EISCAT RadarSchool 2005 : Basic ScatteringTheory A generic radar system The radar equation Basic scattering theory Thomson scattering Electron radar cross.

EISCAT RadarSchool 2005 :

Basic ScatteringTheory

• A generic radar system• The radar equation• Basic scattering theory• Thomson scattering• Electron radar cross

section• Scattering from many

electrons• Scattering from plasma• Ion-acoustic waves

Gudmund Wannberg, EISCAT HQ

RADAR

• acronym for RAdio Detection And Ranging:

• Generic name for measuring systems deriving information about distant objects (radar targets) by illuminating them with RF energy and recording the reflected and/or back-scattered energy,

• In the early days, only the presence of, and range to, the radar targets could be inferred, (that is how the acronym became RADAR...)

• Today, many radar systems also allow spectral analysis and (sometimes) target imaging.

A generic radar system

Transmitting antenna: GT

Receiving antenna: Ar

A/DRX

Power: PTX

Timing & Control

To computer

Signal generator

Radar target:

Physics

Engineering

The Radar Equation for hard targets A transmitter at the origin generates pulses of EM energy with power = P. The pulses are radiated f rom a gain antenna with gain GT that concentrates the transmitted power into a narrow beam, rather than spreading it equally in all directions. At a distance R f rom the origin, the antenna beam illuminates an area , where the power density [power per unit area] becomes = P GT / 4R2 R <= A A small radar target located somewhere inside the border of and having a radar cross section scatters some of the power hitting it:

Scattered power = P GT / 4R2 A f raction of the scattered power returns towards the general location of the transmitter and reaches it af ter travelling a distance R. When it gets there, obviously:

Scattered power density = P GT / [(4R2) (4R2)] At the transmitter, there is a receiving antenna with surface area = Ar which collects the scattered power: Received scattered power Pr = P GT Ar / [(4R2) (4R2)] which is the fundamental f orm of the radar equation.

Basic Radar Equation:

Pr = P GT Ar /[(4R2) (4R2)]

Most radars are monostatic, i.e. transmitter and receiver are co-located and share a common antenna with gain G.

From antenna theory:

G = 4 A/2

Monostatic Radar Equation:

Pr = P G2 2 /[ (4)3 R4 ]

The physics is contained in ...

We first consider a single electron:

Scattering from a single electron

Let us assume that the periodic electric field (generated e.g. by a radar transmitter) at the location of the electron is E(t):

E(t) = Re (E0 . exp (–i0t)), (1)

where E0 is a complex electric field. E(t) exerts a time-varying force on the electron. If 0 >> 0 , its equation of motion becomes:

-e E0 exp (–i0t) = me . dve/dt(2)

where e = 1.602 10-19 Coulomb,

me = 9.110 10-31 kg, and

0 is the electron gyrofrequency.

The velocity of the electron is vreal = Re (v). Solving for v(t):

v0 = -i e E0 / (me 0) (3)


The current density associated with the motion of this electron becomes:

j (r,t) = -e v(t) (r – re(t)) (4)

where re(t) is the position of the electron and (r) is a spatial delta function.

The vector potential at the receiver due to this current becomes:

A (r,t) = (0 / 4) (j’/ r – r’) d (r’)(5)

where the integral is evaluated at the retarded time t’ = t - r – r’/c

Using the current density defined by (4), we can integrate (5):

A (r,t) = (0 e / 4) v (t’) / (r – re(t’)) (6)


Assuming non-relativistic conditions and choosing the origin at the location of the electron, we can rewrite (6) as:

A (r,t) = - (0 e / 4) v (t’) / r

= - [0 e2 / (4 me 0)] E0 (1/ r ) exp –i (0 (t - r /c)) (7)

We now introduce the wave vector k1, defined through

k1• R = 0 r /c

where k1 = 0 /c and the direction of k1 is along r :

A (r,t) =- [0 e2/ (4 me 0)] E0 (1/ R1) exp –i(0t - k1 . r ) (8)

where R1 = r


Now compute the magnetic induction B = (1/ 0 ) H at r (the location of the receiver) in the far field approximation:

B (r, t) = x A

- [0 e2/ (4 me 0)] [(k1 x E0) /R1] exp –i(0 t - k1• r ) (10)

The Poynting vector at the receiver due to this induction becomes

Sr = ½ H 2 = ½ [ e2 / (4 me 0)]2 [ k1 x E0 / 4 ] 2 (11)

where = (0 /0) ½ = 376.7 ohms is the impedance of free space.


Introducing the power density incident on the electron, Sin ,

Sin = ½ E0 2/

and the polarisation angle, χ

sin χ = k1 x E0 / ( k1 . E0 )

we can finally compute Sr, the power density at the receiver:

Sr = ¼ [e2/ (0 mec2)]2 sin2 χ Sin /(4 R12) = (4 r0

2) sin2 χ Sin /(4 R1

2)

where r0 is the classical electron radius, defined as

r0 = e2/ (4 0 mec2) = 2.81 • 10-15 m (12)


The total power scattered by the electron is found by integrating (12) over the whole sphere :

PT = Sr . R2 d =

= T Sin • 3/2 • 1/4 sin2 χ d = T Sin (13)

where T = 8/3 r02 = 6.6 10-29 m2, the electron Thomson cross

section, is the ratio of total scattered power to incident power density. T is a constant, independent of 0

The scattered power density along the r direction is

Sr = (4 R12)-1 [3/2 T Sin sin2 χ ]

where the factor in brackets is the electron radar cross section, 0 When sin2 χ = 1 (backscatter), 0 = 3/2 T 10-28 m2

Scattering from many electrons

Consider an ensemble of electrons contained in a small box of volume V, sufficiently far away from the radar that a plane wave approximation is OK.

The transmitter-generated field in the volume is:

E1 (r,t) = Є0 p exp –i (0 t - k0 . r ) (27)

where r is some arbitrary position within the volume, relative to an origin somewhere in the volume. The scattered field from an electron at position rp(t) is:

Er (R1,t) = - (r0/R1) p Є0 exp –i (0 t’ - k0 . rp(t’)) (28)

where the retarded time t’ is = t – (1/c) R1 – rp(t’)


Since V is far away from both transmitter and receiver, we can approximate:

t’ ~ t + (1/c) [R1 + n1 . rp(t’)](29)

Here, n1 is a unit vector from the origin within the volume towards the receiver.

We now introduce a time-dependent electron density N(r,t). The total number of electrons in a volume element d(r) is then

N(r,t) . d(r)

and the field at the receiver due to these electrons is

dEr (R1,t) = - (r0/R1) p Є0 N(r,t’) exp –i (0 t’ - k0 . r) d(r) (30)

In the ionosphere, the electron density is typically some 1011 - 1012 m-3 – a very large swarm of mosquitos indeed ! As with mosquitos, the electron density will fluctuate with time due to random thermal motions and/or density waves of one kind or another.

Rather than treating so many electrons individually, we expand the density time variation in a Fourier integral:


We expand the density variation vs. time in a Fourier integral:

N (r,t) = (1/2 ) N (r, ) exp -it d

T

where N (r, ) = N(r,t) exp it dt

0

After some substitutions, we can compute the Fourier component of the elementary contribution from the volume element d(r) to the received field Er:

dEr (R1, ) = - (r0/R1) p Є0 d(r) . (1/2) •

• d’ N(r, ’) exp i [(’ + 0) R1/c - (k1- k0) rN] •

• dt exp it ( - ’ - 0)

where rN is a unit vector along r and the integral over t is taken from 0 to T.

After a further bit of algebra (omitted due to lack of time), we arrive at the following expression for the scattering cross section per unit volume and frequency interval:

V(1) = 4 r02 sin2 χ . (1/T) (1/V) < NT (k1- k0 , 1- 0)2>

where k0 = 0 /c and k1 = 1 /c


The scattering is a three-wave interaction, which satisfies a Bragg condition:

k = k1 – k0

= 1- 0

Scattering volume

I ncident

EM wave, k0 Scattered EM wave, k1

RECEI VER

Scattering k-vector;

k = k1- k0

TRANSMI TTER

Only the spatial Fourier component of the density distribution with wave-vector k = k1- k0 contributes to the scattering at (k1, 1) ; i.e. exists.

We can illustrate this using a stationary model:

If a small, randomly distributed, stationary ensemble of electrons is illuminated, they will scatter in different relative phases and the effective cross section will depend on the look direction...

Scattering from a plasma

• Plasma is electrically quasi-neutral, i.e. in any volume there are, on average, equal numbers of electrons and positive ions, and seen from the outside the volume has zero net electric charge.

• Both electrons and ions can scatter EM radiation, BUT

• Ionospheric ions (H+, O+, NO+, O2+) are 1.8 103 – 5.8 104

times heavier than electrons and therefore scatter that much more weakly, as we can see from the equation of motion:

• e E0 exp (–i0t) = mi dvi/dt = - me dve/dt ; dvi/dt = - (me/ mi) dve/dt

• The current density and induction due to ions are thus negligible

• BUT: In plasma, electrons no longer move freely; they are bound to the ions by electric polarisation fields.

• The plasma dispersion function has a low-frequency branch, the ion-acoustic branch, which is largely controlled by the ions...

Scattering from ion-acoustic waves

The ion-acoustic dispersion relation in the one-dimensional case is:

(/)2 = mi-1 (kTe + k Ti) = cs

2

Here designates the wave vector and k is Boltzmann’s constant.

The large ion mass determines the period of the associated ion acoustic waves.

Note that in the ionosphere, the electron and ion temperatures can be (and probably are) unequal !

Scattering from ion-acoustic waves in plasma

We can re-write the dispersion relation as

f = (1/2 ) [mi-1 k (Te + Ti)] ½ = Λ-1 [mi

-1 k (Te + Ti)] ½

which shows that the ion-acoustic frequency is

- inversely proportional to wavelength,

- directly proportional to mean plasma thermal velocity

In a mono-static radar situation, ion-acoustic waves with Λ = radar/2

will scatter a signal back towards the transmitter !

Scattering from ion-acoustic waves in plasma

Since the power spectral density of an ion-acoustic radar echo is controlled by

- electron density ne

- ion temperature Ti

- electron temperature Te

- ion mass mi

we can derive physical parameters from the echo spectrum !!!

NOTE: The total ion-acoustic scattering cross section

ion ne (1+ Te/Ti) –1

is typically only one-half of that expected from an electron gas of equal charge density; even less when the Te/Ti ratio is >1

Some model IS ion line spectra from different ionospheric regions

LEGEND:

Red – F region (300 km)

ne = 3 .1011 Te = 2000 K O+ Ti = 1000 K

Green - F region (300 km)

ne = 1 .1011 Te = 3000 K O+ Ti = 500 K

Blue – E region (120 km)

ne = 5 .1010 Te = 300 K NO+ / O2

+ Ti = 300 K

Black – topside (1000 km)

ne = 5 .1010 Te = 4000 K 90%O+ 10% H + Ti = 3000 K

Spectra computed for the EISCAT UHF radar wavelength of 0.33 m (930 MHz).

Power spectral density (y-axis) plotted to linear scale

IS and bulk plasma drift

In addition to the internal wave motion, the whole plasma may be drifting in some particular direction under the influence of external forces (e.g. convection electric fields).

In an IS measurement, a bulk drift manifests itself as a Doppler shift of the whole received scatter spectrum. The line-of-sight component of the drift velocity can be determined from:

vdrift = ½ c (f/fo)

where

f = Doppler shift

fo = ISR operating frequency

fo

f

Note: f > 0 drift towards radar !

f

Bulk plasma drift and convection E fields

At altitudes above 200 km, the plasma is essentially collisionless (the mean free path is in the order of tens of meters) and thermal effects tend to drive both electrons and ions in spiral trajectories along the magnetic field lines.

However, if there are electric fields present, these can drive the plasma across the magnetic field lines with a velocity v:

v = E x B / B2

Since B is known to fair accuracy and/or can be extracted from magnetic field models, once we have determined v (which we do by measuring three components of it), we can compute the magnitude and direction of the convection electric field E

EISCAT RadarSchool 2005 : Basic ScatteringTheory A generic radar system The radar equation Basic scattering theory Thomson scattering Electron radar cross.

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