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Prof. David R. Jackson ECE Dept. Spring 2016 Notes 37 ECE 6341 1
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Line Source on a Grounded Slab 2 There are branch points only at (even function of k y1 )
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Steepest-Descent Path Physics There are no branch points in the plane (cos is analytic). Both sheets of the k x plane get mapped into a single sheet of the plane. 3 Steepest-descent transformation:
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Steepest-Descent Path Physics Examine k y0 to see where the plane is proper and improper: 4
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SDP Physics (cont.) P: proper I: improper 5 II I I P P PP C
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SDP Physics (cont.) Mapping of quadrants in k x plane 6 1 2 3 4 I LWP SWP I I I P P P P 4 1 1 4 3 2 2 3 C
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SDP Physics (cont.) Non-physical “growing” LW poles (conjugate solution) also exist. 7 I LWP SWP I I I P P P P 4 1 1 4 3 2 2 3 C The conjugate pole is symmetric about the /2 line:
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SDP Physics (cont.) SDP: A leaky-wave pole is considered to be physical if it is captured when deforming to the SDP (otherwise, there is no direct residue contribution). 8 SDP LWP SWP C
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SDP Physics (cont.) LWP: 9 SDP: (exists if pole is captured) The leaky-wave field is important if: 1)The pole is captured (the pole is said to be “physical”). 2)The residue is strong enough. 3)The attenuation constant is small. Comparison of Fields on interface ( = / 2 ): (from higher-order steepest-descent method)
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SDP Physics (cont.) LWP captured: Note: The angle b represents the boundary for which the leaky-wave pole is captured (the leaky-wave field exists). 10 SDP LWP
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SDP Physics (cont.) Behavior of LW field: In rectangular coordinates: where 11 (It is an inhomogeneous plane-wave field.)
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SDP Physics (cont.) Examine the exponential term: Hence since 12
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Radially decaying: SDP Physics (cont.) LW exists: Also, recall that 13 LW exists LW decays radially Line source
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Power Flow 14 Power flows in the direction of the vector.
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Power Flow (cont.) Hence Note that Also, 15 Note: There is no amplitude change along the rays ( is perpendicular to in a lossless region).
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ESDP (Extreme SDP) ESDP Fast Slow We can show that the ESDP divides the LW region into slow-wave and fast-wave regions. The ESDP is important for evaluating the fields on the interface (which determines the far-field pattern). 16 The ESDP is the SDP for = / 2.
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ESDP (cont.) (SDP) (ESDP) Recall that To see this: Hence 17
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ESDP (cont.) Fast-wave region: Slow-wave region: Hence Compare with ESDP: 18
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ESDP (cont.) ESDP Fast Slow The ESDP thus establishes that for fields on the interface, a leaky-wave pole is physical (captured) if it is a fast wave. LWP captured LWP not captured SWP 19
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SDP in k x Plane SDP: We now examine the shape of the SDP in the k x plane. The above equations allow us to numerically plot the shape of the SDP in the k x plane. so that 20
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SDP in k x Plane (cont.) (Please see the appendix for a proof.) LW SDP SW 21
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Fields on Interface The leaky-wave pole is captured if it is in the fast-wave region. LW ESDP SW fast- wave region 22 The SDP is now a lot simpler (two vertical paths)!
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Fields on Interface (cont.) The contribution from the ESDP is called the “space-wave” field or the “residual-wave” (RW) field. (It is similar to the lateral wave in the half-space problem.) 23 LW ESDP SW
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Asymptotic Evaluation of “Residual-Wave” Field “Residual-Wave” Field Use 24 - +
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Define 25 Asymptotic Evaluation of “Residual-Wave” Field (cont.) “Residual-Wave” Field (cont.)
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Then Assume Watson’s lemma (alternative form): 26 We then have Asymptotic Evaluation of “Residual-Wave” Field (cont.) “Residual-Wave” Field (cont.)
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It turns out that for the line-source problem at an interface, Hence Note: For a dipole source we have 27 Asymptotic Evaluation of “Residual-Wave” Field (cont.) “Residual-Wave” Field (cont.) Note that the wavenumber is that of free space.
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Discussion of Asymptotic Methods We have now seen two ways to asymptotically evaluate the fields on an interface as x for a line source on a grounded substrate: 1) Steepest-descent ( ) plane 2) Wavenumber ( k x ) plane There are no branch points in the steepest-descent plane. The function f ( ) is analytic at the saddle point 0 = = /2, but is zero there. The fields on the interface correspond to a higher- order saddle-point evaluation. The SDP becomes an integration along a vertical path that descends from the branch point at k x = k 0. 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. 28
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Summary of Waves LW SW RW Continuous spectrum 29
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Interpretation of RW Field The residual-wave (RW) field is actually a sum of lateral-wave fields. 30
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Proof of angle property: Hence Appendix: Proof of Angle Property The last identity follows from or 31
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As Hence Proof (cont.) On SDP: or (the asymptote) SDP SAP 32
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ESDP: Hence To see which choice is correct: In the k x plane, this corresponds to a vertical line for which Proof (cont.) 33
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