# Does all slowed light becomes circularly polarized?

When un-polarized light goes through a quarter wave plate the one of the wavelength is retarded by 90° phase, resulting in circularly polarized light. Does it means that all circularly polarized light are traversing less than speed of light in vacuum?

Imagine you want to describe a linearly polarized plane wave propagating in z Direction.

$$u(z,t)=\vec p e^{i(kz-\omega t)}$$,

where $\vec p$ is a polarization vector (Jones vector). Using this formalism we can describe a quater waveplate using the following matix (i stands for a retardation of 90°).

$$W=\left(\begin{array}{aa} 1 & 0\\ 0 & i \end{array}\right)$$

Let's have a look at a linearly polarized state.

$$\vec p=\frac{1}{\sqrt 2}\left(\begin{array}{c}a\\b\end{array}\right)$$

The quater wave plate now acts on this state.

$$\vec p'=M\vec p=\vec p=\frac{1}{\sqrt 2}\left(\begin{array}{c}a\\ib\end{array}\right)$$ This is a circularly polarized light. It can be rewritten in the following way. $$\vec p'=\frac{1}{\sqrt 2}\left(\begin{array}{c}a\\0\end{array}\right)+\frac{1}{\sqrt 2}\left(\begin{array}{c}0\\b\end{array}\right)e^{i\frac{\pi}{2}}$$ $$u'(z,t)=\vec p' e^{i(kz-\omega t)}$$

So it turns out that the circularly polarized light is just a superposition of two linearly polarized states (as you mentioned already) with a phase-shift of 90°. There is no mechanism involved changing the speed of light in the vacuum whatsoever.

Hence the answer is: No, circularly polarized light propagates at the same speed in vacuum as un-polarized light, namely at the speed of light.

• Frankly speaking I was intimidated by the math however thanks for clarifying my doubts on polarization. – user6760 Sep 18 '16 at 4:41

Light of any polarization has always the same speed c in vacuum. The quarter wave 90 degrees "retardation" is only a phase shift. It has no influence on the propagation of light in vacuum. The wave velocities (phase velocity, group velocity, signal velocity) in an optical medium can be reduced depending on the refractive index and it's dispersion.