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Figure from:Optimal reception of partially polarized waves

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where the polarized P and unpolarized U parts are uniquely determined? by the relations The completely polarized component of R can be written in spherical coordinates of the Stokes subspace as?4,!4

Figure 1 where the polarized P and unpolarized U parts are uniquely determined? by the relations The completely polarized component of R can be written in spherical coordinates of the Stokes subspace as?4,!4

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The key step here is to realize that an unpolarized wave can always be represented by an incoherent sum of any twc orthogonal completely polarized waves of equal intensity.’ Let us represent the received wave as 1/2(H + H,) where H , is the Stokes vector orthogonal to H. The H part will be absorbed by the receiver, and the H, part will be totally; rejected, leading to a 50% efficiency for the reception of the unpolarized waves. To illustrate this point, consider the following representation for R: which is an equal mixture of left-hand-circular (LHC)-po- larized waves and right-hand-circular (RHC)-polarized waves. Ifthe receiver is RHC tuned, it will receive 1/2 of the total energy. This argument applies for any h and 7 pair, and we therefore conclude that the total available energy consists of two parts: 100% reception efficiency for P and 50% reception efficiency for U. The former can be affected by changing the polarization state of the receiver, whereas the latter is polarization independent.
The key step here is to realize that an unpolarized wave can always be represented by an incoherent sum of any twc orthogonal completely polarized waves of equal intensity.’ Let us represent the received wave as 1/2(H + H,) where H , is the Stokes vector orthogonal to H. The H part will be absorbed by the receiver, and the H, part will be totally; rejected, leading to a 50% efficiency for the reception of the unpolarized waves. To illustrate this point, consider the following representation for R: which is an equal mixture of left-hand-circular (LHC)-po- larized waves and right-hand-circular (RHC)-polarized waves. Ifthe receiver is RHC tuned, it will receive 1/2 of the total energy. This argument applies for any h and 7 pair, and we therefore conclude that the total available energy consists of two parts: 100% reception efficiency for P and 50% reception efficiency for U. The former can be affected by changing the polarization state of the receiver, whereas the latter is polarization independent.
Table 2. Optimal versus Standard Stokes Vectors for Case 1 which has six (real and complex) roots, as is shown in Table 3. Whereas the norm test shows that the complex roots are valid, they must be ignored, as the solution vector must always be real by the construction of Eqs. (4) and (19b). All numerical experiments have shown the existence of at least two real roots that correspond to the maximum and the minimum of the adjustable intensity. In this case, despite the presence of four complex roots, the extrema are found correctly, and they correspond to the two real roots, as is shown in Table 3. It is interesting to note that our maximum value of adjust- able intensity (0.9677) exceeds the maximum gain for the total intensity in the beam (calculated to be 0.89 in Ref. 18). This apparently paradoxical result prompted us to check the degree of polarization of the output beam [0.8938, —0.5287, 0.3410, 0.73], which is 1.08. Thus even an experimental Mueller matrix may be slightly unrealizable (the violations are less than 8%, however, and only for an extremely small
Table 2. Optimal versus Standard Stokes Vectors for Case 1 which has six (real and complex) roots, as is shown in Table 3. Whereas the norm test shows that the complex roots are valid, they must be ignored, as the solution vector must always be real by the construction of Eqs. (4) and (19b). All numerical experiments have shown the existence of at least two real roots that correspond to the maximum and the minimum of the adjustable intensity. In this case, despite the presence of four complex roots, the extrema are found correctly, and they correspond to the two real roots, as is shown in Table 3. It is interesting to note that our maximum value of adjust- able intensity (0.9677) exceeds the maximum gain for the total intensity in the beam (calculated to be 0.89 in Ref. 18). This apparently paradoxical result prompted us to check the degree of polarization of the output beam [0.8938, —0.5287, 0.3410, 0.73], which is 1.08. Thus even an experimental Mueller matrix may be slightly unrealizable (the violations are less than 8%, however, and only for an extremely small
Table {. Roots and Intensities for Case | part of the Stokes space). We emphasize that the last result is entirely independent of our optimization procedure and probably reflects a slight measurement error.2° Indeed, one need only use Eq. (6) with the optimal Stokes vector to recover the above violation.
Table {. Roots and Intensities for Case | part of the Stokes space). We emphasize that the last result is entirely independent of our optimization procedure and probably reflects a slight measurement error.2° Indeed, one need only use Eq. (6) with the optimal Stokes vector to recover the above violation.
Fig. 1. Polarization dependence of the adjustable intensity. The general topology of the surface is determined by the six stationary points corresponding to the six roots of Eq. (20) with the global maximum at 0 (tilt) = 75.1 deg and e (ellipticity) = 24.1 deg and with the global minimum at 0 (tilt) = 72.4 deg and ¢ (ellipticity) = —17.3 deg.14
Fig. 1. Polarization dependence of the adjustable intensity. The general topology of the surface is determined by the six stationary points corresponding to the six roots of Eq. (20) with the global maximum at 0 (tilt) = 75.1 deg and e (ellipticity) = 24.1 deg and with the global minimum at 0 (tilt) = 72.4 deg and ¢ (ellipticity) = —17.3 deg.14
Table 4. Optimal versus Standard Stokes Vectors for Case 2
Table 4. Optimal versus Standard Stokes Vectors for Case 2
Table 3. Roots and Powers for Case 2
Table 3. Roots and Powers for Case 2