# polellip

Parameters of ellipse traced out by tip of a polarized field vector

## Syntax

``tau = polellip(fv)``
``````[tau,epsilon] = polellip(fv)``````
``````[tau,epsilon,ar] = polellip(fv)``````
``````[tau,epsilon,ar,rs] = polellip(fv)``````
``polellip(fv)``

## Description

example

````tau = polellip(fv)` returns the tilt angle, in degrees, of the polarization ellipse of a field or set of fields specified in `fv`. `fv` contains the linear polarization components of a field in either one of two forms: (1) each column represents a field in the form of `[Eh;Ev]`, where `Eh` and `Ev` are the field’s horizontal and vertical linear polarization components or (2) each column contains the polarization ratio, `Ev/Eh`. The expression of a field in terms of a two-row vector of linear polarization components is called the Jones vector formalism.```

example

``````[tau,epsilon] = polellip(fv)``` returns, in addition, a row vector, `epsilon`, containing the ellipticity angle (in degrees) of the polarization ellipses. The ellipticity angle is the angle determined by the ratio of the length of the semi-minor axis to semi-major axis and lies in the range `[-45°,45°]`. This syntax can use any of the input arguments in the previous syntax.```

example

``````[tau,epsilon,ar] = polellip(fv)``` returns, in addition, a row vector, `ar`, containing the axial ratios of the polarization ellipses. The axial ratio is defined as the ratio of the lengths of the semi-major axis of the ellipse to the semi-minor axis. This syntax can use any of the input arguments in the previous syntaxes.```

example

``````[tau,epsilon,ar,rs] = polellip(fv)``` returns, in addition, a cell array of character vectors, `rs`, containing the rotation senses of the polarization ellipses. Each entry in the array is one of `'Linear'`, `'Left Circular'`, ```'Right Circular'```, `'Left Elliptical'` or ```'Right Elliptical'```. This syntax can use any of the input arguments in the previous syntaxes.```

example

````polellip(fv)` plots the polarization ellipse of the field specified in `fv`. This syntax requires that `fv` have only one column. Unlike the returned arguments, the size of the drawn ellipse depends upon the magnitude of `fv`.```

## Examples

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Create an input field that is linearly polarized by setting both the horizontal and vertical components to have the same phase. Then, compute the tilt angle.

```fv = [2;1]; tau = polellip(fv)```
```tau = 26.5651 ```

For linear polarization, `tau` is computed using `tau = atan(fv(2)/fv(1))*180/pi`.

Start with an elliptically polarized input field (the horizontal and vertical components differ in magnitude and in phase). Choose the phase difference to be 90°.

```fv = [3*exp(-i*pi/2);1]; [tau,epsilon] = polellip(fv)```
```tau = 1.3156e-15 ```
```epsilon = 18.4349 ```

The tilt vanishes because of the 90° phase difference between the horizontal and vertical components of the field.

Start with an elliptically polarized input field (the horizontal and vertical components differ in magnitude and in phase). Choose the phase difference to be 60°.

```fv = [2*exp(-i*pi/3);1]; [tau,epsilon,ar] = polellip(fv)```
```tau = 16.8450 ```
```epsilon = 21.9269 ```
```ar = -2.4842 ```

The nonzero tilt occurs because of the 60° phase difference. The negative value of the axial ratio indicates left elliptical polarization.

Start with an elliptically polarized input field (the horizontal and vertical components differ in magnitude and in phase). Choose the phase difference to be 60°.

```fv = [2*exp(-i*pi/3);1]; [tau,epsilon,ar,rs] = polellip(fv)```
```tau = 16.8450 ```
```epsilon = 21.9269 ```
```ar = -2.4842 ```
```rs = 1x1 cell array {'Left Elliptical'} ```

A nonzero tilt occurs because of the 60° phase difference. The rotation sense is `'Left Elliptical'` indicating that the tip of the field vector is moving clockwise when looking towards the source of the field.

Draw the figure of an elliptically polarized field. Begin with an elliptically polarized input field (the horizontal and vertical components differ in magnitude and in phase) and choose the phase difference to be 60 degrees.

```fv = [2*exp(-i*pi/3);1]; polellip(fv)``` The rotation sense is `'Left Elliptical'` as shown by the direction of the arrow on the ellipse. The filled circle at the origin indicates that the observer is looking towards the source of the field.

## Input Arguments

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Field vector in linear component representation specified as a 1-by-N complex-valued row vector or 2-by-N complex-valued matrix. Each column contains an instance of a field specification. If `fv` is a matrix, each column in `fv` represents a field in the form of `[Eh;Ev]`, where `Eh` and `Ev` are the field’s linear horizontal and vertical polarization components. If `fv` is a row vector, then the row contains the ratio of the vertical to horizontal components of the field `Ev/Eh`. For a row vector, the value `Inf` is allowed to designate the case when the ratio is computed for `Eh = 0`. `Eh` and `Ev` cannot both be set to zero.

Example: `[1;-i]`

Example: ```2 + pi/3*i```

Data Types: `double`
Complex Number Support: Yes

## Output Arguments

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Tilt angle of polarization ellipse returned as a 1-by-N real-valued row vector. Each entry in `tau` contains the tilt angle of the polarization ellipse associated with each column of the field `fv`. The tilt angle is the angle between the semi-major axis of the ellipse and the horizontal axis (i.e. xaxis) and lies in the range `[-90,90]°`.

Ellipticity angle of the polarization ellipse returned as 1-by-N real-valued row vector. Each entry in `epsilon` contains the ellipticity angle of the polarization ellipse associated with each column of the field `fv`. The ellipticity angle describes the shape of the ellipse and lies in the range `[-45°,45°]`.

Axial ratio of the polarization ellipse returned as a 1-by-N real-valued row vector. Each entry in `ar` contains the axial ratio of the polarization ellipse associated with each column of the field `fv`. The axial ratio is the signed ratio of the major-axis length to the minor-axis length of the polarization ellipse. Its absolute value is always greater than or equal to one. The sign of `ar` carries the rotational sense of the vector – a negative sign denotes left-handed rotation and a positive sign denotes right-handed rotation.

Rotation sense of the polarization ellipse returned as a 1-by-N cell array of character vectors. Each entry in `rs` contains the rotation sense of the polarization ellipse associated with each column of the field `fv`. The rotation sense can be one of `'Linear'`, `'Left Circular'`, ```'Right Circular'```, `'Left Elliptical'` or ```'Right Elliptical'```.

 Mott, H., Antennas for Radar and Communications, John Wiley & Sons, 1992.

 Jackson, J.D. , Classical Electrodynamics, 3rd Edition, John Wiley & Sons, 1998, pp. 299–302

 Born, M. and E. Wolf, Principles of Optics, 7th Edition, Cambridge: Cambridge University Press, 1999, pp 25–32.