# Linearly and Circularly Polarized Electromagnetic Waves

## Table of contents

## Video - Electromagnetic Wave Equation Simply Explained

Subscribe on YouTube Consider a *plane*, *periodic* electromagnetic wave in vacuum. It has an **electric field** \(\boldsymbol{E}\) and a **magnetic field** \(\boldsymbol{B}\).

For **polarization** only the E-field \(\boldsymbol{E} = (E_{\text x}, E_{\text y}, E_{\text z}) \) is relevant. The individual E-field components of a plane wave are:

**E-field vector components**

Here \(\boldsymbol{E}_0 = (E_{0 \text x}, E_{0 \text y}, E_{0 \text z}) \) is the **amplitude** of the E-field, \(\omega\) the **angular frequency**, \(k\) the **wave number** and \(\alpha, \beta, \gamma\) are **phases** to get the possible phase shift between the vector components.

Since the electromagnetic wave is *plane*, Eq. 1

depends only on one position coordinate (here it is the \(z\) coordinate). And, since the wave is *periodic*, it is described by a sine or cosine function (here it is cosine). Furthermore, the wave propagates in the \(z\) direction.

Light, i.e. an electromagnetic wave, can be polarized with a polarization filter, for example. Mathematically, this means that the E-field components 1

are linked to certain conditions, depending on the type of polarization. For this purpose, let's look at two important types of polarization and their conditions, namely *linear* and *circular* polarization.

One condition that both types of polarization must meet is:

Thus the \(E_{\text z}\) component of the E-field is zero:

**E-field vector components without z-component**

## Linear polarized plane wave

A *linearly* polarized electric wave must also satisfy the following condition besides condition #1:

You wonder why it has to be that way? Because this is a definition! If the conditions #1 and #2 are fulfilled, then we speak of linear polarized plane waves.

According to condition #2, the phases \(\omega \, t - k\,z + \alpha\) and \(\omega \, t - k\,z + \beta\) must be equal. For this, \( \alpha = \beta \) must be satisfied. For simplicity, let's set \( \alpha \) and \(\beta\) equal to zero (the important part is that they are BOTH equal to zero):

**Vector components of a linearly polarized EM wave**

Of course, we can write down this E-field vector compactly and get:

**Linearly polarized plane wave**

## Circularly polarized wave

For a *circularly* polarized wave, the phase shift \( \beta - \alpha \) between the two E-field components is not zero, as it is for a linearly polarized wave, but \(\pm \pi/2\) (i.e., 90 degrees).

Let us apply the definition to the E-field 2

:

**E-field components phase-shifted by 90 degrees**

Since cosine and sine are also 90 degrees out of phase, the second E-field component in 5

can be replaced with sine:

**E-field components of a plane wave expressed with cosine and sine**

Another condition that a circularly polarized wave must meet is:

With the third condition E-field 6

becomes:

**Right-circularly polarized plane wave**

The E-field 7

corresponds exactly to the polar representation. Thus, when the **time** \(t\) changes, the E-field vector \(\boldsymbol{E}\) rotates in the \(x\)-\(y\) plane (see illustration 2). This is where the term "circular" comes from. Along the \(z\)-axis the E-field vector thus spirals.

If the circularly polarized plane wave is viewed orthogonal to the \(x\)-\(y\) plane in the propagation direction, the E-field vector rotates *rightwards* for the observer. Therefore the E-field vector 6

is called a **right-circularly** polarized wave (or short: \(\sigma^{+}\) wave).

If cosine and sine are interchanged in 6

, the field vector turns *left* for the observer. This wave is called a **left-circularly** polarized wave (or short: \(\sigma^{-}\) wave):

**Left-circularly polarized wave**

Now you should have a theoretical understanding of the definitions of linearly and circularly polarized plane waves.