7. introduction to 3D scattering 8. ISAR. 9. antenna theory (a) antenna examples (b) vector and scalar potentials (c) radiation in the far field

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Alvin Summers
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1 .. Outline 7. introduction to 3D scattering 8. ISAR 9. antenna theory (a) antenna examples (b) vector and scalar potentials (c) radiation in the far field 10. spotlight SAR 11. stripmap SAR

2 Dipole antenna

3 Conical monopole

4 Helical antenna

6 Log-periodic antenna

7 Horn antennas

8 Corrugated horn

11 Microstrip antennas

13 Bowtie antenna Toothed bowtie

14 Fractal antennas

15 Array antennas

17 Conformal antennas (usually microstrip)

19 Slotted waveguides

20 Slotted waveguide arrays Shuttle imaging radar antenna

21 .. Outline 7. introduction to 3D scattering 8. ISAR 9. antenna theory (a) antenna examples (b) vector and scalar potentials (c) radiation in the far field 10. spotlight SAR 11. stripmap SAR

22 Vector and scalar electromagnetic potentials E = t B H = J + t D D = ρ B = 0 B = A (3) (1) (2) vector potential (1) E + t A = 0 (E + t A) = 0 E + t A = φ scalar potential plug into (2), use B = µ 0 H, D = ɛ 0 E 1 µ 0 ( A) = J t ɛ 0 ( φ + t A) }{{} ( A) 2 A

23 2 A µ 0 ɛ 0 2 t A + [ ] ( A) + µ 0 ɛ 0 t φ = µ 0 J A and Φ are not unique! We get the same B, E, H, D if { A A + Ψ φ φ t Ψ gauge invariance Two commonly used gauges: { A + µ0 ɛ 0 t φ = 0 Lorentz gauge A = 0 Coulomb gauge We use the Lorentz gauge to obtain 2 A µ 0 ɛ 0 2 t A = µ 0 J in frequency domain: 2 A + k 2 A = µ 0 J

24 .. Outline 7. introduction to 3D scattering 8. ISAR 9. antenna theory (a) antenna examples (b) vector and scalar potentials (c) radiation in the far field 10. spotlight SAR 11. stripmap SAR

25 Strategy (frequency domain) 1. model antenna as current density J 2. solve 2 A + k 2 A = µ 0 J for A: A(x) = e ik x y 4π x y µ 0J(y)dy eik x 4π x µ 0 e ik ˆx y J(y)dy } {{ } F (k ˆx)=F[J](k ˆx) 3. recall that E = iωa Φ and the Lorentz gauge condition is iωφ = c 2 0 A ( ) E = iω A c2 0 ( A) (iω) 2 = iω [ A + k 2 ( A) ] 4. compute E using far-field approximation use expressions for and in spherical coordinates

26 E = iω A(x) eik x 4π x µ 0F (k ˆx) [A + 1k 2 ( A) ] = iωµ 0 e ik x 4π x [F + 1k 2 (ik ˆx(ik ˆx F )) ] + O ( 1 ) (k x ) 2 E(x) = iωµ 0 e ik x 4π x [F ˆx(ˆx F )] + O ( 1 ) (k x ) 2 A (B C) = B(A C) C(A B) E(x) = iωµ 0 e ik x 4π x [ˆx (ˆx F )] + O ( 1 ) (k x ) 2 Note E ˆx

27 F (k ˆx) = Example: uniform current density on a line a a = Iê e ik ˆx (sê) Iêds = Iê e ikaˆx ê e ikaˆx ê 2i sin(kaˆx ê) ik ˆx ê width of main lobe is obtained from ka }{{} ˆx ê sin θ (θ measured from normal) ik ˆx ê = 2aIê sinc(kaˆx ê) = π If sin θ θ, then beamwidth is 2π ka = 2π (2π/λ)a = λ a = 2 number of wavelengths that fit on antenna ( k = ω c and ν = c λ k = 2π λ )

29 Example: uniform current density on a rectangle F (k ˆx) = a a b b e ik ˆx (s 1ê 1 +s 2 ê 2 ) I ds 1 ds 2 = I (2b sinc(kbˆx ê 1 )) (2a sinc(kaˆx ê 2 ))

30 Key features of antenna beam Main lobe of beam is perpendicular to antenna Antenna has length L = 2a Angular width of main lobe 4π kl = 2λ L width of antenna footprint = 4π kl R = 2λ L R. Get a more focused beam with: bigger antenna higher frequency x 2 x 1

31 Phased arrays Take J(k, y) = Ie ik ˆµ y for y = s 1 ê 1 + s 2 ê 2 (phase depends on position): F (k ˆx) = a a b b e ik(ˆx ˆµ) (s 1ê 1 +s 2 ê 2 ) I ds 1 ds 2 same as above, but beam points in direction ˆx = ˆµ. Electronically Steered Array (ESA)

32 Real-Aperture Imaging Use big antenna or high frequencies to form small spot on ground Scan the spot over the ground For each spot position on a map, assign a color corresponding to the amplitude of the received energy Synthetic-Aperture imaging Use a smaller antenna Illuminate same region from different antenna locations γ(s) Use mathematics to form high-resolution image

33 An array of point antennas F (k ˆx) = I = N 1 j=0 e ik ˆx y δ(y djê)dy = I 1 eikdn ˆx ê 1 e ikdˆx ê = eikdn ˆx ê/2 e ikdˆx ê/2 N 1 j=0 ik ˆx êdj e [ ] sin(kdˆx ên/2) } sin(kdˆx ê/2 {{ } array factor periodic with period kdˆx ê = 2π grating lobes (extra main lobes) when kd is sufficiently big (an issue for sparse apertures)

34 Far-field expansion when antenna is at γ(s) Center of antenna is at γ(s) Point on antenna is y = γ(s) + q x is on the ground x y = x γ(s) }{{} R s,x + γ(s) y = R s,x q }{{} q!(s) q y R x q << R s,x R s,x q = R s,x R s,x q + O( R s,x 1 )

35 Field from antenna at γ(s) ˆx R s,x F (k ˆx) F (k R s,x ) = e ik R s,x q J(ω, s, q)dq E(ω, x) iωµ 0 4π e ik R s,x R s,x [ )] F R s,x ( R s,x F

36 Vector model 2 A µ 0 ɛ 0 2 t A = µ 0 J 2 A + k 2 A = µ 0 J Scalar model 2 E µ 0 ɛ 0 2 t E = µ 0 t J =: j 2 E + k 2 E = iωµ 0 J A(ω, x) = e ik x y 4π x y µ 0J(ω, y)dy E = iω [ A + k 2 ( A) ] E(ω, x) = e ik x y 4π x y (iωµ 0)J(ω, y)dy E(x) iωµ 0 e ik x 4π x [F ˆx(ˆx F )] E(x) iωµ 0 eik x 4π x F F (k ˆx) = e ik ˆx y J(ω, y)dy F (k ˆx) = e ik ˆx y J(ω, y)dy

37 s rec (t) = antenna Antenna reception E sc (t t, x 0 y) W(t, y) }{{} dydt weighting factor depending on antenna shape in the frequency domain: S rec (ω) = E sc (ω, x 0 y)w (ω, y)dy = antenna e ik x0 y z 4π x 0 V (z)e(ω, z)dzw (ω, y)dy y z use far-field expansion for x 0 z >> y : e ik x0 z S rec (ω) 4π x 0 V (z)e(ω, z)dz e ik x 0 z y W (ω, y)dy z }{{} F[W ](ω, x 0 z)

38 Antenna research questions How can antennas be designed to achieve certain desired far-field patterns? Antennas must be possible to manufacture and feed. Ideally antennas should be small. What can be done with array antennas in which each element is activated with a different waveform? What can be done with distributed networks of antennas? Is it possible to send a message only to a desired location, and send different messages in other directions? (Applications to secure wireless communications)

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