βi β r medium 1 θ i θ r y θ t β t

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1 W.C.Chew ECE 350 Lecture Notes Date:November 7, Reections and Refractions of Plane Waves. Hr Ei Hi βi β r Er medium θ i θ r μ, ε y θ t μ, ε medium x z Ht β t Et Perpendicular Case (Transverse Electric or TE case) When an incident wave impinges on a dielectric interface, a reected wave as well as a transmitted wave is generated. We can express the three waves as E i = ^ye 0 e ;j ir () E r = ^y? E 0 e ;jrr () E t = ^y? E 0 e ;jtr : (3) The electric eld is perpendicular to the xz plane, and i, r,and t are their respective directions of propagation. The 's are also known as propagation vectors. In particular, i = ^x ix +^z iz (4) r = ^x rx ; ^z rz (5) tx = ^x tx +^z tz : (6) Since E i and E r are in medium, we have + = ix iz =! (7) + = rx rz =! (8)

2 and for E t in medium, we have tx + tz = =! : (9) (7), (8), and (9) are known as the dispersion relations for the components of the propagation vectors. From the gure, we note that ix = sin i iz = cos i (0) rx = sin r rz = cos r () tx = sin t tz = cos t : () To nd the unknown? and?, we need to match boundary conditions for the elds at the dielectric interface. The boundary conditions are the equality of the tangential electric and magnetic elds on both sides of the interface. The magnetic elds can be derived via Maxwell's equations. Similarly, H i = re i ;j! = i E i! =(^z ix ; ^x iz ) E 0! e ;j ir : (3) H r =(^z rx +^x rz )?E 0! e ;jrr (4) H t =(^z tx ; ^x tz )?E 0! e ;jtr : (5) Continuity of the tangential electric elds across the interface implies E 0 e ;j ixx +? E 0 e ;jrxx =? E 0 e ;jtxx : (6) The above equation is to be satised for all x. This is only possible if ix = rx = tx = x : (7) This condition is known as phase matching. From (0), (), and (), we know that (7) implies sin i = sin r = sin t : (8) The above implies that r = i. Furthermore, p sin i = p sin t : If we dene a refractive index n i = q i i 00 n sin i = n sin t, then (9a) becomes (9a) (9b) which is the well known Snell's Law. Consequently, equation (6) becomes +? =? : (0)

3 From the continuity of the tangential magnetic elds, we have ; iz E 0! + rz? E 0! = ; tz? E 0! : () Since r = i, we have iz = rz. Therefore, () becomes Solving (0) and (), we have ;? = tz iz? : ()? = iz ; tz iz + tz (3)? = Using (0), (), and (), we can rewrite the above as iz iz + tz : (4)? = cos i ; cos t cos i + cos t (5)? = cos i cos i + cos t : (6) If the media are non-magnetic so that = = 0, we can use (9) to rewrite (5) as If cos i ; q ; sin i? = : (7) cos i + q ; sin i q sin i >, which is possible if >, when i <, then? is of the form? = A ; jb A + jb (8) which always has a magnitude of. In this case, all energy will be reected. This isq known as a total internal reection. This occurs when i > c where sin c =. or c = sin ; r < : (9) 3

4 When i = c, t = 90 from (9). The gure below denotes the phenomenon. β i less than critical angle larger than critical angle θc at critical angle β t less than critical angle x z When i > c, tz = p ; sin i, or tz =! p 0 ; sin i : (30) The quantity in the parenthesis is purely negative, so that tz = ;j tz (3) a pure imaginary number. In this case, the electric eld in medium is E t = ^y? E 0 e ;jxx;tz z : (3) The eld is exponentially decaying in the positive z direction. We call such a wave an evanescent wave, or an inhomogeneous wave as opposed to uniform plane wave. The magnitude of a uniform plane waveis aconstant of space while the magnitude of an evanescent wave or an inhomogeneous wave is not a constant of space. The corresponding magnetic eld is H t =(^z x +^xj tz )?E 0 e ;jxx;tzz : (33)! 4

5 The complex power in the transmitted wave is S = E t H t =(^x x +^zj tz ) j?j je 0 j! e ;tz z : (34) We note that S x is pure real implying the presence of net time average power owing in the ^x-direction. However, S z is pure imaginary implying that the power that is owing in the ^z-direction is purely reactive. Hence, no net time average power is owing in the ^z-direction. Parallel case (Transverse Magnetic or TM case) In this case, the electric eld is parallel to the xz plane that contains the plane of incidence. Hi Ei β i θ i y θ r θ t β t β r Hr Er medium μ, ε μ, ε medium x z Ht Et as The magnetic eld is polarized in the y direction, and they can be written H i = ^y E 0 e ;j ir (35) E H 0 r = ;^y k e ;jrr (36) H t = ^y k E 0 e ;jtr : (37) We put a negative sign in the denition for k to follow the convention of transmission line theory, where reection coecients are dened for voltages, and hence has a negative sign when used for currents. The magnetic eld is the analogue of a current in transmission theory. 5

6 In this case, the electric eld has to be orthogonal to and ^y, and they can be derived using to be E i = ; i H i! E i = ^y i E 0 e ;j ir =(^x iz ; ^z ix ) E 0 e ;j ir (38) E r =(^x rz +^z rx ) ke 0 E t =(^x tz ; ^z tx ) ke 0 Imposing the boundary conditions as before, we have The above can be solved to give and k = tz ; iz iz + tz k = In (43), k will be zero if e ;jrr (39) e ;jtr : (40) + k = tz iz k (4) ; k = k : (4) iz iz + tz = = cos t ; cos i cos t + cos i (43) cos i cos t + cos i : (44) cos t = cos i : (45) Using Snell's Law, or (9), cos t =; sin i,and(45) becomes Solving the above, we get ; sin i = cos i : (46) sin i = ; ;! : (47) Most materials are non-magnetic in this world so that = 0, then r sin i = : (48) + 6

7 The angle for i at which k = 0 is known as the Brewster angle. It is given by r r ib = sin ; = tan ; : (49) + At this angle of incident, the wave will not be reected but totally transmitted. Furthermore, we can show that sin ib + sin tb = (50) implying that ib + tb = : (5) On the contrary,? can never be zero for = 0 or non-magnetic materials. Hence, aplotof k as a function of i goes through a zero while the plot of j? j is always larger than zero for non-magnetic materials. ρ, or ρ ρ ε < ε ρ 0 90 θ i At normal incidence, i.e., i = 0? = k since we cannot distinguish between perpendicular and parallel polarizations. When i = 90, j? j = k =. On the whole, j? j k for non-magnetic materials. The above equations are dened for lossless media. However, for lossy media, if we dene a complex permittivity = ; j, Maxwell's equations! remain unchanged. Hence, the expressions for?,?, k, and k remain the same, except that we replace real permittivities with complex permittivities. For example, if medium is metallic so that!, then, = q! 0, and? = ;, and? =0. Similarly, k = ; and k =0. 7