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Unformatted text preview: 43 2.4.2 Circular polarization Another important special case is given by E 1 = E 2 , ψ = ± π/ 2 so that E = (ˆ x ± j ˆ y ) E 1 e j ( ωt − kz ) (2.12) In instantaneous notation, this is E ( z,t ) = ℜ bracketleftBig (ˆ x ± j ˆ y ) E 1 e j ( ωt − kz ) bracketrightBig = E 1 [ˆ x cos( ωt − kz ) ∓ ˆ y sin( ωt − kz )] The electric field vector therefore traces out a circle with a radius E 1 such that E 2 x + E 2 y = E 2 1 and an angle with respect to ˆ x given by α = tan − 1 E y /E x = tan − 1 ∓ sin( ωt − kz ) / cos( ωt − kz ) = ∓ ( ωt − kz ) . Imagine holding the spatial coordinate z fixed. At a single location, increasing time decreases (increases) α given a plus (minus) sign in (2.12). In the example in Figure 2.10, which illustrates the plus sign, the angle α decreases as time advances with z held constant, exhibiting rotation with a left-hand sense (left thumb in the direction of propagation, fingers curling in the direction of rotation). The plus sign is therefore associated with left circular polarization in most engineering and RF applications, where the polarization of a wave refers to how the electric field direction varies in time. This text follows this convention. Consider though what happens when time t is held fixed. Then the angle α increases (decreases) with increasing z given the plus (minus) sign in the original field representation (2.12). Referring again to Figure 2.10, the angle α increases as z advances with t held constant, exhibiting rotation with a right-hand sense (right thumb in the direction of propagation, fingers curling in the direction of rotation). In many physics and optics texts, polarization refers to how the electric field direction varies in space, and the plus sign is associated with right circular polarization. This is not the usual convention in engineering, however. 2.4.3 Elliptical polarization This is the general case and holds for ψ negationslash = ± π/ 2 or E 1 negationslash = E 2 . We are left with E ( z,t ) = ℜ bracketleftBig ( E 1 ˆ x + E 2 e ± jψ ˆ y ) e j ( ωt − kz ) bracketrightBig = ˆ xE 1 cos( ωt − kz ) + ˆ yE 2 cos( ωt − kz ± ψ ) At fixed z = 0 , for example, this is E ( z = 0 ,t ) = ˆ xE 1 cos( ωt ) + ˆ yE 2 cos( ωt −± ψ ) which is a parametric equation for an ellipse. With a little effort, one can solve for the major and minor axes and tilt angle of the ellipse in terms of E 1 , 2 and ψ . One still has left and right polarizations, depending on the sign of ψ . The following principles are important for understanding free-space wave polarization: • H is in every case normal to E , with E × H in the direction of propagation and | E | / | H | = Z ◦ ....
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- Spring '13
- The Land, power density, Solid angle