# Jones vector of circularly polarized light

• aftershock
In summary, the jones vector of circularly polarized light is <1,i> because the x and y components of the E-field are represented by complex numbers and have a magnitude and a phase, which leads to a circular polarization.

#### aftershock

Why is the jones vector of circularly polarized light <1,i> ?

Things like <1,0> and <0,1> make perfect sense for linearly polarized light along the x and y axes but what exactly is that i doing there that makes the vector represent a circular polarization?

I never really intuitively understood that.

Thanks for any help guys

The x and y components of the E-field here are being represented by complex numbers. The reason for doing so is because complex numbers have a magnitude and a phase, and as a result they are a natural way of compactly representing the amplitude and phase of sinusoidal, time-varying quantities. In the case of circular polarization, if the x component of the E-field (after normalizing by dividing out the magnitude) is equal to 1, then it is entirely real (its phase angle is 0 in the complex plane). In contrast, if the y component is represented by an i, then although the magnitude of the y component is the same, it is pi/2 out of phase with the x-component. It is therefore entirely imaginary (being rotated by 90 degrees from the x component in the complex plane) Recall that 'i' can be represented in polar form as:

$$i = 1e^{i(\pi/2)}$$

So the amplitude and phase of the y component of the E-field are 1 (after normalization) and pi/2 respectively.

If you think about it, having the x-component and the y-component be 90 degrees out of phase with each other will lead to a total (resultant) E-field vector that rotates in a circle in the plane of polarization.

For example, at t = 0, let's say that Ex is at a maximum in its cycle and Ey is at the point in its cycle where it is 0. Then the resultant E field vector points entirely in the x-direction. But an eighth of an oscillation period later, the Ex vector is now only 1/root(2) of its initial (max) value, and the Ey vector has increased from 0 length to 1/root(2) of the max value. Therefore, the two vectors have the same magnitude, and their resultant points at 45 degrees to the x direction. So, the vector has rotated by this amount. After about a quarter of an oscillation period, the Ex vector has now lessened down to 0 length, and the Ey vector has increased all the way to its max value. The resultant E vector is therefore now entirely in the y direction. It has rotated 90 degrees since it started. If you look at other sample points in the oscillation, you'll find that over a full period, the vector will have traced out a full circle in space.

## 1. What is a Jones vector of circularly polarized light?

A Jones vector is a mathematical representation of the polarization state of light. Specifically, a circularly polarized Jones vector describes the orientation and magnitude of the electric field vector of a circularly polarized light wave at a given point in space.

## 2. How is the Jones vector of circularly polarized light calculated?

The Jones vector of circularly polarized light can be calculated using the following equation: J = [Aexp(iφ), B], where A and B are the complex amplitudes of the electric field components and φ is the phase difference between them.

## 3. What are the properties of the Jones vector of circularly polarized light?

The Jones vector of circularly polarized light has two main properties: magnitude and orientation. The magnitude is represented by the complex amplitudes A and B, while the orientation is represented by the phase difference φ between the electric field components.

## 4. How is the Jones vector of circularly polarized light used in experiments?

The Jones vector of circularly polarized light is commonly used in experiments to analyze and manipulate the polarization state of light. It can also be used to design optical devices that control the polarization of light, such as polarizers and waveplates.

## 5. What are the applications of the Jones vector of circularly polarized light?

The Jones vector of circularly polarized light has various applications in optics, including in telecommunications, medical imaging, and astronomy. It is also used in optical research to study the behavior of light in different media and to understand the properties of materials that interact with polarized light.