FAST ANALYSIS OF ELECTRICALLY LARGE RADOME IN MILLIMETER WAVE BAND WITH FAST MULTIPOLE ACCELERATION

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1 Progress In Eectromagnetics Research, Vo. 120, , 2011 FAST ANALYSIS OF ELECTRICALLY LARGE RADOME IN MILLIMETER WAVE BAND WITH FAST MULTIPOLE ACCELERATION H. F. Meng * and W. B. Dou State Key Laboratory of Miimeter Waves, Southeast University, Nanjing , P. R. China Abstract Radome has strong effects on the radiation performances of the antenna in miimeter wave band. In this paper, the aperture integration-surface integration (AI-SI) method is adopted to anayze the eectricay arge antenna-radome system. The fast mutipoe method (FMM) is proposed to acceerate the aperture integration and inner surface integration in the AI-SI method. An eectricay arge antenna-radome system at W band is anayzed and measured. The radiation patterns of the system cacuated using the AI-SI method with and without the fast mutipoe acceeration and the measured patterns are compared. The cacuated patterns agree very we with each other, and both have the same agreement with the experimenta resuts. However, the computationa time of the proposed anaysis with the fast mutipoe acceeration is reduced significanty. 1. INTRODUCTION A dieectric radome is paced in front of the antenna to protect the system from various environments. However, the presence of the radome aways affects the radiation properties of the encosed antenna, such as distorting the radiation pattern. Therefore, an accurate anaysis of the antenna-radome system is important. Many methods have been used to anayze the antenna-radome system, such as the fu wave method [1 3], the high frequency method [4 7], and the hybrid method [8 10]. As the sizes of the antenna-radome system in miimeter wave band are aways eectricay arge, the efficiency of the fu wave methods, such as method of moment (MoM) [1] and the finite eement method (FEM) [2], are Received 11 August 2011, Accepted 21 September 2011, Schedued 26 September 2011 * Corresponding author: Hongfu Meng (menghongfu@163.com).

2 372 Meng and Dou very ow, even empoying the fast agorithm proposed by [11]. In contrast, the high frequency methods, such as the ray tracing method (RT) [4] and the physica optics method (PO) [5 7], have higher efficiency. However, the accuracy of RT is not very high, especiay for the radome with sma curvature radius. The aperture integrationsurface integration (AI-SI) method, which is based on PO, has a good baance between the accuracy and efficiency [6, 7]. It is suitabe for the anaysis of the eectricay arge antenna-radome system in miimeter wave band. In the AI-SI method, the incident fieds on the inner surface of the radome have two parts. The first part is determined by integrating the aperture currents over the antenna aperture. Some portion of the first incident fieds wi transmit through the radome wa and the remaining wi be refected by the wa. The refected fieds have important effects on the fash obe of the radiation pattern [12 14]. By integrating the refected fieds over the inner surface of the radome, the second incident fieds on the inner surface of the radome can be obtained. In the traditiona AI-SI method [5 7], the aperture integration of the first incident fieds and the inner surface integration of the second incident fieds are a determined by direct integration using the Stratton-Chu formuas. As the antenna aperture and the inner surface of the radome are eectricay arge, these integrations take a ot of time [15]. In MoM, the fast mutipoe method (FMM) is empoyed to acceerate the computation of the mutua coupings between different eements [16 19]. The determinations of the aperture integration and inner surface integration in the AI-SI method are simiar to the computation of the mutua couping in MoM. So, in this paper, FMM is proposed to acceerate the above mentioned aperture integration and inner surface integration in the AI-SI method. Firsty, the genera steps of the AI-SI method for the antennaradome anaysis are given. Then, the appications of the fast mutipoe to acceerate the aperture integration and inner surface integration are presented. Finay, some computationa resuts are compared with the experimenta resuts to show the accuracy and efficiency of the fast mutipoe acceerated anaysis. 2. THE AI-SI METHOD The AI-SI method is a high-frequency method based on PO. It can anayze the eectricay arge antenna-radome system very efficienty with acceptabe accuracy. It was firsty introduced to anayze the radome by Paris [5], and many other researchers have done a ot of work [6, 7]. The genera steps of this method are as foows:

3 Progress In Eectromagnetics Research, Vo. 120, In the antenna-radome mode, an aperture antenna in an ideay back screen is encosed by the dieectric radome [5, 13]. When the equivaent eectromagnetic currents on the aperture of the antenna are known, the first incident fieds on the inner surface of the radome can be obtained by integrating the currents over the antenna aperture using the foowing formuas: ( E i ( r) = jωµ Ī + ) k 2 G( r, r ) J AP ( r )ds where S AP MAP ( r )G( r, r )ds S AP ( H i ( r) = jωε Ī + ) k 2 G( r, r ) M AP ( r )ds + S AP S AP JAP ( r )G( r, r )ds G ( r, r ) = e jk r r 4π r r is the Green s function in free space, J AP, MAP are the eectromagnetic currents on the antenna aperture, and S AP is the antenna aperture. The incident vector at the intersection point on the inner surface of the radome is estabished by the direction of the Poynting vector [5] ( Ŝ i = Re Ei H ) ( i / Re Ei H ) i (3) The incident vector and the norma vector at the intersection point on the inner surface define the pane of incidence. The incident fieds at the intersection point are decomposed into the perpendicuar and parae poarization components to the pane of incidence. After refection and refraction by the radome wa, the refected fieds E r, H r on the inner surface and the transmitted fieds E t, H t on the outer surface of the radome are recombined as [7] ( ( ) E r = Ei v ) i R v r + Ei v // i R // v // r ( ( ) (4) H r = Hi v ) i R // v r + Hi v // i R v // r ( ) Ei v ) i E t = H t = ( T v t + Ei v // i T // v // t ) T v // t ( Hi v i ) T // v t + ( Hi v i // (1) (2) (5)

4 374 Meng and Dou Figure 1. Mode of the AI-SI method for the antenna-radome anaysis. where v and v // are the unit vectors iustrated in Figure 1, the superscripts i, r, and t represent the incident, refected, and transmitted fieds, respectivey. R, R //, T, T // are the refection and transmission coefficients for the perpendicuar and parae poarization components [7, 20]. The refected fieds on the inner surface may bounce in the radome and they have important effects on the fash obe of the antenna-radome system [12 14]. In order to account the mutua interactions between different parts of the radome, the fieds radiated from the refected fieds must be cacuated. So, the equivaent refected currents on the inner surface of the radome are J r = ˆn H r Mr = ˆn E r (6) where ˆn is the norma vector on the inner surface of the radome. As the fange of the aperture antenna is modeed as back screen, the fieds refected to the bottom of the radome are ignored. However, the fieds refected to the inner surface of the radome may form second radiation. Thus, the second incident fieds on the inner surface of the radome can be determined as foows: E i( r) = jωµ S Rin ( Ī + k 2 ) G ( r, r ) Jr ( r )ds Mr ( r ( )G r, r ) ds S Rin

5 Progress In Eectromagnetics Research, Vo. 120, ( H i( r) = jωε Ī + ) k 2 G ( r, r ) Mr ( r )ds S Rin + Jr ( r ( )G r, r ) ds (7) S Rin where G( r, r ) is the Green s function in free space defined in (2) and S Rin is the inner surface of the radome. The second incident fieds are treated the same as the first incident fieds in (5) and the second transmitted fieds E t, H t can be obtained. Finay, the tota fieds on the outer surface of the radome are the vector sum of the first and second transmitted fieds: E tota t = E t + E t H tota t = H t + H t (8) When the tota transmitted fieds on the outer surface are known, the far fied radiation patterns of the antenna-radome system can be determined by integrating the fieds over the outer surface of the radome using the Stratton-Chu formuas [6]. 3. THE FAST MULTIPOLE ACCELERATION As above mentioned, the integration (1) needs to be performed over the antenna aperture S AP and the integration (7) needs to be performed over the inner surface of the radome S Rin. In numerica computation, the integration surface S AP and S Rin are meshed into Figure 2. Schematic of the mesh on the antenna aperture and radome surface.

6 376 Meng and Dou sma trianges with the edge ength much smaer than the waveength as in Figure 2, and then the integrations (1) and (7) are converted into the summations over a the trianges. If the antenna aperture S AP is meshed with M trianges and the inner surface of the radome with N trianges, then the computation of the first incident fieds on the inner surface of the radome is of O(MN) compexity and the computation of the second incident fieds of O(N 2 ) compexity. As the antenna and radome in miimeter wave band are eectricay arge and the numbers M and N are arge, the aperture integration (1) and inner surface integration (7) demand a ong computing time. In MoM, the impedance matrix shows the mutua couping between every triange. FMM is aways adopted to acceerate the matrix-vector product [16 19]. FMM was proposed for static probem by Rokhin in 1987 and for dynamic probems in 1990 [16, 17]. For the matrix-vector product with N unknowns, the two-eve FMM reduces both the memory requirement and numerica compexity from O(N 2 ) to O(N 1.5 ) and the three-eve FMM reduces it to O(N 4/3 ) [16 19]. By using the mutieve fast mutipoe method (MLFMM), the numerica compexity can be further reduced to O(N og N) [21 24]. When appying FMM to acceerate the mutua couping computation, the trianges on the integration surface are divided into groups depending on their position. The key point of FMM is the addition theorem, which is presented as foows: e jk R+ d R + d = jk L =0 ( 1) (2 + 1)j (kd)h (2) (kr)p ( ˆd ˆR) (9) where j is a spherica Besse function of the first kind, h (2) is a spherica Hanke function of the second kind and P is a Legendre poynomia. The expansion of the product j P in propagating pane waves is ( ) ) 4π( j) j (kd)p ˆd ˆR = e jkˆk d P (ˆk ˆR d 2ˆk (10) Appying the Gaussian-Legendre quadrature to (10) and substituting (9) and (10) into (2), the Green s function can be expressed as G ( r, r ) = e jk R+ d 4π R + d [ ω p e j k p d = jk (4π) 2 L 4π K p=1 ( j) (2 + 1)h (2) (kr)p ( kp ˆR)] (11)

7 Progress In Eectromagnetics Research, Vo. 120, where R is the vector from the source group center to the fied group center, d is the summation of the vectors from the source point to source group center and from the fied group center to fied point, R and d are their ampitudes with R > d, and ˆR and ˆd are the unit vectors, respectivey. In (9), L is an infinite number. However, in numerica practice, L must be truncated with the finite number of modes and the reative error is depending on L with the foowing reationship [18, 21]: L = kd max + γ n (π + kd max ) (12) where d max is the maxima diameter of a the groups. The reative error of (11) can ess than 0.1 when γ = 1 and the arger of γ, the more accurate of (11). In (11), K is the number of integration points over the unit sphere and is aways chosen as K = 2L 2 (13) When kr > L, the source group and fied group are caed far groups. The interactions between the trianges in the far groups in the integras (1) and (7) can be cacuated as foows: the fieds from a the trianges in the source group are aggregated to the source group center firsty, then, the fied information is transformed from the source group center to the fied group center, and finay, the fied effect is disaggregated to every triange in the fied group. The integration formua between the far groups can be repaced by: E far jk K ( r) = jωµ (Ī ˆkˆk) (4π) 2 [ S ω p e j k p d ( jk) [ S ω p e j k p d L jk (4π) 2 p=1 ( j) (2 + 1)h (2) (kr)p ( kp ˆR) J( r ) L K ˆk p=1 ( j) (2 + 1)h (2) (kr)p ( kp ˆR) M( r ) ] ] (14) When kr is comparabe or smaer than L, the source and fied group are near groups. At this time, the Equation (11) can not get desired accuracy and the effects between the trianges in the near groups must be cacuated using the traditiona direct integration.

8 378 Meng and Dou Finay, the integras (1) and (7) can be expressed as E( r) = E near ( r) + E far ( r) (15) in which E near ( r) is cacuated with the traditiona direct integration in the near group trianges. When appying (14) and (15) to acceerate the aperture integration (1), the symbos E( r), J( r ) and M( r ) represent E i ( r), J AP ( r ) and M AP ( r ), and when appying them to (7), they are E i ( r), J r ( r ) and M r ( r ), respectivey. For the aperture integration (1), the integration ony needs to cacuate the mutua interactions from the aperture trianges to trianges on the inner surface of the radome. As the radome is severa waveengths away from the antenna as shown in Figure 3, a the groups between the aperture and the radome are far groups, then the fast mutipoe acceeration wi be very significant. For the inner surface integration of (7), the schematic of the appication of the fast mutipoe acceeration is shown in Figure 4. As the distances between most groups are arger than L, FMM wi reduce the computing time between the far groups very significanty. However, in the top of the radome, where the distances between the groups are comparabe with the group size, FMM wi not be used. Figure 3. Schematic of the appication of FMM to acceerate the aperture integration. Figure 4. Schematic of the appication of FMM to acceerate the inner surface integration.

9 Progress In Eectromagnetics Research, Vo. 120, Figure 5. radome. The photo and geometrica sizes of the miimeter wave 4. RESULTS To confirm the vaidity and efficiency of the fast mutipoe acceeration, an eectricay arge antenna-radome system at W band are fabricated and measured. The radiation patterns of the antenna-radome system determined by the AI-SI method with the fast mutipoe acceeration are compared with the cacuated and measured resuts in the pervious work [7]. The photo and geometrica sizes of the miimeter wave radome are shown in Figure 5. The radome has a height of 200 mm and a base diameter of 156 mm. In the front of the radome, there is an arc with the curvature radius of 8 mm. The radome is made of Tefon with the reative permittivity of 2.1 and the thickness is 5 mm. A conica horn with the aperture diameter of 20 mm is encosed by the radome. The horn can rotate around the gimba center, which is ocated at the base center of the radome. The antenna-radome system is operating at 94 GHz. Firsty, the first incident fieds on the radome are cacuated by direct integration using (1) and by integration with the fast mutipoe acceeration using (14) and (15). In FMM, the trianges on the aperture and radome are divided into groups with the maxima diameter d max = 1.25λ. In order to obtain enough accuracy, the parameters in (12) are chosen as γ = 1 and L = 10, thus the reative error of FMM resut can be ess than 0.1. The Gauss-Legendre integration number is K = 2L 2 = 200. The distributions of the eectrica fied components E y and E z on the inner surface of the radome are compared in Figure 6.

10 380 Meng and Dou (a) (b) (c) (d) Figure 6. The distributions of the eectrica fied components on the inner surface of the radome, (a) E y computed by direct integration, (b) E y computed by integration with the fast mutipoe acceeration, (c) E z computed by direct integration, (d) E z computed by integration with the fast mutipoe acceeration. As the conica horn is inear poarized in y direction, the eectrica fied component E x is very sma and is not given here. It is cear that the eectrica fied distributions cacuated by direct integration and by fast mutipoe acceerated integration have good agreement with each other. Then, the radiation patterns of the antenna-radome system are computed. In order to shown the accuracy of FMM, the radiation patterns determined by FMM with different the number of modes (L = 8, 10 and 12) are compared with the direct integration resut in Figure 7. It is cear that, the resut of L = 8 has obvious differences with the direct integration resut, however, the resuts of L = 10 and 12 agree very we with it. As discussed in (12), when L = 8, the

11 Progress In Eectromagnetics Research, Vo. 120, Figure 7. Cacuated H pane radiation patterns using AI-SI with fast mutipoe acceeration compares with the resut in [7] cacuated by direct integration. reative error of (11) wi be very significant. However, when L = 10, it wi be ess than 0.1, and 0.01 for L = 12. The resuts agree with the concusion in [21] as compared in Figure 7. For this antenna-radome system at W band, the direct integration of AI-SI needs about 1 h to cacuate the aperture integration and 19 h for the inner surface integration. However, by appying the fast mutipoe acceeration with L = 10, it takes ony 12 m for the aperture integration and 6 h for inner surface integration. The fast mutipoe acceeration reduces the computationa time of the antennaradome anaysis significanty, especiay the time for the aperture integration. When the number of modes L in FMM becomes arge, the computationa time wi increase quicky, for exampe, 28 m for the aperture integration and 13 h for the inner surface integration with L = 12. However, as in Figure 7, the accuracy is improved a itte. In the foowing antenna-radome anaysis, the number of modes is L = 10. The cacuated radiation patterns of the antenna-radome system using the fast mutipoe acceerated AI-SI with the number of modes L = 10 are compared with the measured radiation patterns in Figure 8. It can be seen that the resuts with fast mutipoe acceeration have some agreements with the measured patterns. They have the same accuracy with the direct integration method as in [7]. Finay, the antenna tits 10 in the H pane as in [7]. The cacuated radiation pattern with the fast mutipoe acceeration and the resuts in [7] are iustrated in Figure 9. The radiation patterns cacuated with and without FMM acceeration agree very we with each other and they both have the same agreement with the measured resut.

12 382 Meng and Dou (a) Figure 8. Measured and cacuated radiation patterns of the conica radome encosed conica horn, (a) H pane, (b) E pane. (b) Figure 9. Measured and cacuated H pane radiation patterns of the conica radome encosed conica horn when the horn tits 10 in H pane.

13 Progress In Eectromagnetics Research, Vo. 120, Up to now, the fu wave methods (MLFMM, MoM and FEM) have high accuracy for antenna-radome anaysis, but their ony suitabe for eectricay sma system. The RT method has higher efficiency and ower accuracy. However, the proposed FMM acceerated AI-SI reduces the computationa compexity of PO based method significanty and keeps an acceptabe accuracy. It is more accuracy and more efficiency than the other methods for the anaysis of eectricay arge radome in miimeter wave band. 5. CONCLUSION In this paper, an eectricay arge antenna-radome system in miimeter wave band is anayzed using the AI-SI method. The fast mutipoe method is adopted to acceerate the aperture integration from the antenna to the radome and the surface integration on the inner surface of the radome. The cacuated radiation patterns have good agreements with the experimenta resuts and direct integration resuts. The efficiency of the present method is much higher than the traditiona direct integration method and the accuracies of the two methods are comparabe. It is suitabe for fast anaysis of the antenna-radome system in miimeter wave band. ACKNOWLEDGMENT This work is supported by the Nationa Natura Science Foundation of China under grant and the State Key Laboratory of Miimeter Waves under grant Z REFERENCES 1. Arvas, E., A. Rahhaarabi, U. Peke, et a., Eectromagnetic transmission through a sma radome of arbitrary shape, IEE Proceedings-H Microwaves, Antennas and Propagation, Vo. 137, No. 6, , Povinei, M. J. and J. D Angeo, Finite eement anaysis of arge waveength antenna radome probems for eading edge and radar phased arrays, IEEE Transactions on Magnetics, Vo. 27, No. 5, , Nie, X.-C., N. Yuan, L.-W. Li, T. S. Yeo, and Y.-B. Gan, Fast anaysis of eectromagnetic transmission through arbitrary shaped airborne radomes using precorrected-fft method, Progress In Eectromagnetics Research, Vo. 54, 37 59, 2005.

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