2.9   POLARIZATION

If we examine the electrical state of matter on a microscopic level, we discover that
it consists of charges, the distribution of which depends on the presence or absence
of external fields. If we assume that for every positive charge there is a negative
charge, then we may think that each positive–negative charge constitutes an electric
dipole. The electric moment of a dipole at some distance is a function of distance
and charge density. Now, for a distribution of electric dipoles, the electric dipole
moment per unit volume is defined as the polarization vector P.

Two relations describe the propagation of light in nonconducting media:

and

where ε₁ and ε₂ are two constant unit vectors that define the direction of each field,
k is the unit vector in the direction of propagation, and E₀ and H₀ are complex amplitudes,
which are constant in space and time.

Assuming a wave propagating in a medium without charges, then ΔE = 0 and ΔH = 0.
Based on this, the product of unit vectors is

That is, the electric (E) and the magnetic (H) fields are perpendicular to the direction
of propagation k (Figure 2.2). Such a wave is called a transverse wave.

Polarization of electromagnetic waves is a complex subject, particularly when
light propagates in a medium with different refractive indexes in different directions.
As light propagates through a medium, it enters the fields of nearby dipoles
and field interaction takes place. This interaction may affect the strength of the electric
and/or magnetic fields of light differently in certain directions so that the end
result may be a complex field with an elliptical or a linear field distribution. For example,
the electric field e becomes the linear combination of two complex fields E_0x
and E_0y, the two componets in the x and y directions of a Cartesian coordinate system,
such that

This relationship implies that the two components, E_0x and E_0y, vary sinusoidally,
they are perpendicular to each other, and there may be a phase betwen them, φ.
From this relationship, a vector is defined known as the Jones vector; the Jones vector
J = [J₁, J₂] is related to the radiation aspects of the wave:

In this case, the dielectric quantity ε is described by a tensor that, in general, has
different values in the three axes:

Now, from

and

one obtains

or

where I is the identity matrix. The latter is a vector equation equivalent to a set of
three homogeneous linear equations with unknowns the components of E₀, E_0x, E_0y,
and E_0z. In a typical case, the component E_0z along the axis of propagation is equal
to zero.

This vector equation determines a relationship between the vector k (k_x, k_y, k_z),
the angular frequency ω, and the dielectric constant ε (ε_x, ε_y, ε_z), as well as the polarization
state of the plane wave.

Now, the term [k × (k × I) + μ₀εω²] describes a three-dimensional surface. As
the (complex) electric field is separated into its constituent components, each component
may propagate in the medium at a different phase. The phase relationship as
well as the magnitude of each vector defines the mode of polarization.

If E_0x and E_0y have the same magnitude and are in phase, then the wave is called
linearly polarized.

If E_0x and E_0y have a phase difference (other than 90°), then the wave is called elliptically
polarized.

If E_0x and E_0y have the same magnitude but differ in phase by 90°, then the wave
is called circularly polarized. For example, in circularly polarized light the wave
equation (propagating in the z direction) becomes

Then, the two real components (in the x and in the y directions) are

and

These equations indicate that at a fixed point in space the fields are such that the
electric vector is constant in magnitude but it rotates in a circular motion at a frequency
ω. The term ε_x + jε_y indicates a counterclockwise rotation (when facing the
oncoming wave), and this wave is called left-circularly polarized or a wave with
positive helicity. The term ε_x − jε_y indicates a clockwise rotation (when facing the
oncoming wave), and this wave is called right-circularly polarized or a wave with
negative helicity.

Using the notion of positive and negative helicity, E can be rewritten as

where E₊ and E₋ are complex amplitudes denoting the direction of rotation.

Now, if E₊ and E₋ are in phase but have different amplitude, the last relationship
represents an elliptically polarized wave with principal axes of the ellipse in the directions
ε_x and ε_y.

Then, the ratio of the semimajor to semiminor axis is (1 + r)/(1 – r), where E₋ /E₊
= r.

If the amplitudes E₊ and E₋ have a difference between them, E₋/E₊ = re^jα, then
the ellipse traced out by the vector E has its axes rotated by an angle φ/2.

When E₋/E₊ = r = ±1, then the wave is linearly polarized. Thus, we have come to
the same definition of polarization modes. The above discussion for the electric
field E can also be repeated for the magnetic field H.

It can be shown that the displacement vector yields a vector equation similar to
the one above as well as a refractive index with an ellipsoidal distribution or an ellipsoid
of revolution around the z axis (the axis of propagation). The ellipsoidal distribution
is the result of the displacement vector D and the electric field E, which
are related by

where ε is a tensor (a 3 × 3 matrix with elements ε_xx, ε_xy, ε_xz, ε_yx, ε_yy, ε_yz, ε_zx, ε_zy,
ε_zz). Because of the conservation of energy, the tensor is symmetric, that is ε_xy = ε_yx,
ε_xz = ε_zx, and ε_yz = ε_zy, the mathematical derivation of the tensor expression in the
three displacement components is greatly simplified:

Based on this, the energy, W, stored in the electric field is then expressed by

In this relationship, the parenthetical term is of the form (x²/a + y²/b + z²/c), which
defines an ellipsoid, and, hence, the name of the ellipsoidal refractive index of nonlinear
optical crystals. Now, remember that a cross section through the center of an
ellipsoid can yield an ellipse or a circle. In addition, a sphere for which a = b = c is
a special case of an ellipsoid (this is, also the definition of isotropy).

In simple terms, this can be summarized as follows. The electromagnetic wave
nature of monochromatic light implies that the electric and the magnetic fields are
in quadrature and in time phase. When created light propagates in free space, the
two fields change sinusoidally and are perpendicular to each other. When light enters
matter, then, depending on the displacement vector distribution in matter (and
hence the dielectric and the refractive index), light (its electric and/or magnetic
field) interacts with it in different ways. If the two planes (of the electric or magnetic
field) are fixed in a Cartesian coordinate system, then light is linearly polarized.
If on the other hand, the planes keep changing in a circular (or helicoidal)
motion and the fields remain at the same intensity, then light is circularly polarized.
If the intensity of the field changes monotonically, then light is elliptically
polarized. Now, if we consider that light is separated into two components, one
linearly polarized, I_P, and one unpolarized, I_u, then, the degree of polarization, P,
is defined as

Light may be polarized when it is reflected, refracted, or scattered. In polarization
by reflection, the degree of polarization depends on the angle of incidence and on
the refractive index of the material, given by Brewster’s law:

where n is the refractive index and I_P the polarizing angle.