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					6.013 - Electromagnetics and Applications

Fall 2005

Lecture 9 - Oblique Incidence of Electromagnetic Waves
Prof. Markus Zahn October 6, 2005

I. Wave Propagation at an Arbitrary Angle


√ kz � = kx x + kz z, kx = k sin(θ), kz = k cos(θ), k = ω µ� � � � � � ¯ ˆ ˆ E (x, z, t) = Re Eej(ωt−kz ) ¯y = Re Eej(ωt−kx x−kz z) ¯y i i � � ∂Ey ¯ ∂Ey 1 1 ¯ ¯ ¯ ¯ � × E = −jωµH ⇒ H = − �×E =− −¯x i + iz jωµ ∂z ∂x jωµ � � 1 ˆ ¯ H=− j ˆ¯ j ˆ¯ −j(kx x+kz z) � kz Eix − � kz Eiz e jωµ � ˆ E = − [cos(θ)¯x − sin(θ)¯z ] e−j(kx x+kz z) i i η � � ˆ E j(ωt−kx x−kz z) ¯ H (x, z, t) = Re − (cos(θ)¯x − sin(θ)¯z ) e i i η In general: ¯ k = kx¯x + ky¯y + kz¯z is the wave vector i i i ¯x + y¯y + z¯z is a position vector r = xi ¯ i i
¯ e−jk·r = e−j(kx x+ky y+kz z) � � ¯¯ ¯¯ ¯ ¯¯ � e−jk·r = −j (kx¯x + ky¯y + kz¯z ) e−jk·r = −j ke−jk·r i i i ¯

z � = x sin(θ) + z cos(θ)

¯ � → −j k

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ˆ ˆ ¯ ¯ � × E = −jωµH ⇒ ˆ ˆ ¯ ¯ � × H = jω�E ˆ ¯ �·E =0 ˆ =0 ¯ �·H

ˆ ˆ ¯ ¯ ¯ −jk × E = −jωµH ˆ ˆ ¯ ¯ ¯ k × E = ωµH ˆ ˆ ¯ ¯ ¯ ⇒ −jk × H = jω�E ˆ ˆ ¯ ¯ ¯ k × H = −ω�E ˆ ˆ ¯ ¯ ¯ ¯ ⇒ −jk · E = 0 (k ⊥ E ) ˆ =0 ˆ ¯ ¯ ¯ ¯ ⇒ −jk · H (k ⊥ H)

ˆ ¯ ¯ ¯ k× k×E

�

�

� � �0 ˆ ˆ ˆ ˆ ¯ k� ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = k �· E − E (k · k ) = ωµ k × H = −ω 2 �µE � �

2 2 2 ¯ |k |2 = kx + ky + kz = ω 2 �µ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ A × (B × C ) = B (A · C ) − C (A · B )

1 ¯ ˆ ˆ ˆ ˆ 1ˆ ¯ ¯ ¯ ¯ ¯ (k × E ) S = E × H ∗, H = 2 ωµ � � � � �0 1¯ 1 ¯ ˆ∗ 1 ∗ ∗ ¯� ˆ ˆ ˆ E ¯ ¯ ˆ ˆ ¯ ¯ ¯ ¯ ¯ ˆ S= E× k×E = k(E · E ) − E (� · k ) 2 2ωµ ωµ ¯ ˆ2 ¯ ˆ k|E| ¯ S= 2ωµ ˆ ¯ ¯ (S in the direction of k )

II. Oblique Incidence Onto a Perfect Conductor ¯ A. E Field Parallel to Interface (TE - Transverse Electric) � � ¯ ˆ Ei = Re Ei ej(ωt−kxi x−kzi z)¯y i � � ˆ Ei ¯ Hi = Re (− cos(θi )¯x + sin(θi )¯z )ej(ωt−kxi x−kzi z) i i η kxi = k sin(θi ), kzi = k cos(θi ), k = ω �µ, η = √ � µ �

� � ¯ ˆ Er = Re Er ej(ωt−kxr x+kzr z)¯y i � � ˆ Er j(ωt−kxr x+kzr z) ¯ Hr = Re (cos(θr )¯x + sin(θr )¯z )e i i η kxr = k sin(θr ), kzr = k cos(θr ) Boundary conditions require that

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ˆ ˆ ˆ Ey (x, z = 0) = 0 = Eyi (x, z = 0) + Eyr (x, z = 0) ˆ ˆ = Ei e−jkxi x + Er e−jkxr x = 0 ˆ ˆ ˆ Hz (x, z = 0) = 0 = Hzi (x, z = 0) + Hzr (x, z = 0)
 � �
 1 ˆ −jkxi x ˆ Ei e sin(θi ) + Er e−jkxr x sin(θr ) = 0
 η � � angle of incidence = kxi = kxr ⇒ sin(θi ) = sin(θr ) ⇒ θi = θr
angle of reflection

ˆ ˆ Er = −Ei � � � � ˆ ˆ Ei = Ei (real) ⇒ Ey (x, z, t) = Re Ei e−jkz z − e+jkz z ej(ωt−kx x) = 2Ei sin(kz z) sin(ωt − kx x) �E � � � � � � ˆi ¯ cos(θ) −e−jkz z − e+jkz z ¯x + sin(θ) e−jkz z − e+jkz z ¯z i i H (x, z, t) = Re η � · ej(ωt−kx x) 2Ei � − cos(θ) cos(kz z) cos(ωt − kx x)¯x i = η � + sin(θ) sin(kz z) sin(ωt − kx x)¯z i

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Ky (x, z = 0, t) = −Hx (x, z = 0, t) =

2Ei cos(θ) cos(ωt − kx x) η

� � 2E 2 � � 1 i ˆ ˆ ¯ ¯ ¯ S = Re E × H ∗ = sin(θ) sin2 (kz z)¯x i 2 η ¯ B. H Field Parallel to Interface (TM - Transverse Magnetic) � � ¯ ˆ Ei = Re Ei (cos(θi )¯x − sin(θi )¯z ) ej(ωt−kxi x−kzi z) i i � � ˆ Ei j(ωt−kxi x−kzi z)¯ ¯ e Hi = Re iy η � � ¯ ˆ Er = Re Er (− cos(θr )¯x − sin(θr )¯z ) ej(ωt−kxr x+kzr z) i i � � ˆ Er j(ωt−kxr x−kzr z)¯ ¯ Hr = Re e iy η ˆ ˆ Ex (x, z = 0, t) = 0 ⇒ Ei cos(θi )e−jkxi x − Er cos(θr )e−jkxr x = 0 kxi = kxr ⇒ sin(θi ) = sin(θr ) ⇒ θi = θr ˆ ˆ Ei = Er � � � � � � � � ˆ ¯ ˆ Ei = Ei (real) ⇒ E = Re Ei cos(θ) e−jkz z − e+jkz z ¯x − sin(θ) e−jkz z + e+jkz z ¯z ej(ωt−kx x) i i = 2Ei [cos(θ) sin(kz z) sin(ωt − kx x)¯x − sin(θ) cos(kz z) cos(ωt − kx x)¯z ] i i � � � ˆ Ei � −jkz z ¯ e + e+jkz z ej(ωt−kx x)¯y i H = Re η = 2Ei cos(kz z) cos(ωt − kx x)¯y i η

2Ei cos(ωt − kx x) η σs (x, z = 0) = −�Ez (x, z = 0) = 2�Ei sin(θ) cos(ωt − kx x) Kx (x, z = 0) = Hy (x, z = 0) = Check: Conservation of Charge ¯ ∂σs = 0 ⇒ ∂Kx + ∂σs = 0 �Σ · K + � �� � ∂t ∂x ∂t

surface
 divergence


� � 2E 2 � � 1 i ˆ ˆ ¯ ¯ ¯ S = Re E × H ∗ = sin(θ) cos2 (kz z)¯x i 2 η

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III. Oblique Incidence Onto a Dielectric


¯ A. TE (E � Interface) Waves � � ¯ ˆ Ei = Re Ei ej(ωt−kxi x−kzi z)¯y i � � ˆ Ei j(ωt−kxi x−kzi z) ¯ Hi = Re (− cos(θi )¯x + sin(θi )¯z ) e i i η1 � � ¯ ˆ Er = Re Er ej(ωt−kxr x+kzr z)¯y i � � ˆ Er j(ωt−kxr x+kzr z) ¯ Hr = Re (cos(θr )¯x + sin(θr )¯z ) e i i η1 � � ¯ ˆ Et = Re Et ej(ωt−kxt x−kzt z)¯y i � � ˆ Er ¯ Ht = Re (− cos(θt )¯x + sin(θt )¯z ) ej(ωt−kxt x−kzt z) i i η2 5


kxi = k1 sin(θi ) kxr = k1 sin(θr ) kxt = k2 sin(θt ) kzi = k1 cos(θi ) kzr = k1 cos(θr ) kzt = k2 cos(θt ) √ √ ω ω k1 = c1 = ω �1 µ1 k2 = c2 = ω �2 µ2 c1 = √�1 µ1 c2 = √�1 µ2 �1 �2 µ1 2 η1 = �1 η2 = µ2 � ˆ ˆ ˆ Ey (z = 0− ) = Ey (z = 0+ ) ⇒ Ei e−jkxi x + Er e−jkxr x = Et e−jkxt x � � 1 ˆ ˆ Hx (z = 0− ) = Hx (z = 0+ ) ⇒ −Ei cos(θi )e−jkxi x + Er cos(θr )e−jkxr x η1 1 ˆ = − Et cos(θt )e−jkxt x η2 kxi = kxr = kxt ⇒ k1 sin(θi ) = k1 sin(θr ) = k2 sin(θt ) θi = θr k1 ωc c2 �2 sin(θt ) = sin(θi ) = sin(θi ) = sin(θi ) (Snell’s Law) k2 ωc c1 �1 √ �µ c0 √ Index of refraction: n= =√ = �r µr c �0 µ0 n1 sin(θi ) sin(θt ) = n2 η2 η1 ˆ Er η2 cos(θi ) − η1 cos(θt ) cos(θ ) − cos(θi ) Reflection Coefficent: R= = η2 t = η1 ˆ η2 cos(θi ) + η1 cos(θt ) Ei cos(θt ) + cos(θi ) ˆt 2η2 E 2η2 cos(θi ) � �= Transmission Coefficent: T = = ˆ η2 η2 cos(θi ) + η1 cos(θt ) Ei cos(θ ) + η1
t cos(θt ) cos(θi )

B. Brewster’s Angle of No Reflection R = 0 ⇒ η2 cos(θi ) = η1 cos(θt ) � � c2 2 2 2 2 2 η2 cos2 (θi ) = η2 (1 − sin2 (θi )) = η1 cos2 (θt ) = η1 (1 − sin2 (θt )) = η1 1 − 2 sin2 (θi ) c2 1 � 2 2 � η1 c2 2 2 2 sin2 (θi ) − η2 = η1 − η2 c2 1 � � µ1 �µ1 µ2 �1 µ1 µ2 � 2 sin (θi ) − = − �1 �1 �2 �2 � � �2 µ2 1 − �2 µ1 �1 µ2 2 2 sin (θi ) = sin (θB ) = � �2 1 − µ1 µ2 θB is called the Brewster angle. There is no Brewster angle for TE polarization if µ1 = µ2 . C. Critical Angle of No Power Transmission If c2 > c1 , sin(θt ) can be greater than 1: c2 sin(θt ) = sin(θi )
 c1
 c1 θi = θc ⇒ sin(θi ) = (Real solution for θi if c1 < c2 ) c2 6

θc is called the critical angle. At the critical angle, θt = For θi > θc , sin(θt ) > 1 ⇒ cos(θt ) = ¯ Et ¯ Ht These are non-uniform plane waves.

π 2

⇒ kzt = k2 cos(θt ) = 0.

� 1 − sin2 (θt ) ⇒ −jα = kzt � � ˆ = Re Et ej(ωt−kxt x) e−αz¯y i �	 � ˆ Et = Re	 (− cos(θt )¯x + sin(θt )¯z ) ej(ωt−kxt ) e−αz i i η2

�	 � � � ˆ ˆ∗ 1 1 Et Et ∗ −2αz ˆ ˆ∗ �Sz � = − Re Ey Hx = − Re (− cos(θt )) e η2 2 2 � � jα =0 cos(θt ) = − k2 ¯ D. TM (H � interface) Waves �	 � ¯ ˆ Ei = Re Ei (cos(θi )¯x − sin(θi )¯z ) ej(ωt−kxi x−kzi z) i i �	 � ˆ
 Ei j(ωt−kxi x−kzi z)¯ ¯ Hi = Re e iy
 η1
 �	 � ¯ ˆ Er = Re Er (− cos(θr )¯x − sin(θr )¯z ) ej(ωt−kxr x+kzr z) i i �	 � ˆ Er j(ωt−kxr x+kzr z)¯ ¯ Hr = Re	 e iy
 η1
 �	 � ¯ ˆ Et = Re Et (cos(θt )¯x − sin(θt )¯z ) ej(ωt−kxt x−kzt z) i i �	 � ˆ Et j(ωt−kxt x−kzt z)¯ ¯ Ht = Re	 e iy
 η2
 ˆ ˆ ˆ Ex (x, z = 0− , t) = Ex (x, z = 0+ , t) ⇒ Ei cos(θi )e−jkxi x − Er cos(θr )e−jkxr x = Et cos(θt )e−jkxt x � � 1 ˆ −jkxi x ˆ −jkxr x 1 ˆ Hy (x, z = 0− , t) = Hy (x, z = 0+ , t) ⇒ E1 e + Er e = Et e−jkxt x η2 η1 kxi = kxr = kxt ⇒ θi = θr c2 sin(θt ) = sin(θi ) (Snell’s Law) c1 ˆ Er η1 cos(θi ) − η2 cos(θt ) R= = ˆi η2 cos(θt ) + η1 cos(θ1 ) E ˆ Et 2η2 cos(θi ) T = = ˆi η2 cos(θt ) + η1 cos(θi ) E

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Brewster’s Angle: R = 0 ⇒ η1 cos(θi ) = η2 cos(θt )
2 η1 cos2 (θi )

=

2 η1	(1

2 2 − sin (θi )) = η2 cos2 (θt ) = η2 (1 − sin2 (θt )) � � c2 2 2 2 = η2 1 − 2 sin (θi ) c1 � 2 2 � η c 2 2 2 sin2 (θi ) 2 2 2 − η1 = η2 − η1 c1 � � µ2 � 2 � �1 µ1 − µ1 = µ2 − µ1 sin (θi ) �2 �2� µ2 �2 �1 �1 � �1 µ1 1 − �2 µ2 sin2 (θi ) = sin2 (θB ) =	 � �2 1 − �1 �2

2

If µ1 = µ2 : 1	 sin (θB ) = ⇒ tan(θB ) = 1 + �1 �2 π θB + θt = 2 1 1 = + 1 ⇒ θ C > θB sin2 (θB ) sin2 (θC )
2

�

�2 �1

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