Exploring Elliptical Polarization of Plane Waves

In summary, the conversation discusses the properties and equations of plane-polarized harmonic plane waves and their superposition. It also touches on the concept of propagation constant and its relation to phase constant and amplitude. The main focus is on showing that the superposition of two perpendicular plane waves with different phases results in an elliptically polarized plane wave. The conversation also mentions the attempt to recast the resulting equation into the form of an ellipse, but the correct form for an oblique ellipse is needed.
  • #1
schrodingerscat11
89
1

Homework Statement


Hi! The entire problem is this:

(a) Two plane-polarized harmonic plane waves having the same propagation constant are polarized, respectively, along two perpendicular directions. Show that if the phases of the two waves are different, their superposition yields generally an elliptically polarized plane wave.
(b) Show that the time-average Poynting vector of an elliptically polarized plane wave is equal to the sum of the time-average, Poynting vectors of the two orthogonal plane-polarized waves into which it can be decomposed.

Homework Equations


Plane waves
Def: a constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector (Wikipedia).
[itex]A(x,t)=A_ocos(kx-\omega t +\phi)[/itex]
[itex]A(\mathbf{r},t)=A_o cos(\mathbf{k} \cdot \mathbf{r}-\omega t +\phi)[/itex]
[itex]A(\mathbf{r},t)=A_o e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t +\phi)}[/itex]

where
[itex]A(x,t)[/itex] is the wave height at position x and t.
[itex]A_o[/itex] is the amplitude
[itex]k[/itex] is the wave number
[itex]\phi[/itex] is the phase constant
[itex]\omega[/itex] is the angular frequency

Propagation constant:

[itex]\frac{A_o}{A_x}=e^{\gamma x} [/itex]
[itex]\gamma=\alpha+i\beta[/itex]
[itex]\beta=k=\frac{2\pi}{\lambda}[/itex]
where
[itex]A_x[/itex] and [itex]A_o[/itex] are the amplitude at position x and the amplitude at source of propagation, respectively.
[itex]\gamma[/itex] is the propagation constant
[itex]\alpha[/itex] is the attenuation constant
[itex]\beta[/itex] is the phase constant

Equation of an ellipse:
[itex]\frac{x^2}{a}+\frac{y^2}{b}=1[/itex]
whose parametric equations are
[itex]x=a ~ cos ~t[/itex]
[itex]y=b ~sin ~t[/itex]

The Attempt at a Solution



So far these are the things that I am not sure:
  • I now know that plane waves have mathematical forms as given above. My question is how will they change if they become harmonic?
  • I assume that plane polarization means that if [itex]\mathbf{A}(\mathbf{r},t)[/itex] is a vector, the disturbance is along a certain direction only. That is,[itex]\mathbf{A}(\mathbf{r},t)=A_o e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t +\phi)}\mathbf{\hat{x}}[/itex] is said to be plane polarized along the x direction. Right?
  • If the propagation constant is the same, I assume the phase constant is also the same which means that k is the same for both plane waves. Also by the definition of propagation constant above, the amplitude of the two plane waves are equal any time. Right?
  • I am utterly confused on which among these quantities are complex and which are real. Hence, I don't know how to manipulate the exponential parts or if I can apply Euler's formula to simplify these.
My attempt for (a):
Let the first plane wave be
[itex]\mathbf{A_1}(\mathbf{r},t)=A_o e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t +\phi)}\mathbf{\hat{x}}=A_o e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t )}e^{\phi}\mathbf{\hat{x}}[/itex]
and the second plane wave be
[itex]\mathbf{A_2}(\mathbf{r},t)=A_o e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t +\psi)}\mathbf{\hat{y}}=A_o e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t )}e^{\psi}\mathbf{\hat{y}}[/itex]

Taking their superposition:
[itex]\mathbf{A}=\mathbf{A_1}+\mathbf{A_2}[/itex]
[itex]\mathbf{A}=A_o e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t )}e^{\phi}\mathbf{\hat{x}}+A_o e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t )}e^{\psi}\mathbf{\hat{y}}[/itex]
[itex]\mathbf{A}=A_o e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t )}(e^{\phi}\mathbf{\hat{x}}+e^{\psi}\mathbf{\hat{y}})[/itex]
[itex]1=\frac{A_o}{\mathbf{A}}e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t )}(e^{\phi}\mathbf{\hat{x}}+e^{\psi}\mathbf{\hat{y}})[/itex]

I want to recast this to the form of equation of an ellipse (see relevant equations above) but I'm stuck.

Thank you very much.
 
  • #3
Thanks. No info yet. :( But I think the equation of ellipse above is wrong. I think I should put the equation of an oblique ellipse. That's all.
 

FAQ: Exploring Elliptical Polarization of Plane Waves

1. What is elliptical polarization?

Elliptical polarization of plane waves refers to the orientation of the electric field vector of a wave as it propagates through space. Unlike linear polarization, where the electric field vector oscillates in a single direction, elliptical polarization involves a combination of two perpendicular directions of oscillation. This results in an elliptical shape of the wave's electric field vector.

2. How is elliptical polarization different from linear polarization?

As mentioned, elliptical polarization involves a combination of two perpendicular directions of oscillation, while linear polarization involves only one direction. This means that the electric field vector of an elliptically polarized wave rotates as it propagates, while the electric field vector of a linearly polarized wave remains in a fixed orientation.

3. What causes elliptical polarization?

Elliptical polarization can occur when a linearly polarized wave passes through a medium with different refractive indices in different directions. This results in the oscillations of the wave becoming out of phase with each other, causing the electric field vector to rotate and become elliptically polarized.

4. How is the degree of elliptical polarization measured?

The degree of elliptical polarization is measured using the parameter called ellipticity, which is the ratio of the major and minor axes of the ellipse formed by the electric field vector. An ellipticity of 1 indicates perfect circular polarization, while an ellipticity of 0 indicates linear polarization.

5. What are the applications of exploring elliptical polarization of plane waves?

Understanding and manipulating elliptical polarization of plane waves has various applications in fields such as optics, telecommunications, and radar technology. It can also provide insights into the properties and behavior of different materials and media, as well as improve the accuracy and efficiency of various scientific instruments and devices.

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