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

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.

[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.

[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.

[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.

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

Tilt Angle for Linearly Polarized Field

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.

Tilt Angle and Ellipticity for Elliptically Polarized 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 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.

Tilt Angle, Ellipticity and Axial Ratio for Elliptically Polarized 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.

Tilt Angle, Ellipticity, Axial Ratio and Rotation Sense for Elliptically Polarized 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,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.

Polarization Ellipse

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

fv — Field vector in linear component representation

1-by-N complex-valued row vector or 2-by-N complex-valued matrix

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

tau — Tilt angle of polarization ellipse

1-by-N real-valued row vector

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]°.

epsilon — Ellipticity angle of the polarization ellipse

1-by-N real-valued row vector

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°].

ar — Axial ratio of the polarization ellipse

1-by-N real-valued row vector

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.

rs — Rotation sense of the polarization ellipse

1-by-N cell array of character vectors

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'.

References

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

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

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

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Does not support variable-size inputs.

Version History

Introduced in R2013a