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The Stokes vector can be defined operationally as follows:
|
(1) |
where Iu is the irradiance of the beam measured with a detector insensitive to the polarization of ER, Il and Ir are irradiances of the linearly polarized beam components respectively parallel and perpendicular to the scattering plane, I/ and I\ are irradiances of the linearly polarized beam components inclined respectively at the angles of +45° and 135° (i.e. -45°) to the scattering plane (the + sign indicates a clockwise rotation of the polarization plane when looking at the ER source), and finally IR and IL are irradiances of the circularly polarized beam components: right- and left-handed, respectively (the electric vector of the right-handed component rotates clockwise when looking at the ER source, making the S4 element of the Stokes vector positive). On the other hand, if one knows the Stokes vector components, for example, from Eq. 1 in Scattering matrix, with a matrix calculated by using an ER scattering (light scattering) theory, one can determine the respective irradiances from Eq. 1.
Equations of Eq. 1 follow from the definitions of the Stokes vector elements (Eq. 1 in Stokes vector) and the representation of the electric field of an EM wave by a sum of electric fields, El and Er of its two orthogonal linearly polarized components (for example, Hecht E 1987), here identified in reference to the scattering plane. Indeed, the first equation in Eq. 1 results from the definition of S1 (the top equation in Eq. 1 in Stokes vector):
| S1 | = <ElEl*> + <ErEr*> | |
| = Il + Ir | ||
| = Iu | (2) |
The second row in Eq. 1 can be obtained in an analogous way and by noting that Il - Ir = Il - Ir + (Il - Il).
The third row in Eq. 1 results from expressing the electric vectors of the two orthogonal polarizations in a new reference frame, rotated by +45° (for example, Bohren CF and Huffman 1983), to obtain the irradiance of a linearly polarized component of the EM wave inclined at +45° in respect of the scattering plane. The "+45°" basis vector of this new frame, relevant here, is expressed as e/ = (1/√2)(el + er). Hence, the irradiance of the "/" component is:
| I/ | = | ½ (<ElEl*> + <ElEr*> + <ElEr*> + <ErEr*>) | |
| = | ½ (Il + S3 + Ir) | ||
| = | ½ (Iu + S3) | (3) |
where the third row in Eq. 1 in Stokes vector has been used in the second line. Equation S3 = 2I/ - Iu (the third row in the present Eq. 1) is obtained by rearranging the above equation. The other operational definition of S3 in Eq. 1, i.e. S3 = I/ - I\, can be obtained by writing an equation for the I\ component (similar to Eq. 3) and making use of the definition of S3 (the third row of Eq. 1 in Stokes vector). Note that the basis vector of the "\" (or a 135°, i.e. -45°) polarization is expressed as e\ = (1/√2)(el - er).
Equation S4 = 2IR - Iu in Eq. 1 can be obtained in a similar way with the relevant unit vector of the new basis expressed as eR = (1/√2)(el + ier), where the factor of i represents a π/2 phase shift between the two orthogonal components of an EM wave, which is required to produce the circular polarization.
| CITATION: Jonasz M. 2006. Stokes vector: Measurements (www.tpdsci.com/Tpc/StoVecMsm.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 20-Mar-2006 Modified: 11-Oct-2006 Peer-reviewed: 11-Mar-2008 |
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