Validity of Image Theorems under Spherical Geometry
aa r X i v : . [ phy s i c s . c l a ss - ph ] J u l Validity of Image Theorems under Spherical Geometry
Shaolin Liao , Sasan Bakhtiari , and Henry Soekmadji Argonne National Laboratory, USA Hamilton Sundstrand, USA
Abstract — This paper deals with different image theorems, i.e., Love’s equivalence principle,the induction equivalence principle and the physical optics equivalence principle, in the sphericalgeometry. The deviation of image theorem approximation is quantified by comparing the modalexpansion coefficients between the electromagnetic field obtained from the image approximationand the exact electromagnetic field for the spherical geometry. Two different methods, i.e., thevector potential method through the spherical addition theorem and the dyadic Green’s functionmethod, are used to do the analysis. Applications of the spherical imaging theorems includemetal mirror design and other electrically-large object scattering.
1. INTRODUCTION
Different image theorems have been widely used for electromagnetic modeling of mirrors and lensantenna [1]-[20]. In [20], Rong and Perkins applied the image theorems to mirror system design forhigh-power gyrotrons. The author also theoretically evaluate the validity of the image theorems inthe cylindrical geometry [1]. In this article, following similar procedures in [1], a closed-form formulafor the discrepancy parameter, which is defined as the ratio of the spherical modal coefficient forimage theorem to that of the exact field, has been derived for the spherical geometry.
2. IMAGE THEOREMS IN THE SPHERICAL GEOMETRY
Fig. 1 shows the spherical geometry for image theorem analysis.
The Vector Potential MethodThe spherical modal expansion
In spherical coordinates, the electrical vector potential F ( r )for M s ( r ′ ) is given as [21], [22], F ( r ) = ǫ Z Z S ′ dS ′ M s ( r ′ ) g ( r − r ′ ) = − jkǫ π Z Z S ′ dS ′ M s ( r ′ ) h (2)0 (cid:0) k [ r − r ′ ] (cid:1) (1)where, h (2)0 is spherical Hankel function of the second kind of order 0. According to the sphericaladdition theorem [21], [22], h (2)0 (cid:0) k [ r − r ′ ] (cid:1) = ∞ X n =0 (2 n + 1) j n ( kr ′ ) h (2) n ( kr ) (2) × n X m =0 (2 − δ m ) ( n − m )!( n + m )! P mn ( θ ′ ) P mn ( θ ) cos m ( φ − φ ′ )where, j n is the spherical Bessel function of the first kind of integral order n; P mn is the associatedLegendre polynomial and δ m is the Kronecker delta function ( δ m = 1 for m=0 and δ m = 0 form = 0). Substituting (2) into (1), the modal expansion of F ( r ) is obtained as, F ( r ) = ∞ X n =0 n X m =0 f M s TE ( n, m ) h (2) n ( kr ) P mn ( θ ) cos mφ sin mφ f m, M s n, TE = χ Z Z S ′ dS ′ M s ( r ′ ) j n ( kr ′ ) P mn ( θ ′ ) cos mφ ′ sin mφ ′ χ = (2 − δ m ) − jkǫ π (2 n + 1)( n − m )!( n + m )! . (3) Figure 1: Image theorem in the spherical geometry: the incident field E i propagates onto spherical surface S ′ , then it may forward-propagate to E + or it could be back-scattered to E − , depending on whether surface S ′ as a fictitious surface where the equivalence theorem applies on a PEC surface. ˆ n + and ˆ n − are theoutward and inward surface normals on spherical surface S ′ respectively. M s and J s are equivalent surfacecurrents for Love’s equivalence theorem. M + s is the image approximation of Love’s theorem and M − s is theimage approximation for the induction theorem. The near field to far field transform of (3) in the spherical coordinate is given as [23], F ( r ) r →∞ = je − jkr kr ∞ X n =0 n X m =0 j n f M s TE ( n, m ) P mn ( θ ) cos mφ sin mφ (4)The duality relation can be used to obtain the magnetic vector potential A ( r ) for the J s ap-proximation as follows, A ( r ) = ∞ X n =0 n X m =0 g M s TE ( n, m ) h (2) n ( kr ) P mn ( θ ) cos mφ sin mφ g m, M s n, TE = χ ′ Z Z S ′ dS ′ J s ( r ′ ) j n ( kr ′ ) P mn ( θ ′ ) cos mφ ′ sin mφ ′ χ ′ = (2 − δ m ) − jkµ π (2 n + 1)( n − m )!( n + m )! . (5) The back-scattered and forward-propagating waves
Similar to the cylindrical geometry, wecan separate (3) into back-scattered and forward-propagating waves as, j n ( kr ′ ) = 12 n h (1) n ( kr ′ ) + h (2) n ( kr ′ ) o (6) f m, M s ± n, TE = χ Z Z S ′ dS ′ M s ( r ′ ) h (1) , (2) n ( kr ′ ) P mn ( θ ′ ) cos mφ ′ sin mφ ′ Since the spherical harmonics is a complete basis set, we can always express the initial incidentelectric field E ( r ′ ) on the initial spherical surface S ′ with radius of r (in Figure 1) as follows, E ( r ) = ∞ X n =0 n X m =0 a m,en,o M m,e + n,o ( r ) + b m,en,o N m,e + n,o ( r ) ψ m,e + n,o ( r ) = h (2) m ( kr ) P mn (cos θ ′ ) cos( mφ ′ )sin( mφ ′ ) L m,e + n,o ( r ) = ∇ ψ m,e + n,o ( r ) M m,e + n,o ( r ) = ∇ × (cid:8) a r rψ m,e + n,o ( r ) (cid:9) N m,e + n,o ( r ) = 1 k ∇ × M m,e + n,o ( r ) . (7)From (3) and noting that M + s ( r ) = 2 E ( r ) × a r , on spherical surface S ′ in Figure 1,˜ E ( r ) = − ǫ ∞ X n =0 n X m =0 (cid:8) L m,e + n,o ( r ) × f m, M s n, TE (cid:9) (8) L m,e + n,o ( r ) = ∇ ψ m,e + n,o ( r )The approximate field ˜ E ( r ) on the initial spherical surface S ′ is obtained from (3) throughimage theorem approximation,˜ E ( r ) = ∞ X n =0 n X m =0 ˜ a m,en,o M m,en,o ( r ) + ˜ b m,en,o N m,e + n,o ( r ) (9)Now the deviation of the spherical coefficients ˜ a m,en,o , ˜ b m,en,o in Eq. (9) from their exact values a m,en,o , b m,en,o in Eq. (7) is defined as the discrepancy parameters ζ , ζ M s TE = ˜ a m,en,o a m,en,o = − j kr h (2) n ( kr ) ∂ [ krj n ( kr )] ∂kr r = r ζ M s TM = ˜ b m,en,o b m,en,o = j kr j n ( kr ) ∂ [ krh (2) n ( kr )] ∂kr r = r and, ζ M s , ± TE = − j kr h (2) n ( kr ) ∂ [ krh (1) , (2) n ( kr )] ∂kr r = r (10) ζ M s , ± TM = j kr h (1) , (2) n ( kr ) ∂ [ krh (2) n ( kr )] ∂kr r = r . Similar expressions exist for J s image approximation, ζ J s TE = ζ M s TM , ζ J s TM = ζ M s TE (11) ζ M s , ± TE = ζ J s , ± TM = [ ζ J s , ± TE ] ∗ = [ ζ M s , ± TM ] ∗ . The Dyadic Green’s Function Method
The magnetic dyadic Green’s function in the sphericalcoordinate is, ¯ G m ( r , r ′ ) = − a r a r k δ ( r − r ′ ) − ∞ X n = −∞ jπ kn ( n + 1) Table 1: Summary of ζ + , − TE, TM ( m s , j s ) for the spherical geometryTE/TM modes and M s / J s The relations ζ + , − TE, TM ( M s , J s ) r → ∞ TE & M s / TM & J s Sphere: back-scattered wave ζ − TE ( M s ) = ζ − TM ( J s ) − jkr h (2) n ( kr ) ∂ [ krh (2) n ( kr )] ∂kr r = r ( − n e − j kr Sphere: forward-propagating wave ζ + TE ( M s ) = ζ + TM ( J s ) − jkr h (2) n ( kr ) ∂ [ krh (1) n ( kr )] ∂kr r = r M s / TE & J s Sphere: back-scattered wave ζ − TM ( M s ) = ζ − TE ( J s ) jkr h (2) n ( kr ) ∂ [ krh (2) n ( kr )] ∂kr r = r − ( − n e − j kr Sphere: forward-propagating wave ζ + TM ( M s ) = ζ + TE ( J s ) [ ζ + TE ( M s )] ∗ / [ ζ + TM ( J s )] ∗ × n X m =0 Q nm (cid:8) M m,en,o ( r ′ ) M m,e + n,o ( r ) + N m,en,o ( r ′ ) N m,e + n,o ( r ) (cid:9) and, Q nm = 2 π ( n + m )!(2 − δ m )(2 n + 1)( n − m )! (12)where M m,en,o ( N m,en,o ) is obtained by replacing h (2) n with j n in M m,e + n,o ( N m,e + n,o ). The approximate field˜ E ( r ) for M + s ( r ′ ) is given as,˜ E ( r ) = −∇ × Z Z S ′ dS ′ M + s ( r ′ ) . ¯ G m ( r , r ′ ) (13)Substituting (12) into (13) and using the orthogonal properties of spherical modal functions, theapproximate field ˜ E ( r ) on initial spherical surface S ′ is obtained as,˜ E ( r ) = ∞ X n = −∞ n X m =0 jπn ( n + 1) Q nm c m,en,o M m,e + n,o ( r ) d m,en,o N m,e + n,o ( r ) (14) × Z Z S ′ dS ′ [ N m,en,o ( r ′ )] ∗ × M m,e + n,o ( r ′ )[ M m,en,o ( r ′ )] ∗ × N m,e + n,o ( r ′ ) · a r ′ . The evaluation of (14) also leads to (9) and (10).
The Analytical Formula for Image Theorems in the Spherical Geometry
Similar to the cylin-drical geometry, ζ M s , J s + TE,TM in (10) and (11) can be considered as theoretical formulas for evaluation ofthe image theorems for narrow-band fields in the spherical geometry. The large argument asymp-totic behaviors of ζ M s , J s + TE,TM for r → ∞ can be obtained by noting that, h (2) n ( kr ) = [ h (1) n ( kr )] ∗ ∼ kr j ( n +1) e − jkr , kr → ∞ ζ M s , J s + TE,TM r →∞ = 1 . (15)
3. RESULTS AND DISCUSSION
TABLE 1 summarizes the properties of ζ M s , J s ± TE,TM , for the back-scattered and forward-propagatingwaves respectively. For r → ∞ , ζ M s , J s TE,TM = ζ M s , J s + TE,TM + ζ M s , J s − TE,TM shows fast oscillations, which can be
Figure 2: The spherical geometry - threshold radii r th Vs. n=0 to 100, for different accuracies, from − −
30 dB (in 10 dB increment, from bottom to top): a) the magnitudes 20 log ( | ζ M s + TE | − [ ℑ ( ζ M s + TE )]. The inset plots in a) are used to make the display clearer. Similarto the cylindrical geometry, imaginary parts ζ M s + TE require larger threshold radii r th for the same accuracy. seen from TABLE 1. Mathematically, the oscillations only appear as modal expansion coefficientsand disappear after the implementation of the double sums in (9). Physically, the oscillations aredue to back-scattered fields, which approach 0 for r → ∞ . For example, consider ζ M s − TE in (10),˜ E − ( r , φ ) = ∞ X n =0 n X m =0 (cid:8) ζ M s − TE c m,en,o M m,e + n,o ( r ) + ζ M s − TE b m,en,o N m,e + n,o ( r ) (cid:9) (16)Changing the variable φ ′ = φ − π and letting r → ∞ , from TABLE 1, (16) reduces to,˜ E − ( r , φ ′ ) r →∞ = ∞ X n =0 n X m =0 e − j kr (cid:8) c m,en,o M m,e + n,o ( r ) − b m,en,o N m,e + n,o ( r ) (cid:9) . (17)Now, the back-scattered field ˜ E − ( r , φ ′ ) r →∞ → e − j kr ,which means that the oscillation in ζ M s − TE doesn’t appear in the actual field evaluation for r → ∞ .Based on the above discussion, ζ M s , J s ± TE,TM is the theoretical formula of interest to evaluate thevalidity of image theorems.It is also helpful to plot the corresponding threshold radius r th with respect to n, for both20 log ( | ζ M s + TE | −
1) and 20 log {ℑ [ ζ M s + TE ] } , with different accuracies ranging from −
60 dB to − −
30 dB for | ζ M s + TE | (with respect to 1), r th ∼ λ and r th ∼ λ for n = 50 and n = 100respectively. However, for the imaginary part ℑ [ ζ M s + TE ], r th ∼ . λ and r th ∼ λ are required for n = 50 and n = 100 respectively, which again implies that the imaginary part ζ M s + TE dominates theaccuracy of image theorems.
4. CONCLUSION
For spherical geometry, the theoretical formulas for evaluation of the image theorems (both M s and J s approximations) have been derived through two equivalent methods - the vector potentialmethod and the dyadic Green’s function method, for both TE and TM modes. The ratio of thespherical modal coefficient of the image theorem to that of the exact field is used as the criterionto determine the validity of the image theorem. REFERENCES
1. Shaolin Liao and R. J. Vernon, “On the Image Approximation for Electromagnetic WavePropagation and PEC Scattering in Cylindrical Harmonics”, Progress In ElectromagneticsResearch, PIER 66, 65-88, 2006.2. Shaolin Liao, “Beam-shaping PEC Mirror Phase Corrector Design,”
PIERS Online , 3(4):392-396, 2007.3. S.-L. Liao and R. J. Vernon, “A new fast algorithm for field propagation between arbitrarysmooth surfaces”, the joint 30 th Infrared and Millimeter Waves and 13 th International Con-ference on Terahertz Electronics , Williamsburg, Virginia, USA, 2005, ISBN: 0-7803-9348-1,INSPEC number: 8788764, DOI: 10.1109/ICIMW.2005.1572687, Vol. 2, pp. 606-607.4. S.-L. Liao and R. J. Vernon, “The near-field and far-field properties of the cylindrical modalexpansions with application in the image theorem,” the 31 st Int. Conf. on Infrared and Millime-ter Waves,
Shanghai, China, IEEE MTT, Catalog Number: 06EX1385C, ISBN: 1-4244-0400-2,Sep. 18-22, 2006.5. S.-L. Liao and R. J. Vernon, “The cylindrical Taylor-interpolation FFT algorithm,” the 31 st Int. Conf. on Infrared and Millimeter Waves,
Shanghai, China, IEEE MTT, Catalog Number:06EX1385C, ISBN: 1-4244-0400-2, Sep. 18-22, 2006.6. S.-L. Liao and R. J. Vernon, “Sub-THz beam-shaping mirror designs for quasi-optical modeconverter in high-power gyrotrons”,
J. Electromagn. Waves and Appl. , scheduled for volume21, number 4, page 425-439, 2007.7. Shaolin Liao and R.J. Vernon, “A new fast algorithm for calculating near-field propagationbetween arbitrary smooth surfaces,” In , volume 2,pages 606-607 vol. 2, September 2005. ISSN: 2162-2035.8. Shaolin Liao, Henry Soekmadji, and Ronald J. Vernon, “On Fast Computation of Electromag-netic Wave Propagation through FFT,” In , pages 1-4, October 2006.9. Shaolin Liao, “Fast Computation of Electromagnetic Wave Propagation and Scattering forQuasi-cylindrical Geometry,”
PIERS Online , 3(1):96-100, 2007.10. Shaolin Liao, “On the validity of physical optics for narrow-band beam scattering anddiffraction from the open cylindrical surface,”
Progress in Electromagnetics Research Sym-posium (PIERS) , vol. 3, no. 2, pp. 158162 Mar., 2007. arXiv:physics/3252668. DOI:10.2529/PIERS06090614231211. Shaolin Liao, Ronald J. Vernon, and Jeffrey Neilson, “A high-efficiency four-frequency modeconverter design with small output angle variation for a step-tunable gyrotron,” In , pages 1-2, September2008. ISSN: 2162-2035.12. S. Liao, R. J. Vernon, and J. Neilson, “A four-frequency mode converter with small outputangle variation for a step-tunable gyrotron,” In
Electron Cyclotron Emission and ElectronCyclotron Resonance Heating (EC-15) , pages 477-482. WORLD SCIENTIFIC, April 2009.13. Ronald J. Vernon, “High-Power Microwave Transmission and Mode Conversion Program,”Technical Report DOEUW52122, Univ. of Wisconsin, Madison, WI (United States), August2015.14. Shaolin Liao,
Multi-frequency beam-shaping mirror system design for high-power gyrotrons:theory, algorithms and methods , Ph.D. Thesis, University of Wisconsin at Madison, USA,2008. AAI3314260 ISBN-13: 9780549633167.15. Shaolin Liao and Ronald J. Vernon, “A Fast Algorithm for Wave Propagation from a Plane ora Cylindrical Surface,”
International Journal of Infrared and Millimeter Waves , 28(6):479-490,June 2007.
16. Shaolin Liao, “Miter Bend Mirror Design for Corrugated Waveguides,”
Progress In Electro-magnetics Research , 10:157-162, 2009.17. Shaolin Liao and Ronald J. Vernon, “A Fast Algorithm for Computation of Electromag-netic Wave Propagation in Half-Space,”
IEEE Transactions on Antennas and Propagation ,57(7):2068-2075, July 2009.18. Shaolin Liao, N. Gopalsami, A. Venugopal, A. Heifetz, and A. C. Raptis, “An efficient iterativealgorithm for computation of scattering from dielectric objects,”
Optics Express , 19(4):3304-3315, February 2011. Publisher: Optical Society of America.19. Shaolin Liao, “Spectral-domain MOM for Planar Meta-materials of Arbitrary Aperture Wave-guide Array,” In , pages 1-4, May 2019.20. Michael P. Perkins and Ronald J. Vernon, Iterative design of a cylinder-based beam-shapingmirror pair for use in a gyrotron internal quasi-optical mode converter, the 29 th Int. Conf. onInfrared and Millimeter Waves, Karlsruhe, Germany, Sep. 27-Oct. 1, 2004.21. Roger F. Harrington,
Time-Harmonic Electromagnetic Fields,
McGraw-Hill, Inc., 1961.22. J. A. Stratton,