The Phase Matrix

As mentioned in Section 2.2.1, the Stokes vector, ${\b I}$, is always defined with respect to a reference plane which is given by the vertical direction and the direction of propagation, the meridian plane. However, the scattering matrix, as discussed in Section 2.3.4, was referenced to the plane of scattering. That is, the plane containing both the incident and scattered directions. Thus, in order to compute the amount of light incident in direction $(\mu',\phi')$ and scattered into direction $(\mu,\phi)$, ${\b I}$ must first be transformed from the incident meridian plane to the plane of scattering so that the scattering calculations can be carried out and then from the plane of scattering to the scattered meridian plane.

The Stokes vector can be rotated through an angle $i(\geq0)$ in the anti-clockwise direction, when looking into the direction of propagation, by the rotation matrix (e.g.: Liou, 1980),

$${\b L}(i) = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & ... ...sin 2i & \cos 2i & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right]$$ (7.1)

such that the rotated Stokes vector is ${\b I}' = {\b L}(i) {\b I}$. The elements I and V are invariant under rotation. The factor of 2 arises as the transformation must be carried out in amplitude space and then converted into intensity space.

  

Figure 4.1: Rotation of the Stokes vector into scattering plane and into scattered meridian plane.

Applying both required rotations and grouping the scattering and rotation matrices together, the phase matrix is arrived at,

$${\b Z}(\tau;\mu,\phi;\mu',\phi') = {\b L}(\pi-i_2) {\b P}(\tau;\mu,\phi;\mu',\phi') {\b L}(i_1)$$ (7.2)

which is also a function of the two rotation angles, i₁ and i₂. The phase matrix effectively replaces the scattering matrix in all source function expressions. Carrying out the matrix multiplication gives the explicit form for the phase matrix,

$${{\b Z}(\tau;\mu,\phi;\mu',\phi') =}$$

$$\left[ \begin{array}{c c c c} \pa & \pb c_1 & \pb s_1 & 0 \\ \pb... ..._1 + c_2 \pc c_1 & -\pd c_2 \\ 0 & -\pd s_1 & \pd c_1 & \pc \end{array}\right]$$

where

$$\cos(-2i_1)$$ c₁ = (7.4)

$$\cos[2(\pi-i_2)]$$ c₂ = (7.5)

$$\sin(-2i_1)$$ s₁ = (7.6)

$$\sin[2(\pi-i_2)]$$ s₂ = (7.7)

The angles i₁ and i₂ can be related to the incident and scattered directions, $(\mu',\phi')$ and $(\mu,\phi)$, respectively, using spherical trigonometry, as illustrated in Figure 4.1,

$$\cos i_1 \frac{-\mu+\mu' \cos\Theta}{\pm(1-\cos^2\Theta)^{1/2} (1-\mu'^2)^{1/2}}$$ = (7.8)

$$\cos i_2 \frac{-\mu'+\mu \cos\Theta}{\pm(1-\cos^2\Theta)^{1/2} (1-\mu^2)^{1/2}}.$$ = (7.9)

The denominators are positive when $\pi<(\phi-\phi')<2\pi$ and negative when $0<(\phi-\phi')<\pi$. The incident and scattered rays have been placed in the same octant in Figure 4.1 for convenience. Recall that the scattering angle, $\Theta$, is given by,

$$\cos \Theta = \mu \mu' + \sqrt{1-\mu^2} \sqrt{1-\mu'^2} \cos(\phi-\phi').$$ (7.10)

For homogeneous, spherical scatterers there are now fourteen non-zero elements as opposed to eight before the rotations were included. These fourteen elements, however, are composed only of the original four independent scattering matrix elements (three for Rayleigh scattering).