# Please verify my derivation on elliptical polarization of EM wave

• yungman
In summary, the conversation is discussing an equation on elliptical polarization and the last line of the equation in the book does not match the derived equation. After checking and not finding any errors, it is suggested that the book may have a typo as it is unlikely for the last result to be true.
yungman
This is not a home work, it is part of the textbook on elliptical polarization. Attached is a page in Kraus Antenna book, I cannot verify the equation on the last line. Here is my work

$$E_y=E_2(\sin{\omega} t \cos \delta \;+\; \cos \omega {t} \sin \delta)$$ , $$\sin\omega {t} =\frac {E_x}{E_1}\;,\; \cos \omega {t} =\sqrt{1-(\frac{E_x}{E_1})^2}$$

$$\Rightarrow\; E_y=\frac {E_2 E_x\cos \delta}{E_1}\;+\;E_2\sqrt{1-(\frac {E_x}{E_1})^2} \;\sin\delta$$

$$\Rightarrow\; \sin \delta \;=\;\frac {E_y}{E_2\sqrt{1-(\frac{E_x}{E_1})^2}}\;-\; \frac{E_x\cos\delta}{E_1 \sqrt{1-(\frac{E_x}{E_1})^2}}$$

$$\Rightarrow\; \sin^2\delta\;=\;\frac{E^2_y}{E_2^2\;(1\;-\;(\frac{E_x}{E_1})^2)}\;-\;\frac{2E_y\;E_x\;\cos\delta}{E_1\;E_2\;(1\;-\;(\frac{E_x}{E_1})^2)}\;+\;\frac {E_x^2\;\cos^2\;\delta}{E_1^2\;(1\;-\;(\frac{E_x}{E_1})^2)}$$

Compare to the last line in the book, I just cannot get the last equation of the book. I checked it a few times and I just cannot see anything wrong with my derivation. Please take a look and see what I did wrong.

Thanks

Alan

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There are times when one has to declare the book wrong. I am saying this because
1. I cannot find anything wrong with your derivation and neither can you.
2. In order to reconcile your expression with the book's it must be true that $$1-\left(\frac{E_x}{E_1}\right)^2=1 ~\rightarrow~\frac{E_x}{E_1}=0$$
That last result is highly unlikely, therefore I think it's a typo which is more likely.

## 1. What is elliptical polarization of an EM wave?

Elliptical polarization refers to the orientation of the electric field vector of an electromagnetic wave. In this type of polarization, the electric field vector traces out an elliptical path as the wave propagates, instead of a linear or circular path.

## 2. How is elliptical polarization different from linear or circular polarization?

Linear polarization occurs when the electric field vector of an EM wave oscillates in a single plane, while circular polarization occurs when the vector rotates at a constant rate. Elliptical polarization combines aspects of both linear and circular polarization, with the vector oscillating in an elliptical path.

## 3. What factors can affect the elliptical polarization of an EM wave?

The orientation and strength of the electric and magnetic fields, as well as the angle of incidence and the medium through which the wave is propagating, can all affect the polarization of an EM wave.

## 4. How is the elliptical polarization of an EM wave mathematically derived?

The derivation of elliptical polarization involves analyzing the electric field components of the wave and solving for the orientation and eccentricity of the resulting elliptical path. This can be done using vector algebra and trigonometric functions.

## 5. What are some real-world applications of elliptical polarization?

Elliptical polarization is commonly used in communication systems to minimize interference and improve signal quality. It is also utilized in various imaging techniques, such as polarized microscopy, and in the design of optical elements like waveplates and polarizers.

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