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S-96.510 Advanced Field Theory Course for Graduate Students Lecture Viewgraphs, Fall Term 2004

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S-96.510 Advanced Field Theory Course for Graduate Students Lecture Viewgraphs, Fall Term 2004 S-96.510 Advanced Field Theory Course for graduate students Lecture viewgraphs, fall term 2004 I.V.Lindell Helsinki University of Technology Electromagnetics Laboratory Espoo, Finland I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 00.01 1 Contents [01] Complex Vectors and Dyadics [02] Dyadic Algebra [03] Basic Electromagnetic Equations [04] Conditions for Fields and Media [05] Duality Transformation [06] Affine Transformation [07] Electromagnetic Field Solutions [08] Singularities and Complex Sources [09] Plane Waves [10] Source Equivalence [11] Huygens’ Principle [12] Field Decompositions Vector Formulas, Dyadic Identites as an appendix I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 00.02 2 Foreword This lecture material contains all viewgraphs associated with the gradu- ate course S-96.510 Advanced Field Theory given at the Department of Electrical and Communications Engineering, fall 2004. The course is based on Chapters 1–6 of the book Methods for Electromagnetic Field Analysis (Oxford University Press 1992, 2nd edition IEEE Press, 1995, 3rd printing Wiley 2002) by this author. The figures drawn by hand on the blackboard could not be added to the present material. Otaniemi, September 13 2004 I.V. Lindell I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 00.03 3 I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 00.04 4 S-96.510 Advanced Field Theory 1. Complex Vectors and Dyadics I.V.Lindell I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.00 5 Complex Vectors • Complex vectors a = ar + jai,(ar, ai real vectors) • Time-harmonic vectors A(t) = A1 cos ωt + A2 sin ωt • Sense of rotation: A1 → A2 shortest way • Correspondence a ↔ A(t) through two mappings • Mapping a → A(t) jωt A(t) = <{ae } = ar cos ωt − ai sin ωt • Inverse mapping A(t) → a a = ar + jai = A(0) − jA(π/2ω) I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.01 6 Special Complex Vectors • Correspondence a = ar + jai ↔ A(t) = ar cos ωt − ai sin ωt • Circularly polarized (CP) vectors a · a = 0 • CP implies ar · ai = 0 and |ar| = |ai| • Linearly polarized (LP) vectors a × a∗ = 0 • LP implies ar × ai = 0 (parallel vectors ar, ai) • Elliptical polarization in general • a and b have same ellipse iff b = ejθa, θ real B(t) = <{bejωt} = <{aej(ωt+θ)} = A(t + θ/ω) I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.02 7 Axial Representation • To find axes of the ellipse of a complex vector a (a · a 6= 0) • Solution through another complex vector b √ | a · a| b = b + jb = √ a = ejθa r i a · a • a and b have same ellipse, same axes (θ real) b · b = br · br + 2jbr · bi − bi · bi = |a · a| > 0 • b · b real and positive ⇒ br · bi = 0, |br| > |bi| • br, bi define the axes of the ellipse of a • br on major axis, bi on minor axis I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.03 8 Helicity Vector 1 • Helicity vector p(a) (’polarization vector’) of a = ar + jai a × a∗ 2a × a p(a) = = i r ∗ 2 2 ja · a |ar| + |ai| • Properties: • p(a) = [p(a)]∗ is a real vector • a → a∗ changes sense of rotation: p(a) → p(a∗) = −p(a) • Linearly polarized vector a × a∗ = 0 ⇒ p(a) = 0 • Circularly polarized vector a · a = 0 ⇒ |p(a)| = 1 • Elliptically polarized vector 0 < |p(a)| < 1 I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.04 9 Helicity Vector 2 • More properties: a × a∗ 2a × a p(a) = = i r ∗ 2 2 ja · a |ar| + |ai| • p(a) orthogonal to plane of a, points RH direction • p(αa) = p(a), α 6= 0, magnitude of a has no effect • |p(a)| = 2e/(e2 + 1), e = ellipticity (axial ratio) • p(a) gives info on ellipticity, plane, and sense of rotation of a • p(a) does not give info on magnitude or orientation of ellipse on its plane I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.05 10 Vector Bases • Three complex vectors a1, a2, a3 form a basis if a1 × a2 · a3 6= 0 • Gibbs’ identity by expanding (a1 × a2) × (a3 × b) in two ways: (a1 × a2 · a3)b = a1(a2 × a3 · b) + a2(a3 × a1 · b) + a3(a1 × a2 · b) • Define reciprocal basis 0 a2 × a3 0 a3 × a1 0 a1 × a2 a1 = , a2 = , a3 = a1 × a2 · a3 a1 × a2 · a3 a1 × a2 · a3 • Expansion for any vector b 3 3 X 0 X 0 b = ai(ai · b) = ai(ai · b) i=1 i=1 I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.06 11 Dyadic algebra • Josiah Willard Gibbs 1884: dyadic algebra • Dyadic = linear mapping from vector to vector • Example 1: projection on line parallel to unit vector u b = u(u · a) = (uu) · a • uu = projection dyadic • Example 2: projection on plane transverse to unit vector u b = a − (uu) · a = (I − uu) · a • It = I − uu projection dyadic (I unit dyadic, mapping to oneself) • uu axial unit dyadic, It transverse (planar) unit dyadic I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.07 12 Dyadic polynomial • Dyad = dyadic product of two vectors ab 6= ba • Dyadic = polynomial of dyads N X A = a1b1 + a2b2 + ··· + aN bN = aibi i=1 • Dyadic as a mapping N N X X A · c = ( aibi) · c = ai(bi · c) = a1(b1 · c) + ··· + aN (bN · c) i=1 i=1 N N X X c · A = c · ( aibi) = (c · ai)bi = (c · a1)b1 + ··· + (c · aN )bN i=1 i=1 N T T X c · A = A · c, transposed dyadic A = biai i=1 I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.08 13 Dyadic expressions • There is no unique expression for dyadics. Two expressions repre- sent same dyadic if they map all vectors in the same way: A1 · c = A2 · c, for all c, ⇒ A1 = A2 • any dyadic can be expressed as a sum of three dyads • Example: take a vector ONB (u1, u2, u3) N N X X A = aibi = (u1u1 · ai + u2u2 · ai + u3u3 · ai)bi i=1 i=1 N N N X X X = u1 (u1 · ai)bi + u2 (u2 · ai)bi + u3 (u3 · ai)bi i=1 i=1 i=1 = u1c1 + u2c2 + u3c3 • (u1, u2, u3) arbitrary ON base ⇒ infinite number of representations I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.09 14 Dyadic classification • Any dyadic can be expressed as a1b1 + a2b2 + a3b3 • Planar dyadic can be expressed as sum of two dyads a1b1 + a2b2 • Linear dyadic can be expressed as a single dyad a1b1 • Dyadic which cannot be expressed as a planar dyadic is complete • Complete dyadic maps volumes to volumes • Planar dyadic maps volumes to plane • Linear dyadic maps volumes to line • (Note: complex vectors ⇒ planes and lines in complex space) • Inverse mapping exists only for complete dyadics I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.10 15 Symmetric dyadics • Symmetric dyadic A = AT = (1/2)(A + AT ) 3 3 3 X 1 X 1 X a b = (a b +b a ) = [(a +b )(a +b )−(a −b )(a −b )] i i 2 i i i i 4 i i i i i i i i i=1 i=1 i=1 • Symmetric dyadic can be expressed as sum of symmetric dyads P cici but not necessarily in three terms • Unit dyadic symmetric I = u1u1 + u2u2 + u3u3 independent of ONB • Examples: I = uxux + uyuy + uzuz = urur + uθuθ + uϕuϕ 3 3 X X I · a = ( uiui) · a = ui(ui · a) i=1 i=1 I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.11 16 Antisymmetric dyadics • Antisymmetric dyadic A = −AT = (1/2)(A−AT ) operates through a vector d(A) 3 3 3 X 1 X 1 X A·c = a (b ·c) = [a (b ·c)−b (a ·c)] = (b ×a )×c = d(A)×c i i 2 i i i i 2 i i i=1 i=1 i=1 3 1 X 1 d(A) = b × a = (b × a + b × a + b × a ) 2 i i 2 1 1 2 2 3 3 i=1 • Denoting a × b = a × (I · b) = (a × I) · b, 3 3 X 1 X A = a b = (b × a ) × I = d(A) × I i i 2 i i i=1 i=1 • Any dyadic of the form a × I = I × a is antisymmetric I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.12 17 Products of dyadics and vectors • Dot products A · c and c · A give vectors X X A · c = ( aibi) · c = ai(bi · c) X X c · A = c · ( aibi) = (c · ai)bi • Example: antisymmetric dyadic A = a × I = I × a A · c = (a × I) · c = a × c, c · A = c · (a × I) = c × a • Cross products A × c and c × A give dyadics X X A × c = ( aibi) × c = ai(bi × c) X X c × A = c × ( aibi) = (c × ai)bi • Note: a · (b × A) = (a × b) · A but 6= a × (b · A)! I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.13 18 Dot-product of dyadics • Dot product A · B gives a dyadic X X X A · B = ( aibi) · ( cjdj) = (bi · cj)aidj i j i,j • Dot product is associative but not commutative like matrix product A · (B · C) = (A · B) · C, A · B 6= B · A (in general) • Powers of dyadics A2 = A · A, An = A · An−1 = An−1 · A, A0 = I • Inverse of a dyadic possible for complete dyadics only A · a = b, ⇒ a = A−1 · b (A · B)−1 = B−1 · A−1 I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.14 19 Double-cross product of dyadics × • Double-cross product A×B gives a dyadic × X × X X A×B = ( aibi)×( cjdj) = (ai × cj)(bi × dj) i j i,j • Double-cross product is commutative but not associative × × × × × × A×B = B×A A×(B×C) 6= (A×B)×C (in general) • Double-cross square 1 A(2) = A×A, I(2) = I 2 × • Inverse of a dyadic can be expressed as A(2)T A−1 = detA I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.15 20 Double-dot product of dyadics • Double-dot product A : B gives a scalar X X X A : B = ( aibi):( cjdj) = (ai · cj)(bi · dj) i j i,j A : B = B : A = AT : BT , A : BT = B : AT • If A antisymmetric and B symmetric, A : B = 0 ab : A = a · A · b, ab : I = a · b, A : B = (A · BT ): I X X A : I = ( aibi): I = ai · bi = trA trI = 3, trace 1 A(2) : I = (A×A): I = spmA 0sum of principal minors0 2 × 1 1 detA = (A×A): A = A(2) : A determinant 6 × 3 I.V.Lindell: Advanced Field Theory, 2004 Helsinki University of Technology 01.16 21 Dyadic identities • Dyadic identities are needed in dyadic analysis.
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