Complex Scattered polarization vector? (Conceptual)

In summary, the conversation discusses the equation for the scattered power per unit solid angle and its relation to the polarization vector. The electric field is described using Euler's formula, with the convention of wrapping phase and amplitude information into the complex vector ##\vec{E}_0##.
  • #1
PhDeezNutz
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Homework Statement
This is more so a conceptual question rather than a specific homework problem. I always thought that the polarization vector (i.e. direction of the ##\vec{E}## - field denoted as ##\vec{\varepsilon}##) in an EM wave was a Euclidean three vector but apparently I am wrong because in one equation in my book they mention the complex conjugate ##\vec{\varepsilon}##. I'm using Zangwill's "Modern Electrodynamics" btw.

The specific part I'm confused about is

Call this equation 1

##\frac{d \sigma_{scatt}}{d \Omega} = \frac{\langle \frac{dP}{d \Omega} \rangle}{\frac{1}{2} \varepsilon_0 c E_0^2} = r^2 \frac{\vec{E}_{rad}^2}{\vec{E}_{0}^2} = \left| \vec{f} \left( \vec{k} \right) \right|^2##

"If our interest is a scattered electric field with a particular polarization ##\vec{\varepsilon}##, the differential scattering cross section (above) generalizes to"

Call this equation 2


##\left. \frac{d \sigma_{scatt}}{d \Omega} \right|_{\vec{\varepsilon}} = r^2 \frac{\left|\vec{\varepsilon}^* \cdot \vec{E}_{rad} \right|^2}{\left| \vec{E}_0\right|^2} = \left| \vec{\varepsilon}^* \cdot \vec{f}\left( \vec{k} \right) \right|^2##

Again I always thought ##\vec{\varepsilon}## was a 3D euclidean vector and I'm trying to reconcile what it would mean for it to be complex.
Relevant Equations
The differential scattering cross section is defined as

##\frac{d \sigma_{scatt}}{d \Omega} = \frac{\langle \vec{S}_{rad} \rangle \cdot r^2 \hat{r}}{\left| \langle \vec{S}_{inc} \rangle \right|}##

In the far field

##\vec{E} = \vec{E}_{inc} + \vec{E}_{rad} = E_0 \left[ \hat{e}_0 e^{i \vec{k}_0 \cdot \vec{r}} + \frac{e^{ikr}}{r} \vec{f} \left( \vec{k} \right)\right]e^{-i \omega t}##
I guess I will show my work for substantiating equation 1 and hopefully by doing so someone will be able to point out where I could generalize.

##\langle \vec{S}_{rad} \rangle = \frac{1}{2 \mu} \mathfrak{R} \left( \vec{E}_{rad} \times \vec{B}^*_{rad}\right) = \frac{1}{2 \mu} \mathfrak{R} \left( \vec{E}_{rad} \times \left( \hat{r} \times \vec{E}_{rad}^*\right) \sqrt{\mu \varepsilon} \right) = \sqrt{\frac{\varepsilon}{\mu}} \mathfrak{R} \left(\vec{E}_{rad} \times \left( \hat{r} \times \vec{E}_{rad}^* \right) \right) = \sqrt{\frac{\varepsilon}{mu}} \mathfrak{R} \left( \hat{r} \left(\vec{E}_{rad} \cdot \vec{E}_{rad}^* \right) - \vec{E}_{rad}^* \left( \hat{r} \cdot \vec{E}_{rad} \right) \right) = \sqrt{\frac{\varepsilon}{\mu}} \frac{E_0^2 \left|\vec{f} \left(\vec{k} \right) \right|^2}{r^2} \hat{r}##

##\langle \vec{S}_{inc} \rangle = \frac{1}{2 \mu} \mathfrak{R} \left( \vec{E}_{inc} \times \vec{B}_{inc}^*\right) = \sqrt{\frac{\varepsilon}{\mu}} \mathfrak{R} \left( \hat{r} \left( \vec{E}_{inc} \cdot \vec{E}_{inc}^*\right) \right) = \sqrt{\frac{\varepsilon}{\mu}} E_0^2##

Therefore

##\frac{d \sigma_{scatt}}{d \Omega} = \left| \vec{f} \left( \vec{k} \right) \right|^2 ##
 
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  • #2
I think I get it. The polarization vector contains information about phase as well as the direction of the electric field.

##\vec{E} = \vec{E}_0 e^{i \vec{k} \cdot \vec{r} - i \omega t}##

##e^{i \vec{k} \cdot \vec{r} - i \omega t}## indicates only that the wave is transverse and provides information about frequency. It says nothing about phase and amplitude.

I think the convention is to wrap up information about phase and amplitude into ##\vec{E}_0## and because ##\vec{E}_0## contains information about phase it must be complex (Euler's formula).
 

Related to Complex Scattered polarization vector? (Conceptual)

1. What is the definition of a complex scattered polarization vector?

A complex scattered polarization vector is a mathematical representation of the polarization state of an electromagnetic wave that has been scattered by a medium or object. It includes both the magnitude and direction of the electric and magnetic fields of the scattered wave.

2. How is a complex scattered polarization vector different from a regular polarization vector?

A regular polarization vector only represents the magnitude and direction of the electric field of an electromagnetic wave, while a complex scattered polarization vector includes both the electric and magnetic fields of the scattered wave. This makes it a more comprehensive representation of the polarization state of the scattered wave.

3. What factors can affect the complex scattered polarization vector?

The complex scattered polarization vector can be affected by the properties of the medium or object that the electromagnetic wave is scattered from, such as its composition, shape, and orientation. The incident angle and wavelength of the wave can also impact the complex scattered polarization vector.

4. How is the complex scattered polarization vector measured or calculated?

The complex scattered polarization vector can be measured experimentally using specialized equipment, such as polarimeters or ellipsometers. It can also be calculated using mathematical models and simulations based on the properties of the scattering medium or object.

5. What applications does the concept of complex scattered polarization vector have?

The complex scattered polarization vector is used in various fields, including optics, remote sensing, and radar. It is particularly useful in characterizing and identifying different types of materials and objects based on their scattering properties. It also has applications in medical imaging and detecting atmospheric pollutants.

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