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1 MIT OpenCourseWare /ESD.03J Electromagnetics and Applications, Fall 005 Please use the following citation format: Markus Zahn, 6.03/ESD.03J Electromagnetics and Applications, Fall 005. (Massachusetts Institute of Technology: MIT OpenCourseWare). (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit:

2 Electromagnetics and Applications Fall 005 Lecture 9 - Oblique Incidence of Electromagnetic Waves Prof. Markus Zahn October 6, 005 I. Wave Propagation at an Arbitrary Angle From Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 987. Used with permission. In general: z = x sin(θ) + z cos(θ) kz = k x x + k z z, k x = k sin(θ), k z = k cos(θ), k = ω µɛ Ee j(ωt kz ) Ē(x, z, t) = Re ˆ ī y = Re Ee ˆ j(ωt kxx kz z) ī y E y + E y E = jωµ H H = jωµ E = jωµ ī x i z z x [ Ĥ = jk z Ei ˆ x jk z Ei ˆ ] z e j(k xx+k z z) jωµ = Ê j(kxx+kz z) [cos(θ)ī x sin(θ)ī z ] e j(ωt k xx k z z) H (x, z, t) = Re E ˆ (cos(θ)īx sin(θ)ī z ) e k = kx ī x + k y ī y + k z ī z is the wave vector r = x ī x + y ī y + z ī z is a position vector e jk r j(kxx+ky y+kz z) = e e jk r = j (k x i + k y + k z ) e jk r x i y i z = j j k ke jk r

3 Ê = jωµ H ˆ jk Ê = jωµ Ĥ k E ˆ = ωµ H ˆ Ĥ = jωɛ E ˆ jk Ĥ = jωɛ Ê k H ˆ = ωɛ E ˆ E ˆ = 0 jk E ˆ = 0 (k Ēˆ ) H ˆ = 0 jk H ˆ = 0 ( k H) ˆ E ˆ = k k 0 E ˆ Ēˆ(k k ) = ωµ k H ˆ k k k = k x + k y + k z = ω ɛµ A (B C ) = B (A C ) C (A B ) = ω ɛµe ˆ S ˆ = ˆ H ˆ, H ˆ = ( Ēˆ) E ωµ k ( ˆ ˆ k( ˆ ˆ Ê ( 0 ) S = E ωµ k Ê = E Ē ) Ê k ) ωµ ˆ Ŝ = k E (S ˆ in the direction of k ) ωµ II. Oblique Incidence Onto a Perfect Conductor A. E Field Parallel to Interface (TE - Transverse Electric) Ē i = Re e j(ωt k xix k zi z) ī y H i = Re ( cos(θi )ī x + sin(θ i )ī z )e j(ωt k xix k zi z) k xi = k sin(θ i ), k zi = k cos(θ i ), k = ω µ ɛµ, = ɛ Ē r = Re Ê r e j(ωt kxr x+kzrz) ī y Ê r j(ωt kxr x+kzrz) H r = Re (cos(θr )ī x + sin(θ r )ī z )e Boundary conditions require that k xr = k sin(θ r ), k zr = k cos(θ r )

4 From Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 987. Used with permission. Ê y (x, z = 0) = 0 = Ê yi (x, z = 0) + Ê yr (x, z = 0) = e jk xix + Ê r e jkxr x = 0 Ĥ z (x, z = 0) = 0 = Ĥ zi (x, z = 0) + Ĥ zr (x, z = 0) Eˆ i e jk xix sin(θ i ) + Ê r e jkxr x sin(θ r ) = 0 angle of incidence = k xi = k xr sin(θ i ) = sin(θ r ) θ i = θ r angle of reflection Ê r = [ ] = E i (real) E y (x, z, t) = Re e jkz z e +jkz z e j(ωt kxx) = E i sin(k z z) sin(ωt k x x) [ ˆ [ ] H (x, z, t) = Re E i cos(θ) e jkz z e +jkz z ī x + sin(θ) e jkz z e +jkz z ī z ] j(ωt k e xx) = E i [ cos(θ) cos(k z z) cos(ωt k x x)ī x + sin(θ) sin(k z z) sin(ωt k x x)ī z ] 3

5 K y (x, z = 0, t) = H x (x, z = 0, t) = E i cos(θ) cos(ωt kx x) E S = Re E ˆ H ˆ i = sin(θ) sin (k z z)ī x B. H Field Parallel to Interface (TM - Transverse Magnetic) Ē i = Re (cos(θ i )ī x sin(θ i )ī z ) e j(ωt k xix k zi z) H i = Re e j(ωt k xi x k zi z) ī y j(ωt kxr x+kzr z) Ē r = Re Ê r ( cos(θ r )ī x sin(θ r )ī z ) e Ê r H r = Re e j(ωt k xr x k zrz) ī y E x (x, z = 0, t) = 0 cos(θ i )e jk xix Ê r cos(θ r )e jkxr x = 0 k xi = k xr sin(θ i ) = sin(θ r ) θ i = θ r = Ê r [ [ ] ] = E i (real) E = Re Êi cos(θ) e jkz z e +jkz z ī x sin(θ) e jkz z + e +jkz z ī z e j(ωt kxx) = E i [cos(θ) sin(k z z) sin(ωt k x x)ī x sin(θ) cos(k z z) cos(ωt k x x)ī z ] ˆ H E i = Re e jkz z + e +jkz z e j(ωt kxx) ī y = E i cos(kz z) cos(ωt k x x)ī y K x (x, z = 0) = H y (x, z = 0) = E i cos(ωt kx x) σ s (x, z = 0) = ɛe z (x, z = 0) = ɛe i sin(θ) cos(ωt k x x) Check: Conservation of Charge σ s K x σ s Σ K + = 0 + = 0 }{{} t x t surface divergence E S = Re E ˆ H ˆ i = sin(θ) cos (k z z)ī x 4

6 III. Oblique Incidence Onto a Dielectric From Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 987. Used with permission. A. TE ( E Interface) Waves Ē i = Re e j(ωt k xix k zi z) ī y H i = Re ( cos(θi )ī x + sin(θ i )ī z ) e j(ωt k xix k zi z) Ē r = Re Ê r e j(ωt kxr x+kzrz) ī y Ê r j(ωt kxr x+kzr z) H r = Re (cos(θr )ī x + sin(θ r )ī z ) e Ē t = Re Ê t e j(ωt kxtx kztz) ī y Ê r H t = Re ( cos(θt )ī x + sin(θ t )ī z ) e j(ωt kxtx kztz) 5

7 k xi = k sin(θ i ) k xr = k sin(θ r ) k xt = k sin(θ t ) k zi = k cos(θ i ) k zr = k cos(θ r ) k zt = k cos(θ t ) ω k = c = ω ɛ µ k = c = ω ɛ µ c = ɛµ c = ɛµ µ µ = ɛ = ɛ E y (z = 0 ) = E y (z = 0 + ) e jk xix + Ê r e jkxrx = Ê t e jkxtx H x (z = 0 ) = H x (z = 0 + ) cos(θ i )e jk xix + Ê r cos(θ r )e jkxr x = Ê t cos(θ t )e jkxtx k xi = k xr = k xt k sin(θ i ) = k sin(θ r ) = k sin(θ t ) θ i = θ r k ωc c sin(θ t ) = sin(θ i ) = sin(θ i ) = sin(θ i ) (Snell s Law) k ωc c c 0 ɛµ Index of refraction: n = = = ɛ r µ r c ɛ0 µ 0 sin(θ t ) = n sin(θi ) ˆ Reflection Coefficent: R = E r cos(θ = t) cos(θ i ) cos(θ i ) cos(θ t ) = ˆ + cos(θ i ) + cos(θ t ) sin (θ i ) c = c [ ] µ ɛ µ µ µ µ sin (θ i ) = ɛ ɛ µ ɛ ɛ ɛ ɛ µ ɛ µ sin (θ i ) = sin (θ B ) = µ µ ω E i cos(θt) cos(θ i ) Ê t cos(θ i ) Transmission Coefficent: T = = = Ê i cos(θt ) + cos(θ i ) + cos(θ t ) B. Brewster s Angle of No Reflection R = 0 cos(θ i ) = cos(θ t ) n cos(θ t) cos(θ i ) cos (θ i ) = ( sin (θ i )) = cos (θ t ) = ( sin (θ t )) = c sin (θ i ) θ B is called the Brewster angle. There is no Brewster angle for TE polarization if µ = µ. C. Critical Angle of No Power Transmission If c > c, sin(θ t ) can be greater than : c sin(θ t ) = c sin(θi ) c θ i = θ c sin(θ i ) = c c 6 (Real solution for θ i if c < c )

8 θ c is called the critical angle. At the critical angle, θ t = π k zt = k cos(θ t ) = 0. For θ i > θ c, sin(θ t ) > cos(θ t ) = sin (θ t ) jα = k zt Ē t = Re Ê t e j(ωt kxtx) e αz ī y Ê t H t = Re ( cos(θt )ī x + sin(θ t )ī z ) e j(ωt kxt) e αz These are non-uniform plane waves. S z = Re Ê te Ê y Ĥ x = Re ˆt ( cos(θ t )) e αz = 0 cos(θ t ) = jα k D. TM ( H interface) Waves Ē i = Re (cos(θ i )ī x sin(θ i )ī z ) e j(ωt k xix k zi z) Eˆ i H i = Re e j(ωt k xi x k zi z) ī y j(ωt kxr x+kzr z) Ē r = Re Ê r ( cos(θ r )ī x sin(θ r )ī z ) e Ê r H r = Re e j(ωt k xr x+k zrz) ī y Ē t = Re Ê t (cos(θ t )ī x sin(θ t )ī z ) e j(ωt kxtx kztz) Ê t H t = Re e j(ωt k xtx k ztz) ī y E x (x, z = 0, t) = E x (x, z = 0 +, t) cos(θ i )e jk xix Ê r cos(θ r )e jkxr x = Ê t cos(θ t )e jkxtx ( ) H y (x, z = 0, t) = H y (x, z = 0 +, t) Ê e jk xix + Ê r e jkxr x = Ê t e jkxtx k xi = k xr = k xt θ i = θ r sin(θ t ) = c sin(θi ) (Snell s Law) c Ê r cos(θ i ) cos(θ t ) R = = cos(θ t ) + cos(θ ) Ê t cos(θ i ) T = = cos(θ t ) + cos(θ i ) 7

9 Brewster s Angle: R = 0 cos(θ i ) = cos(θ t ) cos (θ i ) = ( sin (θ i )) = cos (θ t ) = ( sin (θ t )) = c sin (θ i ) sin (θ i ) c = c [ ] µ ɛ µ µ µ µ sin (θ i ) = ɛ ɛ µ ɛ ɛ ɛ ɛ µ sin (θ i ) = sin ɛ µ (θ B ) = ( ɛ ɛ c ) If µ = µ : sin (θ B ) = tan(θ B ) = + ɛ ɛ π θ B + θ t = = + θ C > θ B sin (θ B ) sin (θ C ) ɛ ɛ 8

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