Scattering by small particles' Antenna model of a chiral scatterer' Calculation of RCS'

1
Scattering by small particles. Antenna model of a
chiral scatterer. Calculation of RCS.

• S-26.300

2
Outline

• Introduction
• Dyadic polarizabilities
• Electromagnetic analysis
• Electrically small particles
• Calculation of CRS
• Conclusions

3
Introduction

• Electrically small chiral shape structures can be
  utilized in producing composite chiral materials.
• Accurate electromagnetic analysis is too
  complicated to allow simple analytical solutions.
• Small helix can be replaced by a wire-and-loop
  structure.
• Microwave response of a single chiral inclusion
  allows predictions of the response of composites.

4
Introduction

• Chiral material has handedness in its structure

5
Dyadic polarizabilities

• Small particles of complex shape can be
  characterized by dyadic electric and magnetic
  polarizabilities which define the bianisotropic
  relations between induced electric- and
  magnetic-dipole moments p, m and external
  electric and magnetic fields E, H.
• Due to the reciprocity principle, the
  polarizability dyadics are subject to the
  relations

6
Dyadic polarizabilities

• The polarizability dyadics can be assumed to have
  the following forms
• Assumptions wire radius is small compared to the
  loop radius and wire length.
• Small chiral elements are considered as small
  loop antennas connected to short electric-dipole
  antennas and antenna theory is employed.

7
Electromagnetic analysis

• Accurate EM analysis of wire and loop antennas
  result complicated expressions for the input
  impedances or admittances.
• If
  analytical expressions for the
  antenna parameters can be used.
• If the terms of order and higher are
  neglected, the input admittance of the linear
  wire antenna can be expressed as

8
Electromagnetic analysis

• Circular-loop antennas can be analyzed with the
  use of the Fourier series expansion of the
  current distribution function. Analytical model
  can be constructed by keeping only the most
  important terms in the expansion resulting
  following input admittance

9
Electromagnetic analysis

• The expression corresponds to the approximation
  of the loop current distribution in the form
• where is the electromotive force of an
  ideal voltage-point source positioned at the
  polar angle ? 0.

10
Polarization by electric field directed along the
wire

• Wire antenna operates as a receptor which
  supplies power to the radiating loop.
• Scattering response is the sum of the power
  radiated by the currents both in the loop and the
  wire portions.
• The current in the center of the particle is
• The corresponding electric dipole moment
  amplitude and polarizability component are

11
Polarization by electric field directed along the
wire

• The electromotive force excites the loop also and
  generates current in it. The uniform part of the
  loop current sets a magnetic dipole along the
  z-axis with the amplitude
• The corresponding cross-polarizability component
  is
• The second Fourier component of the current
  radiates as an electric dipole directed along the
  y-axis. Denoting this as the
  charge distribution can be written as
• The electric dipole moment is then determined by
  integration, which gives the polarizability
  coefficient

12
Polarization by magnetic field directed along the
wire

• The loop antenna operates as a receiver and the
  wire antenna radiates power supplied by the loop
  antenna.
• The uniform part of the receiving loop current is
  given by
• The corresponding magnetic dipole moment and the
  polarizability are

13
Polarization by magnetic field directed along the
wire

• The input current in the wire portion is
• This defines the cross polarizability

14
Polarization by magnetic field directed along the
wire

• As in previous slides the cos component of the
  loop current also sets an electric dipole
  directed along the y-axis with amplitude
• This gives the polarizability

15
Exitation by electric field polarized in the loop
plane

• The incident electric field is in the loop plane
  and the magnetic field lies in the same plane.
• Two electric field polarizations must be
  distinguished since the loop response depends on
  the orientation of the field respect to the
  position of the wire antenna.
• The current exited in the loop by electric field
  along the x-axis is
• This gives polarizability

16
Exitation by electric field polarized in the loop
plane

• The total current in the loop caused by the
  electric field polarized along the y-axis is
• The uniform part of the current generates a
  magnetic dipole along the z-axis and the
  cross-polarizability is
• The terms which vary as cos ? correspond to the
  polarizability component

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18
Electrically small particles

• For electrically small particles the input
  admittances can be modeled by simple
  lumped-element circuits as

19
Polarizabilities of small particles

• Impedances in (14) can be replaced by their
  approximate values and loop-current coefficients
  can be approximated by
• By using (27), (28) and (29), (14) simplifies to

20
Polarizabilities of small particles

• In the same approximation as (31) the magnetic
  polarizability reduces to
• In the present approximation the co- and cross
  polarizabilities are related as

21
Polarizabilities of small particles

• The polarizability coefficients (21) and (25) are
  simplified to
• Dielectric polarizability of the loop becomes
• For the electric field along the y-axis (26)
  becomes

22
Polarizabilities of small particles

• In the same approximation (15) becomes

23
Calculation of CRS

• If the incident electric field is along the
  straight part of the particle the copolarized RCS
  is
• If the incident magnetic field is aligned with
  the dipole antenna portion and the electric field
  is polarized along the x-axis, the copolarized
  RCS is
• If the incident electric field is along the
  y-axis the copolarized RCS is

24
Calculation of CRS

• The cross-polarized RCS is defined as
• If the incident magnetic field is directed along
  the x-axis and the electric field is along the
  wires
• For the orthogonal polarization of the magnetic
  field

25
Comparison to numerical simulations
26
Conclusions

• Analytical model of microwave responses of chiral
  particles was introduced
• RCS calculations based on the presented antenna
  model have good agreement with the numerical
  simulations.
• The model of individual particles can be used in
  prediction of the behavior of microwaves in
  chiral composite material
• Practical applications ?

27
References

• 1 S. A. Tretyakov, F. Mariotte, C. R. Simovski,
  T. G. Kharina, J-P. Heliot, Analytical antenna
  model for chiral scatterers Comparison with
  numerical and experimental data, IEEE
  Transactions on Antennas and Propagation, Vol.
  44, No. 7, Jul. 1996, pp. 1006 1014.

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Homework

• Calculate the copolarized RCS of a wire-and-loop
  element in the case when the incident electric
  field is along the straight part of the particle
  (38) and the background medium is air.