Definition: The polarization of a plane wave is defined as the direction of the electric field vector. The electric field may always remain in the same direction (or in the opposite direction during the negative swing of the time variation) or it may change with time and position in a helical fashion. The direction in either case will be perpendicular to propagation. Note that the magnetic field direction will change in a similar way. A general polarization can be represented by separating the electric field into two orthogonal polarization components and allowing each a magnitude and phase. Consider a plane wave propagating with a propagation vector k = kz.
General time-dependent electric field E x (z, t) ax + E y (z, t) ay = |Ex| cos(- w t + kz + Q x ) ax + |Ey| cos(- w t + kz + Q y ) ay and the corresponding phasor field E x(z) + E y(z) = |Ex| exp(+jkz + j Q x ) ax + |Ey| exp(+jkz + j Q y ) ay The coupled magnetic field phasoris H x(z) + H y(z) = (|Ey|/ h ) exp(+jkz + j Q y ) ax + (|Ex|/ h ) exp(+jkz + j Q x ) ay
Both orthogonal polarization components have the same k for isotropic media.
The polarization type is completely specified by the magnitudes |Ex| and |Ey| and by the phases Q x and Q y .
The situation becomes more complex in anisotropic media. The propagation vector k will be different for different directions within the material. This behavior is particularly common in crystals and is the basis for retardation plates, some beamsplitters, and other devices.
Types: The most general type of polarization is elliptical in which the electric field vector traces an ellipse in the plane of the wavefront. Special cases exist when the locus of points traced by the electric field are circules or lines. Lasers are often linearly polarized to facilitate beam control. Consider a plane wave propagating with a propagation vector k = kz.
The locus of points for the electric field is an ellipse. There is no constraint on the magnitudes and phases except for the special cases mentioned below.
The field moves in a clockwise direction for ( Q y - Q x ) > 0 and a counter-clockwise direction for ( Q y - Q x ) < 0.
The locus of points for the electric field is a circle.
Magnitude Constraint: |Ex| = |Ey|
Phase Constraint : ( Q y - Q x ) = + p /2, + p /2 +2 p , … The field moves in a clockwise direction for + p /2 and a counter-clockwise direction for – p /2.
The locus of points for the electric field is a line.
Magnitude Constraint: None The angle of the line with respect to the x-axis is f = tan-1(|Ey|/|Ex|) for ( Q y - Q x ) = 0, 2 p , ... The angle of the line with respect to the x-axis is f = tan-1(-|Ey|/|Ex|) for ( Q y - Q x ) = p , 3 p , ...
Phase Constraint : ( Q y - Q x ) = 0, p , 2 p , 3 p , ...
The behavior of a plane wave incident on an dielectric-dielectric interface is dependent upon the relative polarization of the wave and the orientation of the interface. The plane of incidence is defined as the plane created by the propagation vector k of the incident plane wave and normal vector for the interface. The k‘s for the reflected and transmitted plane waves will lie in the this plane as well. The waves may be divided into p-polarization components and s-polarization components with respect to this plane. The “p” stands for parallel in German and the “s” stands for perpendicular in German (“senkrecht”). The distinction between these two polarization components are apparent when the electromagnetic boundary conditions are applied for an incident plane wave and its resulting refracted and reflected plane waves.
The p -polarization Component The electric field , i.e. the polarization, of the wave is in the plane of incidence. It can be considered a transverse magnetic component in that the magnetic field of this component is only tangential to the interface and has no normal part.
Boundary Conditions (assuming no charge or current at the interface) HP,tangetial,meda#1 = HP,tangetial,meda#2. EP,tangetial,meda#1 = EP,tangetial,meda#2 and n1 2 Ep,normal.meda#1 = n2 2 EP,normal.meda#2. where n1 and n2 are the indices of the two media.
The s -polarization Component The electric field , i.e. the polarization, of the wave is out the plane of incidence. It can be considered a transverse electric component in that the electric field of this component is only tangential to the interface and has no normal part.
Boundary Conditions (assuming no charge or current at the interface) ES,tangetial,meda#1 = ES,tangetial,meda#2. HS,tangetial,meda#1 = HS,tangetial,meda#2 and HS,normal.meda#1 = HS,normal.meda#2.
The given boundary conditions apply for the total electric and magnetic field in each media. In particular, the total fields in the first media contain the sum of the incident and reflected planes waves. |