FUENTES DE LA NORMATIVIDAD REGULADORA DEL JUICIO DE FALTAS

Generally optical instruments cannot directly measure the electric and mag- netic eld associated with a beam of light, they rather measure the inten- sity of radiation and its polarimetric state. Fundamental examples of these observable quantities are the so-called Stokes parameters. A spherical co- ordinate system, associated with a local right-handed Cartesian coordinate

Scattering, Absorption, and Emission of Light by Small Particles 16

sorbing medium is specified by a unit vector nˆ or, equivalently, by a couplet (_ϑ,_ϕ),

where _ϑ∈[0 ,_π] is the polar (zenith) angle measured from the positive z-axis and

)

2

,

0

[

_π

ϕ∈

is the azimuth angle measured from the positive x-axis in the clockwise

direction when looking in the direction of the positive z-axis. Since the medium is

assumed to be nonabsorbing, the component of the electric field vector along the di-

rection of propagation nˆ is equal to zero, so that the electric field at the observation

point is given by E

₌E

₊E

,where E and

E are the -

_ϑ

and -_ϕ

components of

the electric field vector. The component E

=E

ϑˆ lies in the meridional plane (i.e.,

plane through nˆ and the z-axis), whereas the component E

₌E

_ϕˆ is perpendicular

to this plane; _{ϑˆ and ϕˆ are the corresponding unit vectors such that} nˆ

_=ϑ׈

_ϕˆ. Note

that in the microwave remote sensing literature, E and

E are often denoted as

E

and

E

and called the vertical and horizontal electric field vector components, re-

spectively (e.g., Tsang et al. 1985; Ulaby and Elachi 1990).

The specification of a unit vector nˆ uniquely determines the meridional plane of

the propagation direction except when nˆ is oriented along the positive or negative

direction of the z-axis. Although it may seem redundant to specify _ϕ

in addition to _ϑ

when _ϑ

₌0 or _π, the unit _ϑ and _ϕ

vectors and, thus, the electric field vector com-

ponents E and

E still depend on the orientation of the meridional plane. There-

fore, we will always assume that the specification of nˆ implicitly includes the speci-

fication of the appropriate meridional plane in cases when nˆ is parallel to the z-axis.

To minimize confusion, we often will specify explicitly the direction of propagation

using the angles _ϑ

and ;_ϕ

the latter uniquely defines the meridional plane when

0

=

ϑ

or .π

Consider a plane electromagnetic wave propagating in a medium with constant

ϕ

x

y

O

z

ϑ

Figure 1.2. Coordinate system used to describe the direction of propagation and the polariza- tion state of a plane electromagnetic wave.

Fig. 2.3: Coordinate system used to described the direction of propaga- tion and the polaritation state of a plane electromagnetic wave. Source: Mishchenko et al. (2002).

system with origin in the observation point, is needed with the purpose of dening such parameters. Considering an electromagnetic wave that propa- gates in a homogeneous and non-absorbing medium, the direction of propa- gation can be dened by a unit vector n or by a couple of angles (ϑ and ϕ) (see Figure 2.3). These angles are respectively the zenith angle (ϑ ∈ [0,π]) and the azimuth angle (ϕ ∈ [0,2π)). In such case, being the medium as- sumed as non-absorbing, the component of electrical eld along the direction of propagation in equal to zero, so that the electric eld at the origin of the axes is given by E=Eϑ+ Eϕ. In fact E can be decomposed into two compo-

nents representing the electric vectors parallel and perpendicular to a plane trough the direction of propagation. The plane is usually called the plane of

reference.

Thanks to such division it is possible to dene a set of four quantities, namely the Stokes parameters. Because intensity is proportional to the ab- solute square of the electric eld, neglecting a constant of proportionality, these parameters are calculated as follow:

I =EϑE∗ϑ+EϕE∗ϕ (2.23a)

Q =EϑE∗ϑ−EϕE∗ϕ (2.23b)

U =EϑE∗ϕ+EϕE∗ϑ (2.23c)

V = −i(EϑE∗ϕ−EϕE∗ϑ) (2.23d)

where asterisk stands for the complex conjugate value. The Stokes parame- ters I, Q, U, and V are also dened as the elements of 4x1 column vector I, known as Stokes vector.

I =         I Q U V         (2.24)

The vector components have a physical meaning: I represents the total inten- sity of the electromagnetic wave, Q is the dierence in intensity between the horizontal and the vertical modes, U for the ± 45◦ _{modes (i.e. in a reference}

frame rotated), whereas V stands for intensity dierence of the components in the circular reference frame. Moreover, the Stokes parameters of a plane monochromatic wave are not completely independent but rather related by a quadratic identity. Indeed they satisfy equation (2.25)

I2 = Q2+ U2+ V2 (2.25) and also can be expressed in terms of the geometry dening an ellipse, the so-called polarization ellipse.