Ian Cooper

NUMERICAL ANAYSIS OF OPTICAL AND ELECTROMAGNETIC PHENOMENA: THE VECTORIAL NATURE OF LIGHT: POLARIZATION

cemPol1.m Animated Lissajous figures for the polarization of light as a function of relative phase for orthogonal vibrations of the electric field. The animation of the time evolution of the electric field vector can be saved as an animated gif (flagA = 1) or as an avi file (flagA = 2) or not saved (flagA = 0). The script calls the function arrow.m (downloaded from the Matlab File Exchange). The arrow.m function is a very useful function for adding arrows to your plots.

cemPol2.m (avi) cemPol3.m (gif) [3D] animation of a travelling elliptical polarized wave propagating in the Z direction.

Any state of polarization in a ray of light can be described in terms of elementary components of plane-polarized light. A polarized wave propagating along the Z axis of an orthogonal system of axes maybe analysed in terms of components along the X and Y axes. The two components can be expressed as complex exponential functions.

where is the phase difference between the X and Y components of the electric field . The two components and combine vectorially to give the resultant electric field .

The actual values for the electric field and its components are given by the real of the complex functions.

Phase difference

The X and Y components of the wave vibrate in phase. The resultant electric field is linearly polarized and the oscillation is in the direction and the amplitude of the vibration is constant.

Phase difference

The resultant electric field is linearly polarized and the oscillation is in the direction

Phase difference

The resultant electric field is elliptically polarized. The end-point of the electric field vector traces out an ellipse. The resultant electric field vector rotates clockwise at an angular velocity , as seen by an observer looking towards the source of the radiation, that is, looking back at the source. Such a wave is said to be right-circularly polarized (right-circular light). As the electric field vector rotates, its magnitude also changes. The maximum magnitude is and it occurs at the angle to the X axis where . The angle gives the angle of the major axis of the ellipse.

The vector makes one complete rotation as the wave advances one wavelength.

We can view the elliptical polarized wave propagating along the Z direction in [3D] using the scripts cemPol2.m (avi) or cemPol3.m (gif). The gif animation shows the X and Y components and the resultant electric field for points along the Z axis as functions of time when the phase of the Y components lead the X components by . The electric field has no spatial scale. The electric field is defined at each point in space. A positive charge placed at a point in the oscillating electric field with experience an instantaneous force of varying magnitude and the direction of the force will be in the same direction as the resultant electric field at that instance.

Right-circularly polarized light.

When you view the animation, you see the wave approaching you and the rotation of the resultant electric field vector is clockwise: right-circularly polarized light. The convention we are using states that the wave is right-elliptically polarized. Warning: there are other conventions, so you need to know the rule that you are using in stating whether the wave is right or left polarized.

Phase difference

The resultant wave is elliptically polarized with the rotation of the electric field vector anticlockwise at an angular frequency (left-circularly polarized light). The amplitude of the wave and the direction of the electric field vector vary with time as the electric field vector rotates.

Phase difference

The resultant wave is elliptically polarized with the rotation of the electric field vector clockwise at an angular frequency (right-circularly polarized light). The amplitude of the wave and the direction of the electric field vector vary with time as the electric field vector rotates.

Phase difference

The resultant wave is circularly polarized with the rotation of the electric field vector clockwise at an angular frequency (right-circularly polarized light). The amplitude of the wave is constant.

Phase difference

The resultant wave is circularly polarized with the rotation of the electric field vector anticlockwise at an angular frequency (left-circularly polarized light). The amplitude of the wave is constant.

Lissajous figures as a function of relative phase for orthogonal vibrations

The following animations give a series of Lissajous figures of relative phase from to for orthogonal vibrations of unequal magnitude .

left elliptical polarization (anticlockwise or counter-clockwise rotation)

right elliptical polarization (clockwise rotation)

As far as the mathematical description is concerned, both linear and circular light may be considered as special cases of elliptically polarized light (or more simply, elliptical light).

Matlab script cemPol1.m

It is very easy to model the polarization of light using the Matlab complex number functions.

Input parameters and calculations

% Electric field amplitudes [10] [10]

E0x = 10; E0y = 10;

% Phase of Ey in radians [0.5*pi]

% Try phi [rad]: 0.25*pi 0.5*pi 0.75*pi pi 0 -0.25*pi -0.5*pi -0.75*pi -pi

phi = 0.5*pi;

% Grid points

N = 24;

% CALCULATIONS z = 0 =====================================================

wt = linspace(0,2*pi,N);

% Electric fields

Ex = E0x .* exp(1j*wt);

Ey = E0y .* exp(1j*(wt + phi));

E = real(Ex) + 1j.*real(Ey);

John A Sims

email: john.sims@ufabc.edu.br

Universidade Federal do ABC, Brazil

Ian Cooper

email: matlabvisualphysics@gmail.com