- #1
EmilyRuck
- 136
- 6
Hello!
Talking about propagation of an electro-magnetic field in a non-isotropic medium, I've got some troubles with the expression in object, used to show the Faraday rotation of the polarization of a field.
An electro-magnetic field enters a particular medium, propagating along the [itex]\hat{\mathbf{u}}_z[/itex] direction. In [itex]z = 0[/itex], its electric field is [itex]\mathbf{E} = E_0 \hat{\mathbf{u}}_x[/itex]. It could also be written as a superposition of two circularly polarized waves:
[itex]\mathbf{E} = \displaystyle \frac{E_0}{2} (\hat{\mathbf{u}}_x + j \hat{\mathbf{u}}_y) + \displaystyle \frac{E_0}{2} (\hat{\mathbf{u}}_x - j \hat{\mathbf{u}}_y)[/itex]
The two components have different propagation constants, [itex]β_-[/itex] and [itex]β_+[/itex], in the medium. In a generic [itex]z[/itex] position we could write
[itex]\mathbf{E} = \displaystyle \frac{E_0}{2} (\hat{\mathbf{u}}_x + j \hat{\mathbf{u}}_y)e^{-j β_- z} + \displaystyle \frac{E_0}{2} (\hat{\mathbf{u}}_x - j \hat{\mathbf{u}}_y)e^{-j β_+ z}[/itex]
Rearranging the last expression (this is done in several books, like Pozar), we obtain:
[itex]\mathbf{E} = E_0 e^{-j (β_+ + β_-) \frac{z}{2}} \left\{ \hat{\mathbf{u}}_x \cos \left[ \left( β_+ + β_- \right) \displaystyle \frac{z}{2} \right] - \hat{\mathbf{u}}_y \sin \left[ \left( β_+ - β_- \right) \displaystyle \frac{z}{2} \right] \right\}[/itex]
This is done to show that the polarization is still linear like in the original field [itex]\mathbf{E} = E_0 \hat{\mathbf{u}}_x[/itex], but its "orientation" has changed with the position [itex]z[/itex].
But could this still be called a wave? Its dipendence from [itex]z[/itex] is no more only in the exponential [itex]e^{-j β z}[/itex], but is also contained in the cosine and sine terms.
It apparently does no more satisfy the Helmholtz wave equation, because deriving the [itex]x[/itex] component with respect to [itex]z[/itex] gives a completely different result that that obtained deriving the same component with respect to time (assuming that we are using phasors).
So, how can I interpret this expression? Shouldn't it still satisfy the Helmholtz equation? Shouldn't it be still a wave?
Thank you for having read,
Emily
Talking about propagation of an electro-magnetic field in a non-isotropic medium, I've got some troubles with the expression in object, used to show the Faraday rotation of the polarization of a field.
Homework Statement
An electro-magnetic field enters a particular medium, propagating along the [itex]\hat{\mathbf{u}}_z[/itex] direction. In [itex]z = 0[/itex], its electric field is [itex]\mathbf{E} = E_0 \hat{\mathbf{u}}_x[/itex]. It could also be written as a superposition of two circularly polarized waves:
[itex]\mathbf{E} = \displaystyle \frac{E_0}{2} (\hat{\mathbf{u}}_x + j \hat{\mathbf{u}}_y) + \displaystyle \frac{E_0}{2} (\hat{\mathbf{u}}_x - j \hat{\mathbf{u}}_y)[/itex]
The two components have different propagation constants, [itex]β_-[/itex] and [itex]β_+[/itex], in the medium. In a generic [itex]z[/itex] position we could write
[itex]\mathbf{E} = \displaystyle \frac{E_0}{2} (\hat{\mathbf{u}}_x + j \hat{\mathbf{u}}_y)e^{-j β_- z} + \displaystyle \frac{E_0}{2} (\hat{\mathbf{u}}_x - j \hat{\mathbf{u}}_y)e^{-j β_+ z}[/itex]
Homework Equations
Rearranging the last expression (this is done in several books, like Pozar), we obtain:
[itex]\mathbf{E} = E_0 e^{-j (β_+ + β_-) \frac{z}{2}} \left\{ \hat{\mathbf{u}}_x \cos \left[ \left( β_+ + β_- \right) \displaystyle \frac{z}{2} \right] - \hat{\mathbf{u}}_y \sin \left[ \left( β_+ - β_- \right) \displaystyle \frac{z}{2} \right] \right\}[/itex]
This is done to show that the polarization is still linear like in the original field [itex]\mathbf{E} = E_0 \hat{\mathbf{u}}_x[/itex], but its "orientation" has changed with the position [itex]z[/itex].
But could this still be called a wave? Its dipendence from [itex]z[/itex] is no more only in the exponential [itex]e^{-j β z}[/itex], but is also contained in the cosine and sine terms.
The Attempt at a Solution
It apparently does no more satisfy the Helmholtz wave equation, because deriving the [itex]x[/itex] component with respect to [itex]z[/itex] gives a completely different result that that obtained deriving the same component with respect to time (assuming that we are using phasors).
So, how can I interpret this expression? Shouldn't it still satisfy the Helmholtz equation? Shouldn't it be still a wave?
Thank you for having read,
Emily