## Physics Friday 135

Part 15: Polarization of Electromagnetic Waves

Last week, we began discussing electromagnetic waves in a vacuum, obtaining the fact that light and other electromagnetic waves are transverse waves, with the electric and magnetic fields perpendicular to the direction of propagation. We also established that the fields are perpendicular to each other, and proportional: for waves in vacuum, we have .
So, since the magnetic field is determined by the electric field and the propagation direction, we will focus on the electric field E for the following description, with the magnetic field behaving similarly due to the above.
Given our unit vector in the direction , we may choose a pair of orthogonal unit vectors  and  in the plane with normal , so that the three form an orthonormal basis for space. However, to simplify the discussion, let the wave propagate in the x direction, so that  becomes , while  and  become  and . Now, we can express our electric field in terms of x and y components. Recall that we have electric field , which for our wave propagating in the z direction is
,
or, breaking into components,
,
where Ex and Ey are complex numbers, and we take the real component of the right-hand side above to get the physical field. Specifically, that Ex and Ey are complex means each carries a phase, not necessarily the same.
Suppose, though, that Ex and Ey do have the same phase, so that  is real. Then, taking the real part, we have
,
where φ is the phase. We see then that the electric field vector maintains a constant angle θ with the x axis, , and oscillates sinusoidally in this line. Light in this mode is a linearly polarized wave; and we have retrieved another classic optical phenomenon from Maxwell’s laws.
Now, suppose our phases are different. In particular, let us start with the simple case that the two components have the same magnitude, but differ in phase by π/2 (90°); then , and we have
, or letting Ex set our zero phase,
.
Note that at a particular fixed point in space, then, the electric field vector is of constant magnitude, but rotates in direction with angular frequency ω, sweeping out a circle. Thus, we dub such a wave circularly polarized. Specifically, as we look into the wave, we note that for the upper (+) sign, the vector rotates counter-clockwise, and in the lower (-) sign, the vector rotates clockwise. We say that the former is left circularly polarized, or has positive helicity, while the latter is right circularly polarized, or has negative helicity.
In the case where the phases of Ex and Ey differ by a value other than zero or π/2, or they differ in both phase and amplitude, then the tip of the electric field vector traces an ellipse in the xy-plane, and we call the light elliptically polarized.
Now, note that if we add two circularly polarized waves of equal magnitude and opposite helicity, the result is a linearly polarized wave, with the angle of polarization depending on the phase difference between the circularly polarized waves. More generally, defining the complex-valued vectors ,
we see that left circularly polarized light is given by
,
and right circularly polarized light is given by
,
where E+ and E are complex amplitudes.
Further, any complex vector  may be expressed as a complex linear combination of , and so the general wave may be expressed as
,
and the circular polarizations form a basis: any electromagnetic wave may be decomposed into left and right circularly polarized components. [This is key to the phenomenon of optical rotation.]
For the above basis, let us set , so that r is the ratio of the magnitudes of the components, and φ the phase difference. Then, with a little math, one can show that the ellipse traced by the electric field vector has semi-minor axis to semi-major axis ratio of
, and thus eccentricity of , and that the major and minor axes of this ellipse are rotated from the x and y axes by an angle φ/2.
If r=1, we get, as previously noted, a linearly polarized wave, at an angle θ=φ/2.