Moreover, the discrete model coincides with the structure of practical case such as the one which is shown in Fig. 1, however, the introduced analysis in this study should be extended for these cases where the parabolic dish concentrator is covered with mirrors. Physically, the Fresnel reflecting coefficients depend on the angle of incidence; therefore these coefficients change from a certain ring to another one. In addition the amount of falling radiation on a definite ring is a function of focal length and its vertical distance from the vertex. Based on the arc length of a parabola, the Fig. 1 Parabolic solar dish concentrator with small square mirrors on the base
number of rings on the base would be the integer part of the following equation. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NðfÞ ¼ a 1 þ a þ f a b f b sinh 1 f sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 1 1 þ a 1 f a 1 b f b sinh 1 ð1þ f Equation (1) is based on the arc length of a parabola. Where a, b, f,anda 1 denote the radius of the opening of parabolic dish, the dimensions of a squared element, focal length, and radius of first ring from the vertex respectively. Therefore the number of square elements on jth ring h i is πx j b (integer part) and x j is the radius of jth ring. The Eq. (1) shows that the number of rings is a function of focal length. (a 1 is the radius of the first ring and it may be zero). The following equation describes a simple parabola in two dimensions. yx ðþ¼ x ðþ 4f Thus if the angle of inclination on the jth ring is shown by θ 1j,from() we obtain:(fig. illustrates the definition of θ 1j ). tanθ 1 j ¼ x j f ð3þ There are two different methods of enumerating the number of rings; counting the ring s number from the opening to the vertex and vice versa. By counting rings from the vertex to the opening, x j+1 relates to x j using the following recurrent formula. b x jþ1 ¼ x j þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ x j f j ¼ 1; ; 3; ; NðfÞ ð4þ In the next section, we consider the parabolic boundary surface between two semi-infinite mediums and investigate the amount of F s and at the focal point where F s and denote S and P components of radiation power respectively. Numerical simulation of the reflected energy distribution This section discusses following topics. (a) (b) (c) Parabolic boundary surface between vacuum and glass (the incident electromagnetic wave is in the vacuum) Parabolic boundary surface between glass and vacuum (the incident electromagnetic wave is in the glass) Parabolic boundary surface between vacuum and conductor (the incident electromagnetic wave is in the vacuum) Fig. A linarly polarized electromagnetic wave incident on the parabolic boundary surface between two mediums
Parabolic boundary surface between vacuum and glass (the incident electromagnetic wave is in the vacuum) Assume that the incident plane polarized electromagnetic wave arrives from the first medium (vacuum) along the symmetrical axis on the parabolic boundary surface made of glass (see Fig..) First of all, consider a flat boundary surface and the wave propagates normally through it. Therefore, a fraction of it would pass through the second medium and the remaining would be reflected back into vacuum. In order to calculate the amount of reflected power, the Fresnel reflection coefficients should be determined for S and P polarization [1]. r 1s ¼ n 1cosθ 1 n cosθ n 1 cosθ 1 þ n cosθ r 1p ¼ n cosθ 1 n 1 cosθ n cosθ 1 þ n 1 cosθ ð5þ ð6þ (n 1 and n are refractive indices of first and second medium respectively.) For normal incidence we have: θ 1 ¼ θ ¼ 0 ð7þ Indices 1 and indicate the values of angles for the first and the second medium respectively. Therefore: r 1s ¼ r 1p ¼ n 1 n n 1 þ n ð8þ R s ¼ R p ¼ n 1 n ð9þ n 1 þ n R s,andr p are the reflection coefficients for the S and P polarization respectively. The power which is reflected back to vacuum: F s ð Þ ¼ ð Þ ¼ π a IRs ¼ π a IRp ð10þ a 1 a 1 Equation (10) shows that when the focal length is infinity and a is fixed, then S and P components of reflected power would be the same. Consider the following numerical values for the introduced parameters: n 1 ¼ 1; n ¼ 1:5; I ¼ 1000W=m ; a 1 ¼ 0:1m; a ¼ 1m; b ¼ 0:01 m: Using the Eqs. (9) and (10), the power which is reflected back to vacuum: F s ð Þ ¼ ð Þ ¼ 3:14 ð1 0:01Þ 1 1:5 1 þ 1:5 1000 14:4w ð11þ (Note that parameters are chosen for simplicity.) For a more general case, assume that the linearly polarized electromagnetic plane wave propagates along the symmetry axis of the parabolic boundary surface. Now we divide the parabolic boundary surface to small square elements set on coaxial rings as shown in Fig. 3. The angle between incident wave and small square elements is constant for a particular ring but changes from oneringtoanotherbutitchangesfromacertainringto another one. Using Eqs. 5 and 6 and considering Snells law we obtain: sin 0:5 n 1 cosθ 1 j n 1 n 1 n θ 1 j r 1sj ¼ sin 0:5 ð1þ n 1 cosθ 1 j þ n 1 n 1 θ 1 j n sin 0:5 n cosθ 1 j n 1 1 n 1 n θ 1 j r 1pj ¼ sin 0:5 ð13þ n cosθ 1 j þ n 1 1 n 1 θ 1 j R sj ¼ r 1sj R pj ¼ r 1pj n ð14þ ð15þ Now, we introduce a new parameter φ lj which is theazimuthanglebetweenthelth square element of the jth ring and the electric field vector. (See Fig. 3.) Thus S and P components of the energy flux will be as follows. I slj ¼ Isin φ lj I slj ¼ Isin φ lj It is obvious that: φ lj ðradþ lb x j ð16þ ð17þ ð18þ
Fig. 3 The azimuth angle between the lth square element in the jth ring and electric field when the electric field oscillates in the YOZ plane The total power of S and P polarization reflected back to focal point will be: ½ F s ðf Þ ¼ XN ðfþš X πx j b IR sj b sin φ lj cosθ 1 j ð19þ j¼1 ½ ðf Þ ¼ XN f l¼1 ð ÞŠ X πx j b j¼1 l¼1 IR pj b cos φ lj cosθ 1 j ð0þ Relations (19) and(0) showthatf s and do not have equal part in the total reflected power that is given by the following relation: FðfÞ ¼ ðf Þþ F s ðf Þ ð1þ Calculating F s,, and there sum for various focal lengths in the range of 0.1 m to 3 m given in Fig. 4. To sum up, we can derive the following conclusions. (a) For f 1,F s and have significant difference. The proportion of F s in the total reflected power has the maximum of 19.48 at f=0.4m and this ratio goes to 1asf. It tallies the result given by Eq. 11. (b) As f goes to zero, the power which is reflected back to the focal point. (c) It shows that the total reflected power decreases as the focal length increases. Parabolic boundary surface between glass and vacuum (the incident electromagnetic wave propagates through the glass) In the second case, assume glass as the first medium and vacuum as the second medium.. The incident electromagnetic wave propagates along the symmetrical axis toward the common parabolic boundary surface. Because of the total internal reflection the calculations in this situation will be little different form the previous calculation previously. In this situation, we must consider the total internal reflection and the changing effects with the variation of focal length. The critical angle is given by the following equation. θ c ¼ Arc sin n n 1 ðþ Considering total internal reflection for rings whose θ 1j >θ c, and using (1) and(13) for rings whose θ 1j <θ c, leads to the following results for F s,, total reflection power and F s / ratio which have been shown in Fig. 5. According to Fig. 5, we can conclude that: (a) The selective effect is less than that of the first case. So, we can conclude that the value of F s in the intervals 0<f 0. and 1.4<f is nearly equal to. (b) The selective effect is maximum for f=0.61m, where F s ¼ 9:.
Fig. 4 F s, and total reflection power from parabolic boundary surface between vacuum and glass and the ratio of F s ðn 1 ¼ 1; n ¼ 1:5; I ¼ 1000 W ; b ¼ 0:01 m; a 1 ¼ 0:01 m; a ¼ 1 mþ (c) (d) As f goes to zero, the total incident power tends to reflect completely back to the focal point in the first medium. If f goes to infinity (flat boundary surface), the power which is reflected back to the first medium by (11). This results is expected since the two mediums are interchanged. Parabolic boundary surface between vacuum and nonmagnetic conductor (the incident electromagnetic wave propagates through the vacuum) Assume that the incident plane polarized electromagnetic wave moves along the symmetrical axis from the (vacuum) first medium to the second one (conductor). It should be noted that the conductor is not covered by glass. In order to calculate the reflected power back to the vacuum, Fresnel reflection coefficients should be determined. They are given by the following equations for P and S polarizations. er 1pj ¼ en cosθ 1 j n 1 en cos e θ j en cosθ 1 j þ n 1 en cos e ð3þ θ j er 1sj ¼ n 1 cosθ 1 j en cos e θ j n 1 cosθ 1 j þ en cos e θ j ð4þ Where e θ j is the wave propagation angle in the conductor substrate and en is its complex reflective index and depends on the optical constants: n and k of the conductor. en ¼ n þ ik eκ j ð5þ is defined as the complex propagation wave vector in conductor elements on the jth ring and the complex angle e θ j is defined by the following relations. eκ j :bn ¼ eκ j cos e θ j ð6þ en cos e θ j ¼ β j þ iγ j ð7þ
Fig. 5 F s, and total reflection power from parabolic boundary surface between glass (the first medium) and vacuum and the ratio of F s ðn 1 ¼ 1:5; n ¼ 1; I ¼ 1000W ; b ¼ 0:01m; a 1 ¼ 0:01m; a ¼ 1mÞ Where β j ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 n k n 1 sin θ 1 j þ n k n 1 sin θ 1 j þ 4n k ð8þ γ j ¼ 1 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n k n 1 sin θ 1 j þ n k n 1 sin θ 1 j þ 4n k j ¼ 1; ; 3; ::::; NðfÞ Using Eqs. (3 5) and(7 9) we obtain the following formulas for Fresnel reflection coefficients. h i n k cosθ1 j n 1 β j þ i n k cosθ 1 j γ j n 1 br 1pj ¼ h i n k cosθ1 j þ n 1 β j þ i n k cosθ 1 j þ γ j n 1 ð30þ er 1sj ¼ n 1cosθ 1 j β j iγ j ð31þ n 1 cosθ 1 j þ β j þ iγ j ð9þ Fresnel coefficients for P and S polarizations on the jth ring are: R pj ¼ er 1pj er 1pj R sj ¼ er 1sj er 1sj (The * stands for complex conjugate). ð3þ ð33þ
Consider a situation where the parabolic boundary surface is covered with small square elements of silver as a nonmagnetic reflector. The optical constants of Ag (in visible spectrum) is n=0.05, k=3 [13]. The results after calculations are shown in the Fig. 6. In the discrete model, number of rings and uncovered area of the base are related to the focal length and square element dimensions. Hence, the reflected power not only depends on the electrodynamics effects but also the uncovered area. The number of rings from the outermost ring (near the opening) to the inner most one is calculated by the Eq. 4 (near the vertex). The rings near the opening of parabola contain much more square elements than those near the vertex. Since the number of rings which cover the surface of parabola must be an integer and we put the first ring near the vertex, so using the Eq. (4) leads to an uncovered area near the opening (after the outermost ring) for many focal lengths. Since the reflection coefficient of glass is much less than unity, not taking into account this area did not make a sensible effect on the results of Figs. 4 and 5. However, the reflection coefficient of silver is close to unity, and not considering the outermost uncovered area leads to great jumps in Fig. 6. The solution of this problem is to set rings from the opening to the vertex. In such a case, some area near the vertex would be uncovered that is much less than the preceding when we put rings from the vertex to the opening. Considering the aforementioned effect of omission, we change the recurrent formula so that a part of a ring omits from the vertex as follows: b x jþ1 ¼ x j rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ x j f j ¼ 1; ; 3; ; NðfÞ ð34þ Taking into account this solution is shown in Fig. 7 instead of Fig. 6. Figure 7 shows that: (a) (b) (c) F s and do not have considerable difference. F s has decreasing behavior but and total reflection has a global minimum at f=0.1 m. As f the total reflected power remains constant and its value less than incident power. Fig. 6 F s, and total reflection power from parabolic boundary surface (Ag as nonmagnetic conducting reflector) by counting rings from vertex to the opening (n 1 =1, n =0.05, k =3, I=1000 W, b=0.01 m, a 1 =0.01 m, a =1 m)
Fig. 7 F s, and total reflection power from parabolic boundary surface (Ag as nonmagnetic conducting reflector) and the ratio of Fs,by counting number of rings from opening to the vortex (n 1 =1, n =0.05, k =3, I=1000 W, b=0.01 m, a 1 =0.01 m,a =1 m) Summary and conclusion In this work, we investigated the reflection of linearly polarized electromagnetic plane waves from parabolic shape boundary surface using simple physical models for three different cases. We consider a case where two mediums are presents which are in contact with each other and make a parabolic boundary surface. Our definition of the first and second medium has been shown in Fig.. The surface of this structure is covered with small square elements that lie in coaxial rings. We studied the S, and P reflected polarization components besides the total reflected power back into the focal point when the opening area of the parabolic boundary was kept constant and the focal distance was changing. First of all, we considered a situation that the first medium was vacuum and the second medium was glass. The incident wave travels along the axis of symmetry of parabola which lies between the vacuum and glass. In the second case, we focused on the parabolic boundary surface which was placed between glass as the first medium and vacuum as the second medium. The incoming wave travels along the axis of symmetry from the semi-infinite glass medium toward the vacuum. The incoming wave is totally or partially reflected to the focal point. Equations (1 and 13) wereusedfor the rings with θ 1j <θ c, and total reflection was considered for those whose θ 1j θ c. The third case considered the incident wave moves along the symmetry axis from the vacuum (first medium) to the common boundary surface with a conductor such as silver. The analysis which introduce in this study can be further developed for real cases where the parabolic dish concentrator is covered with mirrors. According to the above discussions, it seems that results of this investigation can be applied for: first, optimizing the structure of parabolic solar dish concentrators; second, optimizing lens solar concentrators, third, optimizing the non-reflective lenses, fourth, optimizing the performance of communication satellite receivers, and the last, situations where anisotropy is important.
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