# Broadband Spatiotemporal Differential-Operator Representations For Velocity-Dependent Scattering

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1 Broadband Spatiotemporal Differential-Operator Representations For Velocity-Dependent Scattering Dan Censor Ben Gurion University of the Negev Department of Electrical and Computer Engineering Beer Sheva, Israel 84105, * PIER Progress In Electromagnetic Research, Vol. 58, pp , Download from choose file diff-paper.pdf 1

2 1. Introduction 2. Spatiotemporal Differential-Operators 3. The Scattering Algorithm 4. Free-Space Impulsive Plane-Wave Excitation 5. Broadband Scattering by Cylinders At-Rest 6. Broadband Scattering by Moving Cylinders 7. Broadband Scattering by A Half-Plane At-Rest 8. Broadband Scattering by A Moving Half-Space 9. Concluding Remarks References 2

3 1. INTRODUCTION f( R) = q ( d K) f( K) e, q= (2 π ) 4 ikr 4 4 d K = dkxdkydkzdiω c R = r ict = x y z ict K = ( k, iω/ c) = ( k, k, k, iω/ c) /, (, ) (,,, ) x y z 4 i F( ) f( ) = q KR R R ( d K) f( K) F( ik ) e = 0 R = ( r, i t / c), Fi ( K ) = 0 In an inertial reference-frame Γ : r E= tb, r H = td D = ε E, B = µ H D= 0, B= 0 E= E( R) r r 3

4 In an inertial reference-frame Γ : E = B, H = D r t r t D = 0, B = 0, E = E ( R ) r r R R R : Lorentz transformation = [ ] r = U r v = v r 2 ( t), t γ ( t / c ) γ = β β = = 2 1/2 (1 ), v/ c, v, U = I + ( γ 1) vv ˆˆ, vˆ = v/ v Transformation for derivatives = [ ] γ v : R R R 2 r = U ( r + v t / c ), t = ( t + v r) 4

5 F F F : Field transformations = [ ] 2 E = V ( E+ v B), B = V ( B v E/ c ) 2 D = V ( D+ v H/ c ), H = V ( H v D) V = γi + (1 γ) vv ˆˆ Phase-invariance, or K, R Minkowski four-vectors: K R = k r ωt = K R = k r ω t Doppler effect K = K [ K ]: 2 k = U ( k vω/ c ), ω = γ( ω v k) 5

6 2. SPATIOTEMPORAL DIFFERENTIAL- OPERATORS: 1 E = W E = V ( E+ v B) = V ( E v r t E) 1 1 W = V ( I v t r I ), B = t r E 1 H = V ( H v D) = W H, D = t r H 1 E = W E, H = W H, W = V ( I + v I ) t r 6

7 3. THE SCATTERING ALGORITHM ER ( ) ( ) 4 g K i = q ( d ) E KR K e HR ( ) g H ( K ) E ( R) E( R) ( ) 4 i g = = q E K KR W R ( d K) e W K H ( R ) H ( R ) g H ( K ) ( ) 4 i g = q ( d ) e E K K R K W K, β = v/ vph, vph = ω/ k g H ( K ) 1 W = V ( I v I ), W = V ( I + βvˆ kˆ I ) R t r K 7

8 E ( R ) ( ) 4 i q ( d ) e g = E K K R K W K H ( R ) g H ( K ) E ( R ) 4 i g = q K R E ( d K ') e H ( R ) g H g ( E ge K ) ge( K) = = W K g g ( K ) g ( K) H H H K= K[ K ] [ ] d K' = det K' d K = d K K 8

9 In Γ : E sc ( R ) (, ( ), ) 4 E sc R g = q E K K ( d K ') H sc ( R H sc ( R, g ( ), H K K ) In Γ fields are still expressed in terms of Γ native coordinates: Esc ( R ) E sc ( R ) = W R Hsc ( R H sc ( R (, ( ), ) 4 Esc R g = q E K K ( d K') W R H sc ( R, g ( ), H K K ) 1 W = V ( I + v I ) R t r 9

10 4. FREE-SPACE IMPULSIVE PLANE-WAVE EXCITATION In Γ : E Eˆ e iωτ δτ ( ), δτ ( ) e = = dω/2π ˆ H H h τ = t kˆ r/ c, ωτ = ωt k r ω / k = v = ( µε ), e/ h= Z = ( µ / ε) ph 1/2 1/2 EH ˆ ˆ = Ek ˆ ˆ = Hk ˆ ˆ = 0, Eˆ Hˆ = kˆ D= ε E, B= µ H, ε = ε, µ = µ, ε µ = 1/ c

11 In Γ : E E ˆ e E ˆ e E ˆe = δτ ( ), = W K H Hˆ h Hˆ h Hˆh τ = t kˆ r / c, ωτ = ωτ iωτ iω τ δ ( τ) = e dω/2 π = e pdω /2 π = pδ( τ ) p = dω/ dω = 1/( γ(1 βvk ˆ ˆ )) 11

12 5. BROADBAND SCATTERING BY CYLINDERS AT-REST For E ˆ = z, ˆ vˆ z ˆ = 0 : 1 W = γ(1 + ( v ) ) Z, W = γ(1 βvˆ kˆ ) Z R r t K 1 W γ(1 ( ) ), (1 ˆ ˆ = ), ˆˆ t = γ + β R v r Z W K v k Z Z = zz Scattered wave: E ( R ) = zˆ e dω E ( R, ω )/2π sc ω E ( R, ω ) = e Σ i a ( ω, α ) H ( ρ ) e iρ Cϕ ψ iω t = e g( ψ ) dψ / π, C = cos( ϕ ψ ) ψ = ϕ + ( π /2) i m= =, ρ = ω r / c, Σ =Σ, g( ϕ ) =Σ a e ψ ψ = ϕ ( π /2) + i sc i t m imϕ sc m m m ψ ϕ ψ m m= m m imϕ 12

13 Far-field (asymptotic) approximation: m 1/2 i i H ( ρ ) (2/ iπρ ) e ρ 13 m Far-field scattered wave: E ( R, ω ) zˆ elgs ( τ ), sc 0 τ = t r / c, L= (2/ iπρ ), ρ = ω r / c iωτ S( τ ) = W( ω ) e dω /2π 1/ g = g( ϕ, ω 0) =Σmam( ω 0, α ) e Equation of motion: τ = t r / c= 0 imϕ

14 14 6. BROADBAND SCATTERING BY MOVING CYLINDERS In Γ fields are still expressed in terms of Γ native coordinates: E ( R ) = zˆ e dω E ( R, ω )/2π sc sc i t m imϕ sc ξ m m m ω E ( R, ω ) = e B Σ i a ( ω, α ) H ( ρ ) e ω = e B Σ i b ( ω, α ) H ( ρ ) e i t m imϕ ξ m m m B = γ(1 iβ( C ( S / ρ ) )), C = vˆ xˆ ξ ϕ ξ ρ ϕ ξ ϕ ξ Equation of motion In Γ ( Γ native coordinates): t = γ ( t v r/ c ) = γ ( t 2 vr / c + ( vr / c ) ) = ( r / c) = ( γ( r vt) + r ) ( γ( r vt) + r )/ c 2 2 = ( γ ( r 2 vrt+ v t ) + r )/ c

15 first order effects in v/ c: t = ( t v r/ c ) = ( t 2 vr / c ) = ( r 2 vrt+ v t + r )/ c = ( r 2 vrt+ v t )/ c becomes: t r/ c = 0 7. BROADBAND SCATTERING BY A HALF-PLANE AT-REST α α = π, cos α = kˆ xˆ, cosα = kˆ xˆ 0 0 α α = π, cos α = kˆ xˆ, cos α = kˆ xˆ, xˆ = xˆ

16 iω t iρ Cϕ ψ E ( R, ω ) = zˆ ee e g( ψ ) dψ / π iω t iρ Cϕ ψ =± zˆ ee e g( ψ ) dψ / π =± zˆ ee Σ i a ( α ) H ( ρ ) e 16 sc ψ iω t m imϕ m m 0 m 1/2 iρ iω t ± zˆ e (2/ iπρ ) e g( ϕ ) g( ψ ) = is S /( C + C ) =Σ a ( α ) e ψ α /2 ψ /2 ψ α m m The sign ± applies to y < > 0 sc iρ C imψ iω t 1/2 ϕ α0 ˆ ee ( iπ) [ e F( ρ C( ϕ α0 )/2) ϕ α0 e + F( ρ C )], ρ = (2 ρ ) ( ϕ + α )/2 0 iρ C E + E = z 1/2

17 Reflection zone: E = Eˆ e ( ), t ˆ δτ τ = k r / c ref ref ref ref kˆ = kˆ, kˆ = kˆ, 0< ϕ < π α ref ref Shadow and illuminated zones: ˆ δτ ( ), τ ˆ / Etr = Ee = t k r c α < ϕ < 2 π, α = π + α

18 8. BROADBAND SCATTERING BY A MOVING HALF-SPACE Transmitted wave = - excitation Reflected wave: E = W E = W Eˆ e δτ ( ) = Eˆ e δτ ( ) ref K ref K ref ref ref ref Eˆ = p W Eˆ, τ = t kˆ r/ c, ωτ = ω τ ref W = γ(1 + βvˆ kˆ ) Z, Z = zz ˆˆ ref ref ref ref K ref ref ref ref ref K ref iωτref iωref τref δτ ( ref ) = e dω /2π= e p ref dω p = dω / dω = 1/( γ(1 + βvˆ kˆ )) ref ref ref ref /2π 18

19 Transformation of unit vector: rˆ = r /( r r ) = U ( r vt) /( γ ( r 2 vrt + v t ) + r ) 1/ /2 t r vrc t + v t C = r r /2 ( r v )/( 2 ξ ), ξ / Transformation of arbitrary angle: x rˆ x U r v γ C = ˆ = ˆ ( t)/( ( r 2 vrt + v t ) + r ) /2 ϕ xˆ ( r vt)/ T = ( rc vc t)/ T T = ( r 2 vrc t + v t ) ϕ /2 ξ The half-plane x > 0 : C = 1, C = xˆ r ˆ ( rc vc t) / T ϕ ϕ ϕ ξ ξ 19

20 9. CONCLUDING REMARKS Velocity-dependent scattering of EM waves can be efficiently handled by using the new spatiotemporal differential-operators. This also facilitates the analysis of broadband scattering. Two two-dimensional problems are tackled: Scattering by cylinders, and scattering by a moving half-plane. When the object is moving, intuition suggests that the pulses due to diffraction be located on a series of eccentric circles, whose centers are prescribed by the velocity-dependent location of the cylinder at the time the excitation pulse is scattered. This is valid in the far field and only to the first order in v/ c. Geometrical-optics plane waves, due to the pole in the scattering amplitude are also discussed. The transformation from Γ to the initial Γ, and various zones are also discussed. 20

21 21 THIS IS ALL, FOLKS, THANK YOU