ECE 6341 Spring 16 Prof. David R. Jacson ECE Dept. Notes 37 1
Line Source on a Grounded Slab y ε r E jω A z µ I 1 A 1 e e d y ( ) + TE j y j z 4 j +Γ y 1/ 1/ ( ) ( ) y y1 1 There are branch points only at z ± Z Z TE TE 1 TE ( ) ( ) ( ) + ( ) Z Z TE TE in Γ ( ) TE TE Zin Z TE ( ) 1 tan ( 1 ) Z jz h TE in y (even function of y1 ) ωµ y ωµ y1
Steepest-Descent Path Physics Steepest-descent transformation: sinζ y cosζ There are no branch points in the ζ plane (cosζ is analytic). Both sheets of the plane get mapped into a single sheet of the ζ plane. 3
Steepest-Descent Path Physics Eamine y to see where the ζ plane is proper and improper: cos( ζ ζ ) [ cosζ coshζ jsinζ sinhζ ] + j y r i r i r i Im sinζ sinhζ y r i Proper : Im < y Improper : Im > y 4
SDP Physics (cont.) Proper : Im < y Improper : Im > y Im sinζ sinhζ y r i ζ i P: proper I: improper C I I P P ζ r P P I I 5
SDP Physics (cont.) Mapping of quadrants in plane [ ] sinζ sinζ coshζ + jcosζ sinhζ r i r i i I 3 I ζ i P 1 C 4 SWP P 1 3 4 r LWP ζ r P 3 P 4 I 1 I 6
SDP Physics (cont.) Non-physical growing LW poles (conjugate solution) also eist. ζ i C I 3 I P 1 4 SWP P LWP ζ r P 3 P 4 I 1 I p LW * The conjugate pole is symmetric about the / line: ( ) conj ζ / + / ζ ζ r r r ζ conj i ζ i 7
SDP Physics (cont.) cos ζ θ coshζ 1 SDP: ( ) r i A leay-wave pole is considered to be physical if it is captured when deforming to the SDP (otherwise, there is no direct residue contribution). ζ i SWP C SDP θ LWP ζ r ζ θ 8
SDP Physics (cont.) Comparison of Fields on interface (θ / ): LWP: ( ) LW Res j E j e (eists if pole is captured) z LW β jα SDP: e Ez A j 3/ (from higher-order steepest-descent method) The leay-wave field is important if: 1) The pole is captured (the pole is said to be physical ). ) The residue is strong enough. 3) The attenuation constant α is small. 9
SDP Physics (cont.) LWP captured: θ θ > b ζ i SDP The angle θ b represents the boundary for which the leay-wave pole is captured (the leay-wave field eists). ζ rp LWP θ b ζ ζ + jζ p rp ip ζ r θ Note: b > ζ rp 1
SDP Physics (cont.) Behavior of LW field: ( ) ( ) + j y j z y E F e e d C ( ρ) ( ζ θ) j cos F ζ e cos ζ dζ ( )( ) E jres F ζ cosζ e LW z p p ( ρ) cos( ζ p θ) j ψ In rectangular coordinates: LW z j j y E Ae e p y p (It is an inhomogeneous plane-wave field.) where β jα p LW 11
SDP Physics (cont.) Eamine the eponential term: cos ψ e ( ρ) ( ζ θ) j cos p ( ζ θ) cos ( ζ θ) ζ p rp + j ip ( ) j ( ) cos ζ θ coshζ sin ζ θ sinhζ rp ip rp ip Hence ψ e ( ) sin( rp ) ρ ζ θ sinhζ ip e ( ρ) sinhζip sin( θ ζrp ) since ζ ip < 1
SDP Physics (cont.) Radially decaying: ψ e ( ρ) sinhζip sin( θ ζrp ) θ > ζ rp LW eists: θ > θ b y LW decays radially ζ rp Also, recall that θ b > ζ rp θ b LW eists ε r Line source 13
Power Flow Power flows in the direction of the β vector. β Re Re( ˆ + y ˆ ) y ˆ ( ζ ) ˆ rp jζip y ( ζrp jζip ) ( ˆ sin ζ cosh ˆ cos cosh rp ζip ζrp ζip ) ( ) Re sin + + cos + + y y β θ 14
Power Flow (cont.) ( sin rp cosh ip cos rp cosh ip ) β ˆ ˆ ζ ζ + y ζ ζ Also, ( ˆsin ˆ ycos ) β β θ + θ Note that tanθ β β y tanζ rp y θ ζ rp θ θ b θ Hence ζ rp θ β Note: There is no amplitude change along the rays (β is perpendicular to α in a lossless region). ε r 15
ESDP (Etreme SDP) The ESDP is the SDP for θ /. The ESDP is important for evaluating the fields on the interface (which determines the far-field pattern). ζ i ζ θ / ESDP We can show that the ESDP divides the LW region into slow-wave and fast-wave regions. Fast ζ r Slow 16
ESDP (cont.) To see this: ( ) cos ζ θ coshζ 1 r sinζ coshζ 1 r i i (SDP) (ESDP) Recall that p sinζ p ( ζ jζ ) sin + rp ip Hence β Re p sinζ rp coshζ ip 17
ESDP (cont.) Hence β sinζ coshζ rp ip Fast-wave region: β < 1 sinζ coshζ < 1 rp ip Slow-wave region: β > 1 sinζ coshζ > 1 rp ip Compare with ESDP: sinζ coshζ 1 r i 18
ESDP (cont.) The ESDP thus establishes that for fields on the interface, a leay-wave pole is physical (captured) if it is a fast wave. ζ i SWP ESDP θ / LWP captured Fast ζ r Slow LWP not captured 19
SDP in Plane We now eamine the shape of the SDP in the plane. sinζ ( ζ jζ ) sin + r i so that sinζ coshζ r r i cosζ sinhζ i r i SDP: cos ζ θ coshζ 1 ( ) r i The above equations allow us to numerically plot the shape of the SDP in the plane.
SDP in Plane (cont.) i sin ζ C r LW SW γ SDP γ θ (Please see the appendi for a proof.) 1
Fields on Interface i θ The SDP is now a lot simpler (two vertical paths)! SW r The leay-wave pole is captured if it is in the fast-wave region. LW fastwave region ESDP
Fields on Interface (cont.) E E + E SW CS z z z ( ) E + E + E SW LW RW z z z i θ SW r The contribution from the ESDP is called the space-wave field or the residual-wave (RW) field. LW ESDP (It is similar to the lateral wave in the half-space problem.) 3
Asymptotic Evaluation of Residual-Wave Field ( ) j ( ) RW Ez F e d y EDSP Use i js d j ds r - + s 4
Asymptotic Evaluation of Residual-Wave Field (cont.) ( ) RW j + s E je F js e ds z + ( ) j ( ) s je F js e ds + ( ) RW j + s E je F js e ds z + ( ) j ( ) s je F js e ds Define ( ) + ( ) ( ) H s F s F s 5
Asymptotic Evaluation of Residual-Wave Field (cont.) Then RW j s E je H s e ds z ( ) Ω for Assume H( s) ~ As α as s Watson s lemma (alternative form): We then have E RW z ~ AΓ α + 1 1 α + ( ) j je 6
Asymptotic Evaluation of Residual-Wave Field (cont.) It turns out that for the line-source problem at an interface, Hence α 1/ E RW z ~ j 3 e jaγ 3/ Note that the wavenumber is that of free space. Note: For a dipole source we have E RW z A e j ρ 1 ρ 7
Discussion of Asymptotic Methods We have now seen two ways to asymptotically evaluate the fields on an interface as for a line source on a grounded substrate: 1) Steepest-descent (ζ ) plane There are no branch points in the steepest-descent plane. The function f (ζ ) is analytic at the saddle point ζ θ /, but is zero there. The fields on the interface correspond to a higherorder saddle-point evaluation. ) Wavenumber ( ) plane The SDP becomes an integration along a vertical path that descends from the branch point at. The integrand is not analytic at the endpoint of integration (branch point) since there is a square-root behavior at the branch point. Watson s lemma is used to asymptotically evaluate the integral. 8
Summary of Waves y LW Continuous spectrum RW SW E A e LW z LW β jα LW LW LW LW j E RW z ~ A RW e j 3/ E A e SW z SW SW β SW j SW 9
Interpretation of RW Field y The residual-wave (RW) field is actually a sum of lateral-wave fields. θ c 3
Appendi: Proof of Angle Property Proof of angle property: ( ) γ θ tan γ r i tanζ r r i ( ζ ) i The last identity follows from sinζ coshζ r r i cosζ sinhζ i r i r i tanζ r Hence γ ~ ζ r or γ ~ ζr 31
Proof (cont.) ζ i On SDP: As ζ ζ i r θ + (the asymptote) u > θ u < u θ SDP SAP u < θ + u > ζ r Hence γ ~ θ + or γ ~ θ + θ 3
Proof (cont.) To see which choice is correct: ESDP: θ In the plane, this corresponds to a vertical line for which γ Hence γ θ 33