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IPO: Iterative Physical Optics Image Approximation
Shaolin Liao *, 1 and Lu Ou Abstract —An improved Iterative Physical Optics (IPO) image approximation method has beenpresented to dramatically increase the accuracy of the approximation and extend its applicability toPEC surfaces with smaller radii or larger curvatures. Starting from the first-order conventional POimage approximation, the IPO image approximation method iteratively correct the surface current tocompensate the deviation of the electric field boundary condition on the PEC surfaces, making useof the local plane wave approximation. Numerical validations with two popular PEC surfaces, i . e .,the parabolic dish antennas and the PEC spheres, are carried out and the results show that the IPOapproximation method increases the surface current accuracy by more than two orders of magnitude,compared to the conventional PO image approximation method.
1. INTRODUCTION
Physical Optics (PO) image approximation has been widely used for smooth Perfect Electric Conductor(PEC) surfaces, due to its efficiency and accurate approximation for relatively smooth PEC surfaces[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Compared to other advanced ComputationalElectromagnetics (CEM) methods such as the Method of Moments (MoM), the application ofthe PO image approximation can dramatically reduce the the number of unknowns and memoryrequirements for the electromagnetics scattering problem of electrically large objects [19]. The PO imageapproximation can find important applications in antennas design and analysis [20, 21], beam-shaping[6, 9, 10, 11, 12, 13, 14, 15], and Radar Cross Section (RCS) calculation [22, 23]. In particular, theauthor’s group has applied the PO method to design a beam-shaping mirrors system at the millimeter-wave regime to shape the multi-mode TE waves (TE , /110 GHz, TE , /113 GHz, TE , , >
98% [6, 9, 10, 11, 12].Although useful in the above important applications, the PO image method is only exact for planarPEC surface and is approximate for non-planar but relatively smooth PEC surfaces: the larger radii orthe smaller the curvatures of the surfaces, the better the approximation [1, 2]. So it would be beneficialif the PO image approximation can be extended to PEC surfaces of smaller radii or larger curvatureswith satisfactory accuracy, which is the focus of this letter.
2. PROBLEM FORMULATION
The electromagnetic scattering problem from a PEC surface is shown in Fig. In general, the unknownsurface current can only be solved through rigorous CEM methods such as MoM by imposing zero totalelectric field boundary condition on the metallic surface as follows,ˆ n × (cid:104) E i ( x, y ) + E s ( x, y ) (cid:105) = 0 , (1)where ˆ n is the unit surface normal of the PEC surface; also, the superscripts s and i denote the scatteringand incident electric field, respectively. * Corresponding author: Shaolin Liao ([email protected]). S. Liao is with Department of Electrical and Computer Engineering, Illinois Institute of Technology, Chicago, IL 60616 USA. L.Ou ([email protected]) is with College of Computer Science and Electronic Engineering, Hunan University, Changsha, Hunan,China 410082. a r X i v : . [ phy s i c s . c l a ss - ph ] J u l Liao and Ou
Figure 1.
The improved IPO approximation method for the electromagnetic scattering from relativelysmooth PEC surfaces: a) the parabolic dish antenna; b) the PEC spheres;; and c) the IPO algorithm.For perfect planar surface, the exact surface current can be obtained by the PO as follows, J P O = ˆ n × H = ˆ n × (cid:110) H i + H s (cid:111) = 2ˆ n × H i , (2)and the boundary condition in Eq. (6) reduces to the following,ˆ n × (cid:104) E i ( x, y ) + L (cid:110) J P O (cid:111)(cid:105) = ˆ n × (cid:104) E i ( x, y ) + L (cid:110) n × H i (cid:111)(cid:105) = 0 , (3)where L is the operator that computes the scattering electric field form the surface current.For smooth PEC surfaces, PO is only approximate [1, 2, 12]. So it would be beneficial if the accuracyof the conventional PO image approximation method can be improved so that it can be extended toPEC surfaces with smaller radii or larger curvatures.
3. THE SCATTERING ELECTRIC FIELD
The scattering electric field E s can be expressed in terms of convolution of the surface current J andthe electric dyadic Green’s function G e as follows, E s = L (cid:8) J (cid:9) = G e (cid:126) J = − jωµ (cid:90) (cid:90) S G e ( R ) J ( r (cid:48) ) dS (cid:48) , (4)where (cid:126) denotes the 2D convolution operation; R = r − r (cid:48) ; r = | r | , r (cid:48) = | r (cid:48) | with r and r (cid:48) being theobservation point and source point respectively; ω is the angular frequency; µ is the permeability; andthe dyadic Green’s function is given as Eq (5), G ( r ) = g ( r ) I + 1( k ) ∇∇ g ( r ); g ( r ) = e − jkr πr , r = | r | ; k = | k | = ω √ µ(cid:15), (5)with I being the identity matrix and k being the magnitude of the wave vector k = [ k x , k y , k z ]. PO: Iterative Physical Optics Image Approximation 3
4. THE EFIE
With the scattering electric field given in Eq. (4), the EFIE equation can be obtained from the boundarycondition of Eq. (1) as follows,ˆ n × E i ( x, y ) − jωµ (cid:90) (cid:90) S G e ( R ) J ( r (cid:48) ) dS (cid:48) = 0 , (6)where the surface current J remains to be solved.
5. THE ITERATIVE PO (IPO) APPROXIMATION
The first-order IPO approximation for the surface current J is the PO image theorem given in Eq. (2), J IP O = J P O = 2ˆ n × H i . (7)Now the deviations of the boundary condition of Eq. (1) is given by,ˆ n × δE s = ˆ n × (cid:104) E s − L (cid:110) J IP O (cid:111)(cid:105) . (8)Approximating the local electromagnetic field as local plane wave, the electric field δE s is relatedto the deviation of the deviation of the magnetic field δH s as follows, δE s = ηδH s × ˆ n ; η = (cid:112) µ/(cid:15). (9)Substituting Eq. (9) into Eq. (8), the following is obtained,ˆ n × (cid:2) δH s × ˆ n (cid:3) = 1 η ˆ n × (cid:104) E s − L (cid:110) J IP O (cid:111)(cid:105) → δH s = 1 η ˆ n × (cid:104) E s − L (cid:110) J IP O (cid:111)(cid:105) . (10)Now, to compensate the deviations of the electric field δE s , the surface current can be corrected δJ IP O as follows, δJ IP O = − ˆ n × δH s = − η ˆ n × (cid:110) ˆ n × (cid:104) E s − L (cid:110) J IP O (cid:111)(cid:105)(cid:111) , (11)from which the deviations of the electric field δE s is obtained as follows, δJ IP O = 1 η δE s// = 1 η [1 − ˆ n ] • (cid:104) E s − L (cid:110) J IP O (cid:111)(cid:105) , (12)and δE s// is the tangential electric field projection on the PEC surface.The surface current correction of Eq. (12) is done iteratively until satisfactory result is obtained, δJ i = 1 η [1 − ˆ n ] • (cid:104) E s − L (cid:110) J IP Oi (cid:111)(cid:105) . (13)
6. ALGORITHM
Fig. 1c) shows the algorithm of the improved IPO image approximation: it starts with the first-orderPO image approximation; then it calculates the scattering electric field according to Eq. (4), followedby updating of the surface current according to Eq. (12) and Eq. (13); finally the algorithm ends whensatisfactory result is met.
Liao and Ou
Figure 2.
Parabolic dish antennas: left) Surface current deviation η (cid:12)(cid:12)(cid:12) δJ IP Ox,i (cid:12)(cid:12)(cid:12) at different iterations ofthe IPO approximation method for the focus length of F = 100 λ ; and right) Surface current deviationof the conventional PO image approximation method η (cid:12)(cid:12)(cid:12) δJ P Ox (cid:12)(cid:12)(cid:12) and the IPO approximation method η (cid:12)(cid:12)(cid:12) δJ IP Ox (cid:12)(cid:12)(cid:12) for parabolic dish antennas of different focus lengths F .
7. NUMERICAL VALIDATION
Two PEC surfaces are used to show the efficiency of the IPO image approximation with a Gaussianbeam of waist w = 2 λ as the incidence wave: 1) the parabolic dish antennas with different focus lengths F (Fig. 1 a); and 2) the PEC spheres of different radii R (Fig. 1b). The PEC surface of the parabolicdish antenna [20, 21] and that of the PEC sphere are Z parabolic = X + Y F ; Z spherical = (cid:112) R − X − Y , (14)where F is the focus distance of the parabolic dish antenna and R is the radius of the PEC sphere.The left plot of Fig. 2 shows the surface current deviation η (cid:12)(cid:12)(cid:12) δJ IP Ox,i (cid:12)(cid:12)(cid:12) from the exact surface currentobtained by the MoM at different iterations of the IPO approximation method for the parabolic dishantennas with a focus length of F = 100 λ , showing the convergence of the IPO method. Also, theright plot of Fig. 2 shows the surface current deviations of the conventional PO image approximationmethod η (cid:12)(cid:12)(cid:12) δJ P Ox (cid:12)(cid:12)(cid:12) and that of those of the IPO approximation method η (cid:12)(cid:12)(cid:12) δJ IP Ox (cid:12)(cid:12)(cid:12) for different focuslengths F = [50 , λ , from which it can be seen that more than two orders of magnitude increase inaccuracy has been achieved.In addition, the surface current deviation of the PO image approximation and that of the IPOapproximation for the parabolic dish antenna with a focus length of F = 100 λ have been shown in Fig.3 and Fig. 4 respectively.Similarly, the left plot of Fig. 5 shows the surface current deviation η (cid:12)(cid:12)(cid:12) δJ IP Ox,i (cid:12)(cid:12)(cid:12) at different iterationsof the IPO approximation method for the PEC spheres with a radius of R = 60 λ ; and the right plotof Fig. 5 shows the surface current deviations of the conventional PO image approximation method η (cid:12)(cid:12)(cid:12) δJ P Ox (cid:12)(cid:12)(cid:12) and those of the IPO approximation method η (cid:12)(cid:12)(cid:12) δJ IP Ox (cid:12)(cid:12)(cid:12) for different radii R = [30 , λ , fromwhich it can be seen that the accuracy has been improved by more than two orders of magnitude also. PO: Iterative Physical Optics Image Approximation 5
Figure 3.
PO surface currents deviation for a parabolic dish antenna of a focus length of F = 100 λ (left to right): η (cid:12)(cid:12)(cid:12) δJ P Ox (cid:12)(cid:12)(cid:12) ; η (cid:12)(cid:12)(cid:12) δJ P Oy (cid:12)(cid:12)(cid:12) ; and η (cid:12)(cid:12)(cid:12) δJ P Oz (cid:12)(cid:12)(cid:12) . Figure 4.
IPO surface currents deviation for a parabolic dish antenna of a focus length of F = 100 λ (left to right): η (cid:12)(cid:12)(cid:12) δJ P Ox (cid:12)(cid:12)(cid:12) ; η (cid:12)(cid:12)(cid:12) δJ P Oy (cid:12)(cid:12)(cid:12) ; and η (cid:12)(cid:12)(cid:12) δJ P Oz (cid:12)(cid:12)(cid:12) . Liao and Ou
Figure 5.
PEC spheres: left) Surface current deviation η (cid:12)(cid:12)(cid:12) δJ IP Ox,i (cid:12)(cid:12)(cid:12) at different iterations of theIPO approximation method for the radius of R = 60 λ ; and right) Surface current deviation of theconventional PO image approximation method η (cid:12)(cid:12)(cid:12) δJ P Ox (cid:12)(cid:12)(cid:12) and the IPO approximation method η (cid:12)(cid:12)(cid:12) δJ IP Ox (cid:12)(cid:12)(cid:12) for PEC spheres of different focus radii R . Figure 6.
PO surface currents deviation for a PEC sphere of a radius of F = 60 λ (left to right): η (cid:12)(cid:12)(cid:12) δJ P Ox (cid:12)(cid:12)(cid:12) ; η (cid:12)(cid:12)(cid:12) δJ P Oy (cid:12)(cid:12)(cid:12) ; and η (cid:12)(cid:12)(cid:12) δJ P Oz (cid:12)(cid:12)(cid:12) . PO: Iterative Physical Optics Image Approximation 7
Figure 7.
IPO surface currents deviation for a PEC sphere of a radius of F = 60 λ (left to right): η (cid:12)(cid:12)(cid:12) δJ P Ox (cid:12)(cid:12)(cid:12) ; η (cid:12)(cid:12)(cid:12) δJ P Oy (cid:12)(cid:12)(cid:12) ; and η (cid:12)(cid:12)(cid:12) δJ P Oz (cid:12)(cid:12)(cid:12) .Finally, the surface current deviation of the PO image approximation and that of the IPOapproximation for the PEC sphere with a radius of R = 60 λ have been shown in Fig. 6 and Fig.7 respectively.
8. CONCLUSION
The improved IPO image approximation method has been presented to increase the accuracy of theconventional PO image approximation method. The IPO method iteratively corrects the surface currentto compensate for the deviation of the electric field boundary condition on the PEC surfaces, assuminglocal plane wave approximation. Numerical experiments with parabolic dish antennas and PEC spheresshow that the IPO method can increase the accuracy of the surface current by more than two orders ofmagnitude, compared to the conventional PO image approximation method.
REFERENCES
1. Shaolin Liao and R. J. Vernon. On the Image Approximation for Electromagnetic Wave Propagationand PEC Scattering in Cylindrical Harmonics. Progress In Electromagnetics Research, 66:65-88,2006. Publisher: EMW Publishing.2. Shaolin Liao and Ronald J. Vernon. The Near-Field and Far-Field Properties of the CylindricalModal Expansions with Application in the Image Theorem. In 2006 Joint 31st InternationalConference on Infrared Millimeter Waves and 14th International Conference on TeraherzElectronics, pages 260-260, September 2006. ISSN: 2162-2035.3. Shaolin Liao and R.J. Vernon. A new fast algorithm for calculating near-field propagation betweenarbitrary smooth surfaces. In 2005 Joint 30th International Conference on Infrared and MillimeterWaves and 13th International Conference on Terahertz Electronics, volume 2, pages 606-607 vol.2, September 2005. ISSN: 2162-2035.
Liao and Ou
4. Shaolin Liao, Henry Soekmadji, and Ronald J. Vernon. On Fast Computation of ElectromagneticWave Propagation through FFT. In 2006 7th International Symposium on Antennas, PropagationEM Theory, pages 1-4, October 2006.5. Shaolin Liao and Ronald J. Vernon. The Cylindrical Taylor-Interpolation FFT Algorithm. In 2006Joint 31st International Conference on Infrared Millimeter Waves and 14th International Conferenceon Teraherz Electronics, pages 259-259, September 2006. ISSN: 2162-2035.6. Shaolin Liao. Beam-shaping PEC Mirror Phase Corrector Design. PIERS Online, 3(4):392-396,2007.7. Shaolin Liao. Fast Computation of Electromagnetic Wave Propagation and Scattering for Quasi-cylindrical Geometry. PIERS Online, 3(1):96-100, 2007.8. Shaolin Liao. On the Validity of Physical Optics for Narrow-band Beam Scattering and Diffractionfrom the Open Cylindrical Surface. PIERS Online, 3(2):158-162, 2007.9. 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 2008 33rdInternational Conference on Infrared, Millimeter and Terahertz Waves, pages 1-2, September 2008.ISSN: 2162-2035.10. S. Liao, R. J. Vernon, and J. Neilson. A four-frequency mode converter with small output anglevariation for a step-tunable gyrotron. In Electron Cyclotron Emission and Electron CyclotronResonance Heating (EC-15), pages 477-482. WORLD SCIENTIFIC, April 2009.11. Ronald J. Vernon. High-Power Microwave Transmission and Mode Conversion Program. TechnicalReport DOEUW52122, Univ. of Wisconsin, Madison, WI (United States), August 2015.12. 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: 9780549633167.13. Shaolin Liao and Ronald J. Vernon. A Fast Algorithm for Wave Propagation from a Plane or aCylindrical Surface. International Journal of Infrared and Millimeter Waves, 28(6):479-490, June2007.14. S.-L. Liao and R. J. Vernon. Sub-THz Beam-Shaping Mirror System Designs for Quasi-opticalMode Converters in High-power Gyrotrons. Journal of Electromagnetic Waves and Applications,21(4):425-439, January 2007. Publisher: Taylor & Francis.15. Shaolin Liao. Miter Bend Mirror Design for Corrugated Waveguides. Progress In ElectromagneticsResearch, 10:157-162, 2009. Publisher: EMW Publishing.16. Shaolin Liao and Ronald J. Vernon. A Fast Algorithm for Computation of Electromagnetic WavePropagation in Half-Space. IEEE Transactions on Antennas and Propagation, 57(7):2068-2075,July 2009. Conference Name: IEEE Transactions on Antennas and Propagation.17. 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.18. Shaolin Liao. Spectral-domain MOM for Planar Meta-materials of Arbitrary Aperture Wave-guide Array. In 2019 IEEE MTT-S International Conference on Numerical Electromagnetic andMultiphysics Modeling and Optimization (NEMO), pages 1-4, May 2019.19. Ji Ma, Shu-Xi Gong, Xing Wang, Ying Liu, and Yun-Xue Xu. Efficient Wide-Band Analysisof Antennas Around a Conducting Platform Using MoM-PO Hybrid Method and AsymptoticWaveform Evaluation Technique. IEEE Transactions on Antennas and Propagation, 60(12):6048-6052, December 2012. Conference Name: IEEE Transactions on Antennas and Propagation.20. Chuan Liu, Shiwen Yang, and Zaiping Nie. Design of a parabolic reflector antenna with a compactsplash-plate feed. In 2013 Cross Strait Quad-Regional Radio Science and Wireless TechnologyConference, pages 241-244, July 2013.21. Okan Yurduseven and Ozan Yurduseven. Compact parabolic reflector antenna design withcosecant-squared radiation pattern. In 2011 MICROWAVES, RADAR AND REMOTE SENSINGSYMPOSIUM, pages 382-385, August 2011.