NcStokes Class Reference

Treatment of Stokes parameters for EM polarisation studies. More...

#include "NcStokes.h"

Detailed Description

Treatment of Stokes parameters for EM polarisation studies.

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Copyright(c) 2021 NCFS/IIHE, All Rights Reserved.                           *
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         The Inter-university Institute for High Energies (IIHE).           *                 
                   Website : http://www.iihe.ac.be                          *
           Contact : Nick van Eijndhoven (nickve.nl@gmail.com)              *
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// Class NcStokes
// Treatment of Stokes parameters for EM polarisation studies.
//
// The Stokes parameters provide a convenient way to describe the polarisation
// of electromagnetic radiation based on observed intensities, without the need
// for detailed phase information.
// The Stokes parameters are represented by 4 observables, denoted as components
// of a so-called Stokes vector (S0,S1,S2,S3) also known as (I,Q,U,V).
//
// Since EM radiation is transversely polarized, the polarisation can be described
// by two components of the electric field vector E in a plane perpendicular to
// the direction of propagation.
// These two electric field components may oscillate independently, provided that
// the magnetic field vector B oscillates correspondingly such that it stays
// perpendicular to the electric field in the plane of oscillation.
// This allows a rotating E vector, describing an ellipse as the most generic result,
// with as extreme cases a circle (circular polarisation) or a line (linear polarisation).
//
// A generic polarisation description is obtained via a 2-dimensional complex vector,
// called a Jones vector: J=(E1,E2)=(A1*exp(i*phi1),A2*exp(i*phi2))*exp(i*omega*t).
// Here |E1|=A1 and |E2|=A2 are the amplitudes, and phi1 and phi2 are the phases
// with respect to some orthogonal basis.
// The term exp(i*omega*t) represents the oscillation with angular frequency omega.
// The physical electrical field is the real part of the Jones vector.
//
// The physical interpretation of the Stokes parameters is that each parameter represents
// a net polarisation intensity along a certain coordinate axis of an orthogonal basis.
// The Stokes parameter I represents the total beam intensity, so we need 3 different
// orthogonal bases to describe the net polarisation intensities via the parameters Q, U and V.
//
// The convention used here is to define a right-handed Cartesian coordinate system (x,y,z),
// in which the EM wave moves in the +z direction and the oscillation takes
// place in the x-y plane.
// The right hand rule, with the thumb pointing in the direction of propagation,
// defines the rotation direction with positive helicity.
//
// This Cartesian coordinate system defines the first orthogonal basis (x,y).
// The 2nd orthogonal basis (a,b) is obtained by rotating the (x,y) basis over 45 degrees.
// In other words : a=(x+y)/sqrt(2) and b=(-x+y)/sqrt(2)
// The 3rd orthogonal basis (R,L) is a circular basis with R=(x+iy)/sqrt(2) and L=(x-iy)/sqrt(2).
// Here R and L represent Right handed and Left handed rotation, respectively.
//
// Since intensity=|E|^2 we can obtain the intensities along the various base vectors
// by taking the dot product of J with the corresponding base vector.
// So, for instance I(x)=|<J,x>|^2 where <J,x> denotes the dot product of J with the x base vector.
// Note that for a complex vector E we have |E|^2=E*E^*  where E^* indicates the complex conjugate of E.
//   
// The Stokes parameters represent the following intensities :
//
// I=I(x)+I(y) The sum of the two (x,y) linear polarisation intensities.
// Q=I(x)-I(y) The difference of the two (x,y) linear polarisation intensities.
// U=I(a)-I(b) The difference of the two (a,b) linear polarisation intensities.
//             Note that Q=0 at the axes of the (a,b) frame and U=0 at the axes of the (x,y) frame.
// V=I(R)-I(L) The difference of the right and left circular polarisation intensities.
//
// In other words :
//
// I : represents the total beam intensity.
// Q : represents a net horizontal (Q>0) or vertical (Q<0) linear polarisation component.
// U : represents a net diagonal linear polarisation component.
// V : represents a net circular polarisation (right:V>0  left:V<0) component.
//
// whereas Q=U=V=0 represents an unpolarized beam of EM radiation.
//
// For a fully polarized monochromatic beam of EM radiation in our Cartesian (x,y,z) frame,
// the Stokes parameters are obtained from the projections of the Jones vector as follows :
//
// I=|E1|^2+|E2|^2=A1^2+A2^2
// Q=|E1|^2-|E2|^2=A1^2-A2^2
// U=2*Re(E1*E2^*)=2*A1*A2*cos(phi1-phi2)
// V=2*Im(E1*E2^*)=2*A1*A2*sin(phi1-phi2)
//
// In terms of a polarisation ellipse with semi major axis A, semi minor axis B and
// orientation angle theta between a and the x-axis we have :
//
// I=A^2+B^2
// Q=(A^2-B^2)*cos(2*theta)
// U=(A^2-B^2)*sin(2*theta)
// V=2*A*B*helicity
//
// It is seen that the Stokes parameters reflect intensities without any
// time dependent phase, and as such can be treated via simple addition and subtraction.
// In particular, the above shows that I^2=Q^2+U^2+V^2.
//
// Since a beam of EM radiation may contain both polarized and unpolarized radiation,
// only a fraction f of the total beam intensity I will be polarized.
// Consequently, we should have used only the polarized intensity P=f*I in the above expressions,
// such that P^2=Q^2+U^2+V^2.
// However, the Stokes parameter I always represents the total beam intensity, so that I=P/f
// for a beam with a certain amount of polarisation.
// P=sqrt(Q^2+U^2+V^2) represents the total polarisation intensity and the ratio P/I is called
// the polarisation fraction.
// Furthermore, L=sqrt(Q^2+U^2) is called the linear polarisation intensity,
// whereas |V| represents the circular polarisation intensity.
//
// Inversion of the above equations yields for the Jones vector :
//
// A1=sqrt((P+Q)/2) A2=sqrt((P-Q)/2) and (phi1-phi2)=arctan(V/U)
//
// whereas for the polarisation ellipse parameters we obtain :
//
// A=sqrt((P+L)/2) B=sqrt((P-L)/2) theta=0.5*arctan(U/Q)
//
// This class provides memberfunctions to enter data either directly via the Stokes parameters,
// see SetStokesParameters(), via specification of the Jones vector, see SetJonesVector(),
// or by providing the geometrical parameters of the polarisation ellipse, see SetEllipseParameters().
// Once the data have been entered, the various parameter values may be obtained via GetParameter()
// or listed via the Data() memberfunction.
//
//--- Author: Nick van Eijndhoven, IIHE-VUB, Brussel, August 21, 2021  13:39Z
//- Modified: Nick van Eijndhoven, IIHE-VUB, Brussel, August 27, 2021  10:48Z

Definition at line 12 of file NcStokes.h.

Public Member Functions
	NcStokes ()
	NcStokes (const NcStokes &q)
virtual	~NcStokes ()
void	Data (TString u="rad")
Double_t	GetParameter (TString name, TString u="rad")
void	SetEllipseParameters (Double_t a, Double_t b, Double_t theta, TString u, Int_t h, Double_t p=1)
void	SetJonesVector (Double_t A1, Double_t A2, Double_t phi, TString u, Double_t p=1)
void	SetStokesParameters (Double_t I, Double_t Q, Double_t U, Double_t V)

Protected Attributes
Double_t	fI
Double_t	fQ
Double_t	fU
Double_t	fV

Constructor & Destructor Documentation

NcStokes() [1/2]

NcStokes::NcStokes

(

)

// Default constructor.

Definition at line 156 of file NcStokes.cxx.

~NcStokes()

NcStokes::~NcStokes ( )

virtual

// Default destructor.

Definition at line 170 of file NcStokes.cxx.

NcStokes() [2/2]

NcStokes::NcStokes

(

const NcStokes &

q

)

// Copy constructor.

Definition at line 179 of file NcStokes.cxx.

Member Function Documentation

Data()

void NcStokes::Data

(

TString

u = "rad"

)

// Provide all polarisation related information.
// For details about the various parameters, please refer to the
// general documentation of this class.
//
// The input argument "u" allows to select the units for angles.
//
// u = "deg" --> Angles are given in degrees
//     "rad" --> Angles are given in radians
//
// The default value is u="rad".

Definition at line 387 of file NcStokes.cxx.

GetParameter()

Double_t NcStokes::GetParameter	(	TString	name,
		TString	u = "rad" )

// Provide the value of the parameter with the specified name.
// For details about the various parameters, please refer to the
// general documentation of this class.
//
// u = "deg" --> Angles are returned in degrees 
//     "rad" --> Angles are returned in radians
//
// Supported parameter names are :
// -------------------------------
// I (or S0) : The total beam intensity I(x)+I(y) or equivalently I(a)+I(b)
// Q (or S1) : The net linear polarisation intensity I(x)-I(y)
// U (or S2) : The 45 degree rotated net linear polarisation intensity I(a)-I(b)
// V (or S3) : The net circular polarisation intensity I(R)-I(L)
// P         : The total polarisation intensity
// L         : The linear polarisation intensity
// C         : The circular polarisation intensity
// a         : The semi major axis of the polarisation ellipse
// b         : The semi minor axis of the polarisation ellipse
// e         : The eccentricity of the polarisation ellipse
// theta     : The polarisation angle (=orientation angle of the ellipse)
// beta      : The ellipticity angle of the polarisation ellipse, i.e. arctan(b/a)
// helicity  : The helicity of the polarisation (+1=right handed  -1=left handed)
// A1        : Amplitude |E1| of the Jones field vector (E1,E2)=(A1*exp(i*phi1),A2*exp(i*phi2))
// A2        : Amplitude |E2| of the Jones field vector (E1,E2)=(A1*exp(i*phi1),A2*exp(i*phi2))
// phi       : Phase difference (phi1-phi2) between the Jones vector components 
//
// The default value is u="rad".

Definition at line 298 of file NcStokes.cxx.

SetEllipseParameters()

void NcStokes::SetEllipseParameters	(	Double_t	a,
		Double_t	b,
		Double_t	theta,
		TString	u,
		Int_t	h,
		Double_t	p = 1 )

// Set the parameters of the polarisation ellipse.
//
// Input arguments :
// -----------------
// a     : semi major axis
// b     : semi minor axis
// theta : angle between the major axis and the x-axis
// u     : "deg" --> theta provide in degrees
//         "rad" --> theta provide in radians
// h     : helicity (+1=right-handed  -1=left-handed)
// p     : total polarisation fraction
//
// The default value is p=1.

Definition at line 215 of file NcStokes.cxx.

SetJonesVector()

void NcStokes::SetJonesVector	(	Double_t	A1,
		Double_t	A2,
		Double_t	phi,
		TString	u,
		Double_t	p = 1 )

// Set the components of the Jones field vector (E1,E2)=(A1*exp(i*phi1),A2*exp(i*phi2))
// with phase difference phi=phi1-phi2.
//
// u : "deg" --> phi is given in degrees
// u : "rad" --> phi is given in radians
// p : total polarisation fraction
//
// The default value is p=1.

Definition at line 261 of file NcStokes.cxx.

SetStokesParameters()

void NcStokes::SetStokesParameters	(	Double_t	I,
		Double_t	Q,
		Double_t	U,
		Double_t	V )

// Set the values of the Stokes parameters I, Q, U and V.
// These parameters are also known as S0, S1, S2 and S3, respectively.

Definition at line 193 of file NcStokes.cxx.

Member Data Documentation

fI

Double_t NcStokes::fI

protected

Definition at line 25 of file NcStokes.h.

fQ

Double_t NcStokes::fQ

protected

Definition at line 26 of file NcStokes.h.

fU

Double_t NcStokes::fU

protected

Definition at line 27 of file NcStokes.h.

fV

Double_t NcStokes::fV

protected

Definition at line 28 of file NcStokes.h.

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The documentation for this class was generated from the following files:

• ncfspack/source/NcStokes.h
• ncfspack/source/NcStokes.cxx