Frequency and time domain characteristics of linearly phased/ sequentially excited periodic arrays of dipoles

Doğrusal fazlı/sırayla uyarılan, periyodik dipol dizilerinin frekans ve zaman domeni karakteristikleri

Abstract

In this study, frequency and time domain radiation characteristics of linearly phased/sequentially excited periodic dipole arrays are investigated. Large periodic arrays of short radiating elements have been increasingly investigated for the last decade or so, since they have a vast range of applications ranging from phased array antenna systems with flexible beam steering and very narrow beam forming capabilities, frequency selective surfaces and diffraction gratings to photonic crystal fibers and metamaterials, etc. The field radiated by a sequentially excited/linearly phased dipole array can be calculated via a superposition over the contributions of individual dipoles in the array. This classical approach, called element by element summation, suffers from two shortcomings: i) its numerical implementation becomes time consuming and inefficient for large size arrays ii) it does not provide a framework for identification of the mechanisms generating different wave objects (propagating, evanescent, diffracted waves) and their relative weights at given observation points. On the other hand, by taking advantage of the periodicity, the problem can be cast into an alternative formulation wherein the contributions of individual elements are represented collectively in terms of Floquet waves, which are augmented with tip diffracted wave constituents to account for the truncation of a strictly periodic sequence at a large but finite number. The phenomenologies associated with the phased array of dipoles are also related to the oblique plane wave scattering from an array of short wire elements. The obliquity of the incident field with respect to the array plane determines the interelement phasing (time delay) of the radiating array. The field radiated by a finite, periodic line array of dipoles at a given observation point can be synthesized via superposition of shifted and properly weighted replicas of the field radiated by semi infinite line arrays of dipoles. In this paper, we propose a synthetic aperture type approach for calculation of the field radiated by a finite line array of length Lz, wherein the contribution of the shifted semi infinite line array is accounted for by the contribution of the original array at an oppositely shifted virtual observation point P’(x,y,z-Lz). The proposed synthetic aperture approach is also used for the efficient modeling of radiation from structures involving line arrays organized spatially to form planar and 3D arrays. In this paper, utilizing the proposed synthetic aperture approach, radiation from linear and planar dipole arrays are calculated and examples of numerical results are presented both in the frequency domain and also in the, heretofore scarcely investigated, time domain. It is shown that when the pulsed excitation has a relatively flat spectra which does not have significant low frequency content, the array response can be calculated in a rather efficient and accurate manner utilizing analytical approximations for the contributions from Floquet wave constituents. High frequency asymptotics can be used for obtaining contributions from all except for the zero indexed Floquet wave which needs to be treated separately and involves the calculation of a convolution. For obtaining the response for wide band dipole excitations, we introduced an accurate analytic approximation technique for calculating the involved convolution integral, which is based on an adaptive windowing scheme wherein the pulse excitation is represented approximately via a set of rectangular pulses with adaptively determined amplitudes and time extends. Following this approach the convolution is synthesized via the contributions from the set of rectangular pulses each of which can be calculated in closed form. The resulting formulation yields substantial reduction in the computation time since the response due to all Floquet wave constituents are calculated analytically, either via high frequency asymptotics or via the weighted sampling approach. The exact solution obtained via element by element summation is utilized as a comparison solution for validating the numerical results calculated via Floquet wave representations. Element by element summation representations suffer from convergence problems which are best illustrated considering the representation for a semi infinite line array. In this context, a method is proposed to substantially improve the convergence properties of the element by element summation approach. In this paper, we have been able to verify that Floquet waves provide a very accurate and efficient representation for calculating time/frequency domain radiation from linearly phased/sequentially excited, large, 1D, 2D and 3D periodic arrays of electric current dipoles.

Özet

Doğrusal fazlı/sırayla uyarılan periyodik dipol dizilerinin frekans ve zaman domeni yanıtları, Floquet dalga ve eleman-eleman toplama gösterilimleri kullanılarak incelenmiştir. Her bir dipolün katkısının ayrı ayrı hesaplanıp toplanması ile bulunan eleman-eleman toplama yönteminin yakınsaklık özelliklerini iyileştiren bir formülasyon geliştirilmiştir. Periyodiklik özellikleri kullanılarak, dizinin toplam ışınım alanı için Floquet dalgaları ve sonsuz periyodik bir dizinin sonlandırılmasının etkilerini karakterize eden uç kırınım bileşenlerini içeren alternatif bir formülasyon elde edilebilir ve biri diğerine göre kaydırılmış ve uygun şekilde ağırlaştırılmış iki yarı-sonsuz çizgisel periyodik dizinin farkını alarak sonlu çizgisel diziler modellenebilir. Yukarıda değinilen yaklaşım literatürde incelenmiştir, bu çalışmada önerilen ve yapay açıklık yaklaşımı adı verilen yöntemde ise, Lz kadar kaydırılmış dizinin bir P(x,y,z) gözlem noktasında oluşturacağı alan, diziyi kaydırmak yerine gözlem noktasını zıt yönde kaydırarak, orijinal kaydırılmamış dizinin bir P’(x,y,z-Lz) sanal gözlem noktasında oluşturacağı alan bulunarak hesaplanmaktadır. Önerilen bu yapay açıklık yaklaşımı, çizgisel dipol dizilerinin süperpozisyonu alınarak elde edilebilen her türlü düzlemsel ve üç boyutlu dipol dizisinin etkin biçimde modellenmesinde de kullanılmıştır. Işınım yapan, yapmayan ve kırınan dalga bileşenleri cinsinden ifade edilen Floquet dalga gösteriliminin etkinliğini göstermek için sayısal örnekler verilmiş ve bu sonuçlar eleman-eleman toplama yöntemi ile karşılaştırılıp doğrulanmıştır. Dipollerin geniş bantlı kaynak fonksiyonlarıyla sürüldüğü durumda karşılaşılan konvolüsyon integrallerinin hesaplanması için kaynak fonksiyonuna genlik ve süreleri adaptif biçimde belirlenen parça parça sürekli bir dizi dikdörtgen darbe ile yaklaşıklık yapılmasına dayanan analitik bir yöntem geliştirilmiştir.