350lect14 - 14 Interference zones, plane waves y...

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14 Interference zones, plane waves Let’s examine the radiation feld oF a 1D array oF N =2 M +1 iden- tical elements located at ( nd, 0 , 0) ,w i th n in the interval - M, ···- 1 , 0 , 1 , ··· M having spherical wave feld phasors x y z D x 0M HPBW = λ D x 2 D x 2 D 2 x λ Fraunhoffer region Fresnel region ˜ E n ( r )= o I n sin θ n e - jk | r - ˆ xnd | 4 π | r - ˆ xnd | ˆ θ n where cos θ n z · r - ˆ xnd | r - ˆ xnd | . The total feld phasor ˜ E ( r M ± n = - M ˜ E n ( r ) oF the array will have di±erent types oF spatial variations in di±erent interference zones or regions : 1. The region | r | ± 2 D 2 x λ , where D x Md is the physical length oF the array, is known as Fresnel region or the near-feld radiation zone — in this zone paraxial approximation cannot be used and the radiation feld is highly structured having a prominent magnitude directly above the array (i.e., For - ± x ± ). 1
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2. The region x y z D x 0M HPBW = λ D x 2 D x 2 D 2 x λ Fraunhoffer region Fresnel region | r | ± 2 D 2 x λ is known as Fraunhofer region or the ±ar ²eld —th isisthezonein which paraxial approximation works well, and spherical waves arriving from individual array elements merge to become a single spherical wave of a higher directivity. The concept of antenna beam applies only in the Fraunho±er region. Abeamw itha²xedangu larw idthemergesouto ftheF resne lreg ion as Fraunho±er region is approached, as shown in the cartoon in the margin (in which an “unphased” broadside array has been assumed in sketching the far-²eld beam). In addition, it should be noted that the region | r | ² few λ will include strong storage ²elds, whereas for | r 2 D 2 x λ ,d e epinF raunho±e rr eg ,sph e r i ca lw av e sw i l l “locally” look like plane waves. We will next examine the transition between Fresnel and Fraunho±- ±er regions and then examine how spherical waves can be treated as plane waves over limited regions o± space in the ±ar-²eld. 2
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Consider the “phase-delay” of signals arriving from individual elements of a broadside array on the x -axis to a location (0 ,r, 0) on the y -axis as shown in the margin. x y z D x 0M ray-0 ray-M (0 0) Clearly, the sample “rays” shown in the margin connecting diFerent array elements to (0 0) have diFerent lengths even though in paraxial approximation only one length, r ,wou ldbea s s ignedto all them since nd cos θ x =0 for θ x =90 .
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This note was uploaded on 09/27/2011 for the course ECE 450 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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350lect14 - 14 Interference zones, plane waves y...

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