THE ELECTROMAGNETIC FIELD THEORY II WAVE POLARIZATION …avikb/GNR647/Lec_2_EM Field Theory...

THE ELECTROMAGNETIC FIELD THEORY IIWAVE POLARIZATION

Dr. A. Bhattacharya

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Polarization

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Polarization

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The orientation of the Electric and the Magnetic field vectors define the polarization of the propagating wave

The polarization of the propagating wave are not strictly important in terms of the propagation of radiation in free space

When radiation strikes the ground the response of the surface material can be different for different orientation of the vectors

Need a convention to describe the direction in which the field vectors point

Polarization

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Polarization

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Note that the perpendicular polarized wave is horizontalto the Earth’s surface

In RS it is therefore more often called horizontal polarization

Not strictly correct, parallel polarization is similarly referred to as vertical polarization

The plane of polarization is that in which the Electric field vector oscillates sinusoidally

Polarization

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difference Phase

Polarization

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The plot shows two components of the Electric field as functions of time at a given position in space to illustrate the significance of the phase difference

Polarization

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Pure circular polarization only occurs when the two components have the same amplitude and the phase difference between them is 90°

In the most general case the approaching wave would be an ellipse

There will be left-elliptical polarization and right-elliptical polarization depending on the sign of the phase angle between the components

Circular (phase= 90°) and linear polarization (phase= 0°) are special cases. Their relative amplitudes will determine the orientation of the actual field vector

Polarization

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0

hvaa /Slope

hvaa /Slope

Polarization

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Two properties of the ellipse relate directly to the polarization state of the radiation

Ellipticity or Eccentricity (ε) which describes how different it is from a circle or a straight line

Tilt ( ) or inclination with respect to the horizontal

This shows the explicit relationships between properties of the wave and those of the polarization ellipse

,,vh

aa ,

Polarization (Jones Vector)

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Another way of expressing the Electric field is by using the parameters of polarization ellipse

A Amplitudeζ Absolute phase corresponding to –jβR resulting from propagation path Ej Jones vector

The exponential factor in time is dropped since it applies to all fields


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je j

0 if 90

We can absorb the sign into eccentricity correspond to left elliptical polarization

sinj


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We can transform the vector so that it applies to more general case of the inclined ellipse by rotating the axes clockwise by the inclination angle

The Electric field vector in the most general case


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Some common Jones vectors


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Circular polarization as a basis vector system

We have considered a travelling wave as a combination of horizontal and vertical components It is possible to choose right circularly polarized and left circularly polarized fields components as the basis

The unit vectors and are unit vectors rotate around the unit circle in their respective directions carrying the relevant field magnitude with them


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Purely left circular polarization wave will have ER =0 The horizontal and vertical field have the same amplitude and the vertical component leads

has a positive phase angle of 900

The magnitudes are assumed to be unity and El will have the same dependence on time and position as its two components

which can be removed as a common factor

Since and are unit vectors


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In a similar manner a right circularly polarized wave will have its unit vector


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In matrix form :

This indicates how the linear fields can be computed from the circular components

The circular components in terms of the linear components


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A horizontal polarized wave is made up of right and leftcircularly polarized waves starting in phase (and contra-rotating)

A vertical polarized wave is made up of the two contra-rotating starting in anti-phase

The j is a time phase term common to both components advancing them by 900

Polarization (Stokes Vector)

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Stokes parameters provide a very convenient means to describe the power density relationships in an EM wave in radar

For a single frequency signal (monochromatic) Stokes parameters are defined as


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S0 The amplitude squared or intensity of the actual field vector. It is directly proportional to the power density being carried by the wave

S1 This indicates whether the wave is more horizontally than vertically polarized

S2 & S3 Indicate the ellipticity of the wave’s polarization

If we have linear polarization and S3=0 If S2=0 and the polarization ellipse will aligned vertically or horizontally It will be circular if the magnitudes of the vertical and horizontal components are also equal

0090


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Complex phasorrepresentation

Representation in terms of principle angles of the polarization ellipse

Relative phase representation


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The Stokes vector can be represented in different forms represented as:

Coherency vector

Polarization (Stokes Vector)Question ?

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Is the absolute phase ( ) or ( ) preserved in Stokes vector representation of waves ?

x

y

Polarization (Stokes Vector)Quasi-monochromatic and Partially polarized waves

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In Synthetic Aperture Radar (SAR) remote sensing the wave is transmitted in a narrowband

The transmitted and received waves are narrowband about the central frequency

The wave may still be interpreted as a plane wave is said to be quasi-monochromatic


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The components and of the real vector is

Mean frequency

Mean wave vector

Slowly varying in comparison with the periodic term

1E

2E )(E

))(cos()(),())(cos()(),(

tzkttatzEtzkttatzE

yyy

xxx

k

)(),(),(),( tttatayxyx

)exp( tj


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The receiving antenna measures the target scattered narrowband wave during an interval of time (Azimuth integration time)

If

The wave then behaves in the time interval like a monochromatic wave with mean frequency

The Jones vector or the Stokes vector can be used to characterize the polarization of the monochromatic wave that is said to be a completely polarized wave

T

constant Assumed and

time coherenceTarget

)(),(),(),( tttata

T

yxyx

T


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For a longer time interval the Electric field components and the phases are time varying

Wave is partially polarized

In this case the parameters that characterizes the polarization wave should be time averaged

The correlation between the time Electric field components is necessary to characterize partially polarized waves Coherency matrix measurement

Tyx

aa ,yx

,

conditionsergodic andty stationarisense - widesignal of conditionthe Under ?


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The coherency matrix is an interesting tool that permits observable parameters of a partially polarized wave to be measured

To deal with observables quantities, 2 quadratic forms of the quadratic products of 푬 and 푬∗푻 are considered

Time averaged total intensity 푬∗푻.푬

Coherency (2x2) Hermitian matrix 푱 = 푬.푬∗푻

… Ensemble average ?


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푱 is Hermitian positive semi-definite Real non-negative eigenvalues

The fact that 푱 = 푱 ∗푻 makes 푱 an observable quantity

The trace 풔ퟎ of the matrix 푱 is the total intensity of the wave

The coherency matrix 푱 is equivalent to the density matrix of Von Neuman that is widely used in Quantum Mechanics

푠 = ‖E‖ = 푡푟푎푐푒 퐽


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The four elements of the coherency matrix 푱 are uniquely associated with the wave

The unique set is intimately related to the appropriate degree of the Electric fields in the two orthogonal direction


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If the axes 풙,풚 are rotated about the direction of propagation the coherency matrix changes

However, the determinant 푱 of 푱 , the two real non-negative eigenvalues 흀ퟏ and 흀ퟐ, as well as the trace of the Hermitian coherency matrix 푱 remains rotation invariant

Combination of these entities leads to rotation invariant parameter of the wave Degree of Polarization (DOP)


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The rotation invariant parameter DOP has a physical significance

The wave is considered to be completely polarized if 푱 = ퟎ ⇒ 푫푶푷 = ퟏ

The wave is said to be completely unpolarized if the intensity of its components in any direction perpendicular to the direction of propagation is a constant ⇒ 푱 풊풔풅풊풂품풐풏풂풍풂풏풅풕풉풆풆풍풎풆풏풕풔풂풓풆풊풅풆풏풕풊풄풂풍 ⇒푫푶푷 = ퟎ


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A partially polarized wave can also be related to the elements of the coherency matrix

There is a one-to-one correspondence between the coherency matrix and the Stokes vector


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Radar energy backscattered from landscape will often be polarized

If the scattering is from random scattering media or time-varying scatterers the wave will be either partially polarized or completely unpolarized

For a totally unpolarized wave the two amplitudes fluctuates randomly without any relationship between them Amplitude variation is uncorrelated

The relative phase between the components would be totally random


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If the two orthogonal components (say H and V) are totally random such that there is no preferred polarization

⇒ 풂푯ퟐ = 풂푽ퟐ ⇒ 풔ퟎ = ퟐ 풂푯ퟐ풔ퟏ = ퟎ

풔ퟐ = 풔ퟑ = ퟎ(푻풉풆풕풊풎풆풂풗풆풓풂품풆풐풇풕풉풆풕풓풊품풐풏풐풎풆풕풓풊풄풇풖풏풄풕풊풐풏

풐풇풂풓풂풏풅풐풎풍풚풗풂풓풚풊풏품풂풏품풍풆풘풊풍풍풃풆풛풆풓풐)

풔풖풏풑풐풍 =2 푎

000


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Partially polarized wave results from the addition of unpolarized and polarized component

풔풑풐풍=푠푠푠푠

풂풏풅풔풖풏풑풐풍 =푠

000

풔푻풐풕풂풍 =푠 + 푠

푠푠푠

=푠푠푠푠

푠 = 푠 + 푠 + 푠


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Degree of polarization (DOP)

푫푶푷 =푷풐풘풆풓풅풆풏풔풊풕풚풐풇풕풉풆풑풐풍풂풓풊풛풆풅풑풂풓풕

푻풐풕풂풍풑풐풘풆풓풅풆풏풔풊풕풚

푫푶푷 = 푠

푠 + 푠=

푠 + 푠 + 푠푠

Polarization (Stokes Vector)Poincare Sphere Representation

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A very interesting geometric representation of the Stokes parameters and the state of polarization of a wave emerges from

Equation of a sphere in the coordinate space 321

,, sss


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The sphere has radius of and its surface is the locus of all possible polarization states.

The polarization of a wave can be described by the amplitudes of its two orthogonal components and and their relative phase

The polarization of a wave can alternatively be also described by the angles of the polarization ellipse and

and the wave intensity

0s

Ha

Va

0s


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For partially polarized wave the point lies inside the sphere

The origin represents the case of unpolarized radiation

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THE ELECTROMAGNETIC FIELD THEORY II WAVE POLARIZATION …avikb/GNR647/Lec_2_EM Field Theory...

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