## Using Axial Ratio AR to predict R or L hand polarization.

I take issue with the "Advanced Engineering Electromagnetics" 2nd edition by Balanis again. In Page 156, it claimed for AR=-ve, it is Right Hand rotation, AR=+ve is Left Hand rotation.

For plane wave propagates in z direction and at z=0:

A)Let Ey lag Ex by $\frac{\pi}{2}$
$$\Rightarrow\;\vec E(0,t)=Re[\hat x (E_R+E_L)e^{j(\omega t)}+\hat y (E_R-E_L) e^{j(\omega t -\frac{\pi}{2})}]\;=\;\hat x ( E_R+E_L) \cos \omega t +\hat y (E_R-E_L) \sin \omega t$$
Where $E_{x0}=E_R+E_L \;\hbox { and }\;E_{y0}=E_R-E_L$.
$$AR=\frac{E_{max}}{E_{min}}\;=\;\frac{+(E_R+E_L)}{+(E_R-E_L)}$$
AR is positive

2)But Ey lag Ex by $\frac{\pi}{2}$ can be represented by:
$$\vec E(0,t)=Re[\hat x (E_R+E_L)e^{j(\omega t+\frac{\pi}{2})}+\hat y (E_R-E_L) e^{j\omega t}]\;=\;-\hat x ( E_R+E_L) \sin \omega t +\hat y (E_R-E_L) \cos\omega t$$
$$AR=\frac{E_{max}}{E_{min}}\;=\;\frac{-(E_R+E_L)}{+(E_R-E_L)}$$
AR is negative.

AR is different even if you use different representation of Ey lagging Ex!!!

Also, even if you stay with one convention, Right and Left change between propagation in +z or -z.

This is too important for the book to have a blanket statement, AR cannot predict the direction of rotation of the polarized wave. Can anyone verify this?

Thanks

Alan

What's the matter with this topics? I have 8 EM books, only Balanis get more into this polarization. The book is inconsistent. This is not that hard a topic but I am stuck for like two weeks because every time I turn around, I cannot verify the book. Then information on this is hard to get on the web. I finally get the rotation right, but this AR thing is something again!!!
I am not even talking about difference in conventions, I know Kraus uses different conventions, you either follow Balanis or Kraus. Balanis is more detail, so I follow Balanis. Then Balanis is not consistent in it's own either!!!
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 Anyone? You can see, AR is direction of propagation independent, but rotation is absolutely direction dependent. I also double check EM book by Kraus, it defined AR is always positive and nothing about using AR to predict the direction of rotation, which, should be the correct way. Only problem with Kraus is it uses Poincare circle only and it is not intuitive. Balanis talk about both.