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Lecture 4: Reflection of Optical Waves from Intersections 4.1. Boundary Conditions The simplest case of wave propagation over the intersection between two media is that where the intersection surface can be assumed as a flat perfectly conductive. If so, for a perfectly conductive flat surface the total electric field vector is equal to zero, i.e., E 0 . In this case, as was shown above in Lecture 2, the tangential component of electric field vanishes at the perfectly conductive flat ground surface, that is, E 0 (4.1) Consequently, as follows from Maxwell’s equation E( r) iH( r) (see Lecture 1 for the case of 1 and B H ), at such a flat perfectly conductive ground surface the normal component of the magnetic field also vanishes, i.e., Hn 0 (4.1a) As also follows from Maxwell’s equations (1.1), the tangential component of magnetic field does not vanish because of its compensation by the surface electric current. At the same time the normal component of electric field is also compensated by pulsing electrical charge at the ground surface. Hence by introducing the Cartesian coordinate system, one can present the boundary conditions (4.1)-(4.1a) at the flat perfectly conductive ground surface as follows: E x ( x, y, z 0) E y ( x, y, z 0) H z ( x, y, z 0) 0 4.2. (4.2) Main Reflection and Refraction formulas As was shown above, the influence of a flat terrain on wave propagation leads to phenomena such as reflection. Because all kinds of waves can be represented by means of the concept of the plane waves, let us obtain the main reflection and 2 refraction formulas for a plane wave incident on a plane surface between two media, the atmosphere and the earth, as shown in Fig. 4.1. The media have different dielectric properties which are described above and below the boundary plane z=0 by the permittivities and permeabilities 1 , 1 and 2 , 2 , respectively. Fig. 4.1 Without reducing the general problem, let us consider a plane wave with wave vector k and frequency 2 f incident from a medium described by parameters 1 and 1 . The reflected and refracted waves are described by wave vectors k 1 and k 2 , respectively. Vector n is a unit normal vector directed from medium ( 2 , 2 ) into medium ( 1 , 1 ). According to relations between electrical and magnetic components which follow from Maxwell’s equations, the incident wave can be represented as follows: E E 0 expik x t , H 1 kE 1 | k| (4.3) The same can be done for the reflected wave E1 E01 expik 1 x t , and for the refracted wave H 1 1 k 1 E1 |k1| (4.4) 3 E2 E02 expik 2 x t , H 2 2 k 2 E2 |k 2 | (4.5) The values of the wave vectors are related by the following expressions: | k| | k 1 | k 1 1 , c |k 2 | k 2 22 c (4.6) From the boundary conditions that were described earlier by (4.1)-(4.2), one can easily obtain the condition of the equality of phase for each wave at the plane z=0: (k x) z 0 (k 1 x) z 0 (k 2 x) z 0 (4.7) which is independent of the nature of the boundary condition. Equation (3.37) describes the condition that all three wave vectors must lie in a same plane. From this equation it also follows that k sin 0 k1 sin 1 k 2 sin 2 (4.8) which is the analogue of Snell’s law: 1 1 sin 0 2 2 sin 2 (4.9) Moreover, because | k 0 | | k 1 | , we find 0 1 ; the angle of incidence equals the angle of reflection. It also follows from the boundary conditions that the normal components of vectors D and B are continuous. In terms of the field presentation, these boundary conditions at the plane z=0 can be written as E 1 0 E1 2 E 2 n 0 k E 0 k 1 E 1 k 2 E 2 n 0 E 0 E 1 E 2 n 0 (4.10) 1 1 k 2 E 2 n 0 k E 0 k 1 E1 2 1 Usually in applying these boundary conditions for estimating the influence of the flat ground surface on wave propagation over terrain, it is convenient to consider two separate situations: The first one is when the vector of the wave’s electric field component E is perpendicular to the plane of incidence (the plane defined by vectors k and n ), but the vector of the wave’s magnetic field component H lies in this plane 4 (see Fig. 4.2). The second one is when the vector of the wave’s electric field component E is parallel to the plane of incidence, but the vector of the wave’s magnetic field component H is perpendicular to this plane (see Fig. 4.3). Fig. 4.2 Fig. 4.3 5 In the literature which describes wave propagation aspects, they are usually called the TE-wave (transverse electric) and the TM-wave (transverse magnetic), or waves with vertical and horizontal polarization, respectively. We will derive the reflection and refraction coefficients for the case of an incident plane wave with linear polarization; the general case of arbitrary elliptic polarization can be obtained by use of the appropriate linear combinations of the two results, following the approach presented in Lecture 2. First of all, we consider the incident plane linearly polarized wave with its electric field perpendicular to the plane of incidence (TE-wave), as shown in Fig. 4.2. The orientations of the magnetic field components of the incident, reflected and refracted waves, Hi , i 0, 1, 2, are chosen to give a positive flow of energy in the direction of wave vectors k , k 1 and k 2 , respectively. Since the electric fields are all parallel to the boundary surface, the first boundary condition in (4.10) yields nothing. The third and fourth conditions in (4.10) give E 0 E1 E 2 0 (4.11) 1 E 0 E1 cos 0 2 E 2 cos 2 0 1 2 while the second condition in (4.10), using Snell’s law (4.9), duplicates the third condition. Now, from (4.11), we can obtain the amplitudes of the reflected and refracted waves respectively: 1 ( 2 2 ) 2 ( 1 1 ) 2 sin 2 0 2 | E1 | | E 0 | 1 1 cos 0 1 ( 2 2 ) 2 ( 1 1 ) 2 sin 2 0 2 1 1 cos 0 | E 2 | | E 0 | 2 1 1 cos 0 1 1 1 cos 0 ( 2 2 ) 2 ( 1 1 ) 2 sin 2 0 2 (4.12a) (4.12b) The same results can be obtained from (4.10) for the case of the TM-wave, when the electric field vectors are parallel to the plane of incidence, as is shown in Fig. 4.3. The 6 boundary conditions for the normal component of vector D and for the tangential components of vectors E and H lead to the first, third and fourth equations in (4.10), from which follow: ( E 0 E 1 ) cos 0 E 2 cos 2 0 (4.13) 1 E 0 E1 2 E 2 0 1 2 The continuity of the normal components of the vector D , plus Snell’s law (4.9), merely duplicates the second of equations (4.10). Therefore the amplitudes of the reflected and refracted waves can be written as: 1 ( 2 2 ) 2 cos 0 1 1 ( 2 2 ) 2 ( 1 1 ) 2 sin 2 0 | E1 | | E 0 | 2 1 ( 2 2 ) 2 cos 0 1 1 ( 2 2 ) 2 ( 1 1 ) 2 sin 2 0 2 | E 2 | | E 0 | 2 1 1 2 2 cos 0 1 ( 2 2 ) 2 cos 0 1 1 ( 2 2 ) 2 ( 1 1 ) 2 sin 2 0 2 For the real situation of wave propagation, it is usually permitted to put (4.14a) (4.14b) 1 1. 2 introducing the relative permittivity (with respect to the air), r 2 / 1 , we will obtain, by use of (4.12) and (4.13) the expressions for the complex coefficients of reflection ( ) and refraction (T) for waves with vertical (denoted by index V) and horizontal (denoted by index H) polarization, respectively. For vertical polarization: RV | RV | e jV TV | TV | e jV ' For horizontal polarization: r cos 0 r sin 2 0 r cos 0 r sin 2 0 2 r cos 0 r cos 0 r sin 2 0 (4.15a) (4.15b) 7 RH | RH | e j H TH | TH | e j H ' cos 0 r sin 2 0 cos 0 r sin 2 0 2 cos 0 cos 0 r sin 2 0 (4.16a) (4.16b) Here | V | , | H | , | TV | , | TH | and V , H , V' , 'H are the modulus and phase of the coefficients of reflection and refraction for vertical and horizontal polarization, respectively. Dependence of the coefficient of reflection on the angle of incidence is shown in Fig. 4.4 for two types of field polarization. It is very important to note that for normal incidence of a radio wave on a flat ground surface there is no difference between vertical and horizontal wave polarization. Fig. 4.4. Thus, for 0 0, cos 0 1, sin 0 0, all the formulas above reduce to: | E 1 | | E 0 | | E 2 | | E 0 | r 1 r 1 2 r 1 (4.17) 8 RV RH r 1 r 1 (4.18) 2 TV TH r 1 It should be noted that the results presented by (4.17) are correct only for 1 2 [6]. Moreover, for the reflected wave E 1 the sign convention is that for vertical polarization (4.18). This means that if 2 1 there is a phase reversal of the reflected wave. In the case of vertical polarization there is a special angle of incidence, called the Brewster angle, for which there is no reflected wave. For simplicity we will assume that the condition 1 2 is valid. Then from (4.15) it follows that the reflected wave E 1 limits to zero when the angle of incidence is equal to Brewster’s angle 2 1 0 Br tan 1 (4.19) Another interesting phenomenon that follows from the presented formulas is called total reflection. It takes place when the condition of 2 1 (or n2 n1 ) is valid. In this case from Snell’s law (4.9) it follows that, if 2 1 , then Consequently, when 0 0 kr the reflection angle 1 , where 2 0 kr c sin 1 2 1 1 0 . (4.20) For waves incident at the surface (this case is realistic for ferroconcrete building’s wall surfaces) under the critical angle 0 0kr c there is no refracted wave within the second medium; the refracted wave is propagated along the boundary between the first and second media and there is no energy flow across the boundary of these two media. Therefore, this phenomenon called in literature total internal reflection (TIR), and the smallest incident angle 0 for which we get TIR, is called the critical angle 0 0kr c . 9 4.3. Analysis of Total Internal Reflection in Optics Now, introducing the index of rays refraction for both media which can be defined as n1 1 1 and n2 2 2 , we can in the same manner consider the ray reflection from intersection of two media by introducing instead of propagation parameters (see above). If so, the main reflection formulas (4.12) and (4.14) can be rewritten as: 1) For TE-waves, as shown in Fig. 4.2, we immediately have from (4.12) the amplitude of reflected and refracted rays respectively [1-3]: 1 n22 n12 sin 2 0 2 | E1 | | E 0 | n1 cos 0 1 n22 n12 sin 2 0 2 n1 cos 0 | E 2 | | E 0 | 2n1 cos 0 n1 cos 0 1 n22 n12 sin 2 0 2 (4.21a) (4.21b) For TM-waves, as shown in Fig. 4.3, the amplitudes of the reflected and refracted rays can be written as [1-3]: 1 2 n2 cos 0 n1 n22 n12 sin 2 0 2 | E1 | | E 0 | 1 2 n2 cos 0 n1 n22 n12 sin 2 0 2 2n1n2 cos 0 | E 2 | | E 0 | 1 2 n2 cos 0 n1 n22 n12 sin 2 0 2 where, once more, n12 1 1 and n22 2 2 . (4.22a) (4.22b) 10 We can rewrite now a Snell’s, presented above for 0 1 , as [1-3] (see also the geometry of the problem shown in Fig. 4.1): n1 sin 1 n2 sin 2 (4.23) or sin 0 sin 1 n2 sin 2 n1 (4.24) If the second medium is less optically dense than the first medium which consist the incident ray with amplitude | E 0 | , that is, n1 n2 , from (4.24) follows that sin 0 n2 n1 or n1 sin 0 1 n2 (4.25) The value of incident angle 0 for which (4.25) becomes true is known as a critical angle, which was introduced above. We now define its meaning by use a ray concept [4]. If a critical angle is determined by sin kr sin c n2 n1 (4.26) then for all values of incident angles 0 kr c the light is totally reflected at the boundary of two media. This phenomenon is called in ray theory the total internal reflection (TIR) of rays, the effect, which is very important in light propagation in fiber optics. Figure 4.5 shows effect of total reflection when 0 c . We also can introduce another main parameters usually used in fiber optic communication (see Lecture 8). Thus, the effective index of refraction is defined as: neff n1 sin 0 (4.27) 11 When the incident ray angle 0 90 0 , neff n1 , and when 0 c , neff n 2 . Fig. 4.5 In fiber optics there is another parameter usually used, called numerical aperture of fiber optic guiding structure, denoted as N.A., N . A. n1 sin c sin a (4.28) where 2 a is so-called the angle of existence of full communication [5, 6], when total internal reflection occurs in fiber optic structure. Accounting for cos 2 1 sin 2 , we finally get N . A. n12 n22 1/ 2 (4.29) Sometimes, in fiber optic physics, designers used the parameter, called relative refractive index difference: n 2 1 n22 2 n12 N.A 2 2 n12 (4.30) Using above formulas, we can find relations between these two engineering parameters (we will talk about them in Lecture 8 discussing about fiber optic parameters and properties): N . A. n1 ()1 / 2 (4.31) 12 Example 1: Let us consider that n1 1.45 and 0.02 (2%) . Find: N.A. and 2 a . Solution 1) According to (4.31) N. A. n1 ()1/ 2 sin 1 a Then: N . A. n1 ()1 / 2 1.45 2 0.02 0.29 2) N. A. sin 1 a sin 1 0.29 15.660 Then: 2 a 2 15.660 31.330 Let us now explain the total internal reflection from another point of view. When we have total internal reflection, we should assume that there would be no electric field in the second medium. This is not the case, however. The boundary conditions presented above require that the electric field be continuous at the boundary, that is at the boundary the field in region1 and region 2 must be equal. The exact solution shows that due to total internal reflection we have in region 1 standing waves caused by interference of incident and fully reflected waves, whereas in region 2 a finite electric field decays exponentially away from boundary and carries no power into the second medium. This wave called an evanescent field (see Fig. 4.6). This field attenuate away from boundary as E exp z (4.32) where the attenuation factor equals 2 n12 sin 2 i n22 (4.33) 13 At can be seen from (4.31), at the critical angle i c 0 , and attenuation increases as the incident angle increases beyond the critical angle defined by (4.31). Because is so small near the critical angle, the evanescent fields penetrate deeply beyond the boundary but do so less and less as the angle increases. Fig. 4.6 However, behavior of Fresnel’s formulas (4.21)-(4.22) depends on boundary conditions. Thus if the fields are to be continuous across the boundary, as required by Maxwell’s equations, there must be a field disturbance of some kind in the second media (see Fig. 4.1). To investigate this disturbance we can use Fresnel’s formulas. We, first of all rewrite cos 2 (1 sin 2 2 ) 1/ 2 . For 2 kr we can present sin 2 by use some additional function sin 2 cosh , which can be more than unit. If so, cos 2 j (cosh 2 1) 1/ 2 j sinh . Hence we can write the field component in the second medium to vary as (for nonmagnetic materials 1 2 0 ) x cosh jz sinh exp j t n2 c (4.34a) 14 or x cosh n z sinh exp 2 exp j t n2 c c (4.34b) This formula represents a ray traveling in the x-direction in the second medium (that is, parallel to the boundary) with amplitude decreasing exponentially in the z-direction (at right angles to the boundary). The rate at which the amplitude decreases with z can be written 2z sinh exp 2 (4.35) where is a wavelength of the light in the second medium. As seen, the wave attenuate significantly (~ e 1 ) over distances z of about 2 . Example 2: At the glass-air interface, the critical angle of is krr sin 1 (1 / 15 . ) 418 . 0 . For a light in the glass incident on the glass-air boundary at 60 0 we find that sinh 164 . . Hence the amplitude of the wave in the second medium is reduced by a factor of 5.4 10 3 in a distance of only one wavelength, which is of the order of 1m = 10 -6 m . Even though the wave is propagating in the second medium, it transports no light energy in a direction normal to the boundary. All the light is totally internally reflected (TIR) at the boundary. The fields, which exist in the second medium, give a Poynting vector which averages zero in this direction over one oscillation period of the light wave. The guiding effect is based on TIR phenomenon: all energy transport occurs along the boundary of two media after TIR, without any penetration of light energy inside the intersection. 15 Moreover, according to (4.21)-(4.22) and (4.31), we can note that the totally internal reflected (TIR) wave undergoes a phase change which depends on both the angle of incidence and the field polarization. This directly follows from derivations of Fresnel’s equation (4.21)-(4.22). Namely, for TM wave (i.e., E || -polarization) from (4.22) for 1 2 0 and from above mentioned it follows that E1 jn sinh n2 cos 0 1 E0 jn1 sinh n2 cos 0 (4.36) This complex number provides the phase change on TIR as p where for TM wave ( E || -polarization) we have 2 2 2 p n1 n1 sin 0 n2 tan n22 cos 0 2 1/ 2 (4.37a) and for TE wave ( E -polarization) n12 sin 2 0 n22 s tan 2 n1 cos 0 1/ 2 (4.37b) We note also that there is a close relationship between the light wave phase changes at TIR for both kinds of waves, that is, p tan n12 tan s 2 2 (4.38) and that 2 2 2 p s cos 0 n1 sin 0 n2 tan n1 sin 2 0 2 1/ 2 (4.39) The variations of phase changes, p and s , and their difference, p s , as a function of the incident angle 0 are shown in Fig. 4.7. It is clear that the polarization state of light undergoing TIR will changed as a result of the differential phase change 16 p s . By choosing 0 appropriately and perhaps using two TIRs, it is possible to produce any required final polarization state from any given initial state. It is interesting to note that the reflected ray in TIR appears to originate from a point, which is displaced along the boundary from the point of incidence. Fig. 4.7 This is consistent with the incident ray’s is being reflected from a parallel plane which lies a short distance within the second boundary (see Fig. 4.8). Fig. 4.8 This view is also consistent with the observed phase shift, which is now regarded as being due to the extra optical path traveled by the ray. The displacement is known as 17 the Goos-Hanchen effect and provides an entirely consistent alternative expansion of TIR [4]. Bibliography [1] Jakes, W. C., Microwave Mobile Communications, New York: John Wiley and Son, 1974. [2] Chew, W. C., Waves and Fields in Inhomogeneous Media, New York: IEEE Press, 1995. [3] Lee, W. Y. C., Mobile Cellular Telecommunications Systems, New York: McGraw Hill Publications, 1989. [4] Elliott, R. S., Electromanetics: History, Theory, and Applications, New York: IEEE Press, 1993. [5] Optical Fiber Sensors: Principles and Components. Ed. by J. Dakin and B. Culshaw, Arthech House, Boston-London, 1988. [6] Palais, J. C., Optical Communications, in Handbook: Engineering Electromagnetics Applications, Ed. by R. Bansal, New York: Taylor and Frances, 2006. [7] Kopeika, N. S., A System Engineering Approach to Imaging, Washington: SPIE Optical Engineering Press, 1998.