Pulsed EM Fields: Their Modeling and Potentialities by Martin - PowerPoint PPT Presentation

Pulsed EM Fields: Their Modeling and Potentialities by Martin ˇ Stumpf B rno U niversity of T echnology SIX Research Centre a 3082/12 • 616 00 Brno • The Czech Republic Technick´ • T : +420-5-4114-6539 E : stumpf@feec.vutbr.cz Presentation given at: Hamburg, Germany, 9 January 2015 01 Title and affiliation � 2014 Brno University of Technology c SIX Research Centre

Presentation motivation • ∀ PHYSICAL PHENOMENA ∈ SPACE – TIME • R 3 × R + CAUSALITY Im( s ) CAUSAL f ( t ) FREQUENCY � DOMAIN ANALYTIC ˆ f ( s ) Re( s ) = 0 Re( s ) × 0 s + 0 • THE CONCEPT OF ’FREQUENCY’ = MATHEMATICAL ARTIFACT • ’FREQUENCY DOMAIN’ does away with CAUSALITY UNIQUENESS 02 Motivation � 2014 Brno University of Technology c SIX Research Centre

Presentation overview • Generalized-Ray Theory (GRT) and its extensions • EM Green’s functions in layered media • Applications to bounded domains amd inclusion of relaxation phenomena • Time-Domain RADAR Imaging • Time-Domain Modeling of P/G Structures • Analytical modeling of rectangular P/G structures • Time-Domain Contour Integral Method (TD-CIM) • EM radiation, (self-)reciprocity and mutual coupling • Conclusions 03 Outline � 2014 Brno University of Technology c SIX Research Centre

GENERALIZED-RAY THEORY 04 Generalized-Ray Theory � 2014 Brno University of Technology c SIX Research Centre

Conventions in notation • SUBSCRIPT NOTATION • Latin lower-case = { 1 , 2 , 3 } , Greek lower-case = { 1 , 2 } • position vector: x → x n (tensor of rank 1) • spatial derivative: ∂/∂x n → ∂ n (tensor of rank 1) • Kronecker symbol: δ m,n (symmetrical unit tensor of rank 2) • Levi-Civita symbol: e i,j,k (completely antisymmetrical unit tensor of rank 3) • SUMMATION CONVENTION • δ m,m = � 3 m =1 δ m,m = 3 • e k,l,m e k,l,m = � 3 � 3 � 3 m =1 e k,l,m e k,l,m = 6 k =1 l =1 • ∇ [ ∇ · v ] → ∂ i ∂ k v k = δ i,j ∂ j ∂ k v k • ∇ × [ ∇ × v ] → e i,j,k e k,l,m ∂ j ∂ l v m 05 GRT: Notation � 2014 Brno University of Technology c SIX Research Centre

Problem definition • ND domains described via ǫ N = ǫ N ( x 3 ) and µ N = µ N ( x 3 ) , for N = { 1 , ..., ND } with c N = ( ǫ N µ N ) − 1 / 2 > 0 x 3 receiver { ǫ ND , µ ND } D ND ▽ x 3; ND • the configuration is linear, instantaneously reacting, { ǫ S +1 , µ S +1 } D S +1 x 3; S +1 { ǫ S , µ S } D S source time invariant, shift invariant in the horizontal direction × x 3; S { ǫ 2 , µ 2 } D 2 • the EM field is causally related to the action of x 3;2 { ǫ 1 , µ 1 } D 1 an impulsive source placed at x 3 = x 3; S • a source starts to act at t = 0 and prior to this instant the EM fields vanish throughout the configuration • an arbitrary probe position • no relaxation mechanism incorporated 06 GRT: Problem description � 2014 Brno University of Technology c SIX Research Centre

EM field decomposition † for k = { 1 , 2 , 3 } and p = { 1 , 2 , 3 } : π = { 1 , 2 } − e k,m,p ∂ m H p + ǫ∂ t E k = − J k p = 3 ρ = { 1 , 2 } e j,n,r ∂ n E r + µ∂ t H j = − K j for j = { 1 , 2 , 3 } and r = { 1 , 2 , 3 } : r = 3 − e k,µ,π ∂ µ H π − e k,µ, 3 ∂ µ H 3 − e k, 3 ,π ∂ 3 H π + ǫ∂ t E k = − J k DECOMPOSED EM FIELD EQUATIONS e j,ν,ρ ∂ ν E ρ + e j,ν, 3 ∂ ν E 3 + e j, 3 ,ρ ∂ 3 E ρ + µ∂ t H j = − K j • for a point source: { J k , K j } ( x , t ) = { j k , k j } ( t ) δ ( x 1 , x 2 , x 3 − x 3; S ) ⇒ lim x 3 ↓ x 3; N H π − lim x 3 ↑ x 3; N H π = e 3 ,π,κ j κ ( t ) δ ( x 1 , x 2 ) δ N,S INTERFACE lim x 3 ↓ x 3; N E ρ − lim x 3 ↑ x 3; N E ρ = e 3 ,ι,ρ k ι ( t ) δ ( x 1 , x 2 ) δ N,S CONDITIONS † M. ˇ Stumpf, A. T. de Hoop and G. A. E. Vandenbosch, “Generalized ray theory for time-domain electromagnetic fields in horizontally layered media,” IEEE Trans. AP , vol. 61, no. 5, pp. 2676–2687, May 2013. 07 GRT: EM field decomposition � 2014 Brno University of Technology c SIX Research Centre

Transform-domain field representation † LAPLACE � ∞ ˆ E k ( x , s ) = t =0 exp( − st ) E k ( x , t )d t TRANSFORMATION • Lerch’s sequence: { s ∈ R : s = s 0 + nh, s 0 > 0 , h > 0 , n = 0 , 1 , 2 , ... } ⇒ transformation property: ∂ t → s (for zero initial conditions) WAVE-SLOWNESS FIELD REPRESENTATION � s � 2 � ˆ α 2 ∈ R exp( − i sα µ x µ ) ˜ � E k ( x , s ) = α 1 ∈ R d α 1 E k ( α 1 , α 2 , x 3 , s )d α 2 2 π ⇒ transformation property: ∂ ν → ˜ ∂ ν = − i sα ν † M. ˇ Stumpf, A. T. de Hoop and G. A. E. Vandenbosch, “Generalized ray theory for time-domain electromagnetic fields in horizontally layered media,” IEEE Trans. AP , vol. 61, no. 5, pp. 2676–2687, May 2013. 08 GRT: Transform domains � 2014 Brno University of Technology c SIX Research Centre

Source-type field representations † E π − s 2 γ 2 ˜ 3 ˜ E π = sµ ˜ J π − ˜ ∂ π ˜ ∂ ν ˜ J ν /sǫ − ˜ ∂ π ∂ 3 ˜ J 3 /sǫ − e π,ρ, 3 ∂ 3 ˜ K ρ + e π,ρ, 3 ˜ ∂ ρ ˜ ∂ 2 K 3 H ρ − s 2 γ 2 ˜ K ρ − ˜ ∂ ρ ˜ K π /sµ − ˜ J κ − e ρ,κ, 3 ˜ 3 ˜ H ρ = sǫ ˜ ∂ π ˜ ∂ ρ ∂ 3 ˜ K 3 /sµ + e ρ,κ, 3 ∂ 3 ˜ ∂ κ ˜ ∂ 2 J 3 + TRANSFORM-DOMAIN INTERFACE CONDITIONS AND LINEARITY • transform-domain representations for tangential EM field components VERTICAL ELECTRIC E π ( α 1 , α 2 , x 3 , s ) = ˜ ˜ j 3 ( s ) ˜ ∂ π ∂ 3 ˆ G J ⊥ /sǫ DIPOLE H ρ ( α 1 , α 2 , x 3 , s ) = e ρ,π, 3 ˜ ˜ j 3 ( s ) ˜ ∂ π ˆ G J ⊥ HORIZONTAL ELECTRIC G J � + ˜ DIPOLE E π ( α 1 , α 2 , x 3 , s ) = − sµ ˆ ˜ j π ( s ) ˜ ∂ π ˜ ∂ ν ˆ j ν ( s ) ˜ G Q � /sǫ � ˜ G J � − ˜ G Q � − ˜ ˜ j π ( s ) ∂ 3 ˜ j ν ( s ) e ρ, 3 ,π ˜ H ρ ( α 1 , α 2 , x 3 , s ) = e ρ, 3 ,π ˆ ∂ ν ˆ G J � � � s 2 γ 2 ∂ π ∂ 3 † M. ˇ Stumpf, A. T. de Hoop and G. A. E. Vandenbosch, “Generalized ray theory for time-domain electromagnetic fields in horizontally layered media,” IEEE Trans. AP , vol. 61, no. 5, pp. 2676–2687, May 2013. 09 GRT: Source-type field representations � 2014 Brno University of Technology c SIX Research Centre

Generalized-ray EM field expansion (1) † GENERAL ˜ N exp[ − sγ N ( x 3 − x 3; N )] + W − G = W + N exp[ − sγ N ( x 3; N +1 − x 3 )] SOLUTION • { W + N , W − N } are transform-domain upgoing/downgoing wave amplitudes N = ¯ N + ¯ S + − N W − W + S ++ N W + N − 1 + X + SCATTERING N DESCRIPTION N − 1 = ¯ N + ¯ W − S −− N W − S − + N W + N − 1 + X − N − 1 W + D N W − { ¯ N , ¯ S + − S −− N } = { S + − N , S −− N N N } exp( − sγ N d N ) x 3; N { ¯ N , ¯ S − + N , S − + S ++ N } = { S ++ N } exp( − sγ N − 1 d N − 1 ) W + D N − 1 W − N − 1 N − 1 † M. ˇ Stumpf, A. T. de Hoop and G. A. E. Vandenbosch, “Generalized ray theory for time-domain electromagnetic fields in horizontally layered media,” IEEE Trans. AP , vol. 61, no. 5, pp. 2676–2687, May 2013. 10 GRT: Generalized-ray EM field expansion � 2014 Brno University of Technology c SIX Research Centre

Generalized-ray EM field expansion (2) = ¯ + ¯ S A + − W A − W A + S A ++ W A + N − 1 + X A + HORIZONTAL-SOURCE N N N N N COUPLING N − 1 = ¯ + ¯ W A − S A −− W A − S A − + W A + N − 1 + X A − N − 1 N N N = ¯ + ¯ N − 1 + ¯ + ¯ P B + − W B − C B + − W A − W B + P B ++ W B + C B ++ W A + N − 1 + X B + N N N N N N N N N − 1 = ¯ + ¯ N − 1 + ¯ + ¯ W B − P B −− W B − P B − + W B + C B −− W A − C B − + W A + N − 1 + X B − N − 1 N N N N N N GENERALIZED-RAY x 3;4 (NEUMANN) EXPANSION X + 3 m =0 S m · X + S M +1 · W W = � M × x 3;3 X − 2 ⇒ CAGNIARD-deHOOP x 3;2 PEC INVERSION 11 GRT: Generalized-ray EM field expansion � 2014 Brno University of Technology c SIX Research Centre

Generalized-ray constituent (1) � ∞ � ∞ � s � 2 ˆ GENERIC FORM ˆ R ( x , s ) = Q ( s ) d α 1 Π( α 1 , α 2 ) 2 π α 1 = −∞ α 2 = −∞ � � N �� � × exp − s i α 1 x 1 + i α 2 x 2 + γ K ( α 1 , α 2 ) Z K d α 2 K =1 R ( x , s ) = s 2 ˆ • mapping M : { α 1 , α 2 } → { p, q } with ˆ Q ( s ) ˆ K ( x , s ) � ∞ � i ∞ 1 ˆ ¯ K ( x , s ) = d q Π( p, q ) 4 π 2 i q = −∞ p = − i ∞ PROPAGATION � � �� N � × exp − s FACTOR pr + γ K ( p, q ) Z K ¯ d p K =1 12 GRT: Generalized-ray constituent (1) � 2014 Brno University of Technology c SIX Research Centre

Generalized-ray constituent (2) CAGNIARD-deHOOP PATH pr + � N for { τ ∈ R ; τ > 0 } K =1 ¯ γ K ( p, q ) Z K = τ Im( p ) C BW p -plane C HW Re( p ) 0 HEAD-WAVE PART BODY-WAVE PART 13 GRT: Generalized-ray constituent (2) � 2014 Brno University of Technology c SIX Research Centre

Body-wave part s -DOMAIN PROPAGATOR � Q BW ( τ ) � ∞ � ∂p BW exp( − sτ )d τ 1 � �� K BW ( x , s ) = p BW , q p BW , q ˆ ¯ � � Im Π d q π 2 ∂τ τ = T BW (0) q =0 ⇓ LERCH’S UNIQUENESS THEOREM ⇓ t -DOMAIN PROPAGATOR t -DOMAIN CONSTITUENT � Q BW ( t ) if t < T BW (0) Π ∂p BW � � � 0 K BW ( x , t ) = 1 ⇒ R BW ( x , t ) = ¯ Im d q ( t ) ∗ K BW ( x , t ) if t > T BW (0) ∂ 2 π 2 ∂t t Q ( t ) q =0 14 GRT: Generalized-ray constituent (3) � 2014 Brno University of Technology c SIX Research Centre