The Polarization Ellipse in VS .NET

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2 The Polarization Ellipse
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(b) = /4
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(c) = /2
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(d) = 3 /4
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Fig 22 Ellipse forms are depicted for phase differences x ) D of 0 (a), /4 (b), /2 (c) and 3 /4 (d), with ( y the orthogonal amplitudes, E x0 , E y 0 , equal to each other For D 0, linear polarization ensues; for D /2, with equal component amplitudes, circular polarization results The sense of description of the ellipse gures depends on the sign of the phase difference (see the text)
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Squaring (217) and (218) and adding gives
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2 2 Ey Ex C 2 2 Ex 0 Ey 0
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2E x E y cos y Ex 0 Ey 0
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This last equation represents the polarization ellipse and various forms demonstrating its behaviour according to the phases for the special condition with E x 0 D E y 0 are depicted in Figure 22 Thus the form of the polarization is described by the four parameters E x 0 , E y 0 , x , and y Such quantities, however, are not directly measurable, but it will be demonstrated later that various combinations of them can be formed which are more convenient for describing the polarization ellipse It is important to remember that the polarization ellipse is not static in the x y plane corresponding to z D 0; its locus is described at the frequency of the radiation The directional sense of drawing out the ellipse carries an alternative depending on the magnitude of the phase difference between the orthogonally resolved amplitudes According to the mathematics above, when > y x > 0, the E-vector would be seen to rotate in a clockwise direction as viewed by an observer receiving the radiation; when 0 > y x > , the E-vector would be seen to rotate in an anti-clockwise direction It is traditional to express this by the notion handedness, with the terms righthanded and left-handed being used As it turns out, it is the conventions related to handedness which are a troublesome thorn in general polarimetry (see Clarke, 1973), with the various scienti c disciplines adopting their individual schemes, as does the International Astronomical Union (IAU) itself At this stage it would be confusing to assign handedness to the outcome of the mathematics above for the following reasons Handedness depends on the viewpoint in which right-handed as seen by an observer is left-handed from the point of view of the source Terms such as clockwise and anti-clockwise suffer in the same way The problem is further compounded by
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Fig 23 A snapshot of the helical distribution (left-handed in the gure) of the electric disturbance As time proceeds, the helix is carried along the z-axis without rotation In any x y-plane, the vector will be seen to rotate with a particular sense according to the handedness of the helix According to the viewpoint of the observer, the rotation would be anti-clockwise for the depicted case
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the fact that a sign is sometimes added to the handedness and, across the various cross-disciplinary studies, a given elliptical polarization may be described by one of four differing de nitions: right-handed positive (C), right-handed negative ( ), with right-handed having alternative de nitions according to the viewpoint as just described These issues will be discussed again later in 4 in respect of the de nitions adopted by the IAU There is, however, a unique way of de ning handedness (see Clarke & Grainger, 1970) for it can be readily demonstrated that an instantaneous snapshot of the electric vector, as it is distributed along the z-axis, traces out a helical pattern (see Figure 23) As a helix has a handedness which is de ned independently of the observer s viewpoint, the polarization handedness may be labelled by the type of helix (left- or right-handed) which is present If we consider the motion of the disturbance with time, we see that the helix moves along the direction of propagation without rotation and that its point of intersection with a plane transverse to its direction, say the x y -plane at z D 0, executes the polarization ellipse; the sense of execution, as seen by the observer, is clockwise for a right-handed helix and anti-clockwise for a left-handed helix, and hence the handedness of the helix corresponds to the common de nition of polarization handedness used in general optics In the literature there are, however, con icting comments on this point It is sometimes stated erroneously that when the handedness of the helix is used to describe the polarization handedness, a convention which is opposite to that of tradition results Such a notion perhaps arises by considering the disturbance to be represented by a vector which is rotating as it is propagated Its tip would again describe a helix, but this helix would be xed in space, and not moving without rotation as the equations of the waves imply
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