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34 Poincar Sphere
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The Poincar sphere has its main application in problems relating to phase changes affecting radiation such as those provided by retarders within some instrumenta-
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Fig 33 The procedure for determining the effect of a retarder of phase, , with its principal axis set at an angle, , to the reference frame de ning P, is illustrated with the location of the resulting polarization form, P0 , indicated
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tion Any beam of perfectly polarized light represented by an eigenvector, P, will be modi ed to provide a resultant eigenvector, P0 , its position depending on the orientation of the retarder axis and its phase delay The relationship of P0 with respect of P may be determined according to the following recipe Suppose that radiation described by P and de ned by (I diff , Icos , Isin ) is incident upon a retarder which introduces a retardance, , between perpendicular directions (x 0 , y 0 ) with the fast axis oriented at an angle, , to the x, y directions; the angle is measured anti-clockwise as seen by the observer from (x, y ) to (x 0 , y 0 ) 0 0 0 see Figure 31 The point representing the emergent beam (Idiff , Icos , Isin ) can be determined by two simple steps By considering Figure 33 and starting at the point where the Idiff -axis intersects the surface of the sphere, move round the great circle, which is the intersection of the sphere and the (Idiff Icos )-plane, through an angle, 2 , in the direction Idiff to Icos This de nes a point, A Regarding A as a pole, construct the latitude and longitude circles through the point (Idiff , Icos , Isin ) and then move clockwise as seen from A, round the latitude circle and through an 0 0 0 angle, , of longitude This de nes the point (Idiff , Icos , Isin ) Clearly the method can be extended to evaluate the effect of a series of retarders It must be remembered that the s relating each of the devices must all be measured relative to the original frame of reference The reverse procedure can be used to determine the necessary retarder to achieve a desired result from a known input The normal rules of spherical geometry, as applied to the Poincar representation, constitute an attractive method for investigating retarder problems An excellent example of this is the work of Pancharatnam (1955a, 1955b) who explored the problems of the design of achromatic retarders by this mathematical method
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3 The Algebra of Polarization
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In simple summary, the term polarization refers to the time-dependent behaviour of the electric and magnetic disturbances within some plane set normal to the direction of propagation of the electromagnetic radiation Quantitatively it describes any asymmetry in the azimuthal distribution of the vectors in that plane or any persistent phase coherence between the resolved components of the uctuations Generally in nature, a beam of radiation does not have a persistent classical form but comprises combinations of waves with disturbances persisting for a short time before being replaced by others There are constant uctuations in the phase relationships and in the position angles of the component vibrations The radiation may also encompass a frequency spread as in white light Amongst these statistical uctuations, however, there may be some polarization forms during the integration time of the observation which occur more frequently than others and dominate in the time-averaging process during measurement There are several mathematical formulations which deal with these situations The most favoured involves the use of Stokes parameters and the Mueller calculus These are introduced in the following chapter
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