RADIATION OF WAVES BY 2-D REFLECTOR ANTENNAS FED BY COMPLEX-SOURCE-POINT FEEDS

Alexander I. Alexander I. Alexander I. Alexander I. NosichNosichNosichNosich, , , ,

with inputs from with inputs from with inputs from with inputs from AyhanAyhanAyhanAyhan AltintasAltintasAltintasAltintas and YuriyYuriyYuriyYuriy V. V. V. V. GandelGandelGandelGandel

Institute of Radio-Physics and Electronics NASUwww.ire.kharkov.ua

Bilkent University, Ankarawww.ee.bilkent.edu.tr

Kharkiv National University, Kharkiv, Ukrainewww.univer.kharkov.ua

1. Motivation

>> Metallic quasioptical reflectors are durable and provide very high directivity and small losses

Most common simulation tools and their shortages:

>> GO and PO: fail to characterize resonances, which may appear if near-field environment is complicated

>> MoM: hits "numerical wall" if applied to a single

reflector larger than 20 λ λ λ λ

>> FDTD: not applicable to quasioptical-scale problems because of enormous resources needed

2. Aim of Research | Outline

1.Development of efficient discrete modelbased on the boundary IEs

fast and convergent numerical algorithm with controlled accuracy

2. Computer simulation of 2-D antennas:

Single-reflector: parabolic, elliptic, offset

Dual-reflector: Cassegrain, parabola-cone

3. Frequency analysis of EM field effects

3. 2-D Problem Geometry

2

1 2( , ) : ( ),L x y R x F y y y y= ∈ = < <= ∈ = < <= ∈ = < <= ∈ = < <

Total field in the presence of scatterer:

2-D reflector is a curved strip, assumed to be perfectly

electrically conducting (PEC) and zero-thickness:

0

totU U U= += += += +incident field

scattered field

4. 2-D Problem Formulation

(((( )))) (((( ))))2, 0k U x y∆ + =∆ + =∆ + =∆ + = 2-D Helmholtz equation (off L)

2 21( , )( , ) ,

U x yikU x y o r x y

r r

∂∂∂∂ − = = + → ∞− = = + → ∞− = = + → ∞− = = + → ∞ ∂∂∂∂

Sommerfeld radiation condition

2 22k U U dσσσσ+ ∇ < ∞+ ∇ < ∞+ ∇ < ∞+ ∇ < ∞∫∫∫∫Ω

Local integrability = Edge condition

E polarization

(((( )))) (((( )))) (((( ))))0, , , ,

L LU x y U x y x y L= − ∈= − ∈= − ∈= − ∈

Dirichletb.c.

H polarization

0 ( , )L L

UUx y

n n

∂∂∂∂∂∂∂∂= −= −= −= −

∂ ∂∂ ∂∂ ∂∂ ∂

Neumanb.c.

must satisfy boundary conditions:Uon L,

5. Log-Singular & Hyper-Singular Integral Equations

E-problem reduces to a logarithmic-SIE of the 1st kind, andH-problem reduces to a hyper-SIE of the 1st kind for the unknown current, j(s), induced on the reflector:

E polarization

H polarization

(((( )))) (((( )))) (((( )))) (((( ))))(((( ))))1

0 0 0 0 0( ) ( ) ,

4L

iH k r s r s j s ds U r s s L− = − ∈− = − ∈− = − ∈− = − ∈∫∫∫∫

single layer potential

(((( )))) (((( )))) (((( )))) (((( ))))(((( ))))1

0 0 0 0

1( ) ( )

2x x

Ly xL

j s H k r s r s ds U r sn n n

∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂− = −− = −− = −− = −

∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∫∫∫∫

π

double layer potential

6666. . . . Method of Analytical Regularization via SMethod of Analytical Regularization via SMethod of Analytical Regularization via SMethod of Analytical Regularization via Statictatictatictatic----Part Inversion: from SIE a Part Inversion: from SIE a Part Inversion: from SIE a Part Inversion: from SIE a FredholmFredholmFredholmFredholm 2222----nd Kind IEnd Kind IEnd Kind IEnd Kind IE

Numerical solution of a Fredholm 2-nd kind IE

can be obtained by any not pathologic

discretization scheme (collocation, Galerkin)

leading to the Fredholm 2-nd kind matrices.

Hence, solutions are stable and convergent

kernel decomposition =static

singular part + regular part

H-pol (E-pol) IE for a PEC strip can be integrated (differentiated)

that leads to IE with Cauchy singularity plus auxiliary relation

This is a Fredholm IE of the 2-nd kind,

where

A X = B

X + CX = D

=>

( ) ( ) )(),(4/1

1

tfdttLtthtt

itq sss

s

s =

+

−∫

−

( ) ( ) )(,~)(~ 1

1

tfKdtttNtqtqtsss

−=+ ∫−

∫−

−

−−−=−=

1

1

2/122/12)()1(

4)(),,()1(),(

tx

dxxfx

itfKtthKtttN tstss

π

Singular operator is invertible - see Carleman 1928; Krein 1951; Erdogan&Gupta 1972

)()()(~ tLtqtq =

7. MAR via 7. MAR via 7. MAR via 7. MAR via diagonalizationdiagonalizationdiagonalizationdiagonalization with the aid with the aid with the aid with the aid of the staticof the staticof the staticof the static----part part part part eigenfunctionseigenfunctionseigenfunctionseigenfunctions

( )( ) ( ) ==−

−∫

−

)(,ln)1(

1

12/1

tTtTdtttt

tTnnnss

s

sn σ Chebyshev polynomials

of the 1-st kind

To transform SIE to the Fredholm

2nd kind matrix equation, take full

set of the corresponding Chebyshev

polynomials as a basis (I.e., make

analytical preconditioning)

Green’s function decomposition = static singular part + regular dynamic part

( ) |)|(||ln4

,0 sss

rrkRrri

rrG

−+−=

E-pol.

H-pol.

( )( )

==∂

−∂−∫

−

)(),(ln

)1(1

12

2

2/1 tUtUdtn

rrtUt

nnns

M

ssns

τ

Chebyshev polynomials

of the 2-nd kind

A X = B

X + CX = D

=>

8. Incident Field: Horn Modeling with CSP Beam

r0

x

y

β

2b

x0

y0

φcs

( ) ( )csin

rrkHru

−= 0In 2-D:

Advantages of the CPS field:

- Exact solution of the Helmholtz

equation at any space domain

- Variable beam width controlled by

the parameter kb

Dots and curvy line denote

the branch points and the

cut in the real space.

Branch cut plays the role of emitting aperture.

0,0 1,6 3,2 4,8 6,4 8,0

-4,8

-3,2

-1,6

0,0

1,6

3,2

4,8field intensity

r /λ0

kb = 5

1E-3

0,002796

0,007820

0,02187

0,06115

0,1710

0,4782

1,337

3,739

10,46

29,24

81,77

228,7

639,4

1788

5000

biryxr cscscs

+== 0,

, ββ bSinbCosb =

, 000 yxr =

-180 -120 -60 0 60 120 180

kb = 0.5

kb = 1

kb = 2

( )βϕ

π−−

∞→⋅⋅

−

cos

0

00

2~

kbrrik

ree

rrkiU

Factor, determ

ining the

directiv

e character of th

e CSP

field in

the fa

r zone

9. 2-D Parabolic Reflector Antenna in Free Space

Fed by a Complex-Source-Point Beam

(a) Directivity, and (b) Edge illumination as a function of the feed beamwidth parameter kb - aperture size of horn simulated with CSP

d=20λ, λ, λ, λ, f/d= = = = 0.5CSP feed in GO focus

10. 2-D Parabolic Reflector Antenna in Free Space

Fed by a CSP Beam

Dynamics of the far-field pattern variation with reflector size (left, kb=5), and

edge illumination given by the feed beamwidth parameter kb (right, d=10λλλλ)

11. 2-D Parabolic Reflector Antenna Fed by a CSP Beam

near a Flat Impedance Earth

Normalized radiation (a) and absorption (b) resistance, Efficiency (c), and Gain

(d), as a function of the antenna aiming angle. d=10λλλλ, f/d=0.5, kb=2 (E.I.=-8 dB)

12. 2-D Parabolic Reflector Antenna Fed by a CSP

Beam near a Flat Impedance Earth

Left = directivity as a function of the angle of inclination. Hump

corresponds to the spillover reflection matched with main beam.

Middle and right = far-field patterns for the optimally inclined and

in-zenith looking antennas. Note spillover lobes reflected from earth

d=20λ, λ, λ, λ, f/d= = = = 0.5CSP feed in

GO focus

13. 2-D Parabolic Reflector Antenna Fed by a CSP

Beam in a Circular Dielectric Radome

Directivity (top) and total radiated power (bottom), as a function of the radome

radius, for the matched and mismatched radomes. Note a chance to boost

either directivity or power due to resonances. d=5λλλλ, f/d=0.5, kb=2.6 (- 9 dB E.I.)

14. 2-D Parabolic Reflector Antenna with Resistive

Edges Fed by a CSP Beam in Free Space

Left column = far-field patterns for a uniformly resistive RA. d=20λλλλ , f/d=0.73, kb=5

Right column = far-field patterns for an RA with 1-λλλλ wide “linearly” resistive edge

E

2 2

2

( )( ) ( '( )) ( '( ))

1

v tj t x t y t

t+ =+ =+ =+ =

−−−−

(((( )))) (((( )))) (((( ))))(((( ))))

(((( )))) (((( )))) (((( ))))

1

0 0 02

01

1 1

0

0 02 2

1 1 0

1 1,

4 1

1

4 1 1

i dtK t t v t U t

t t t

dti dtM t v t U t

t t

π

π

−−−−

− −− −− −− −

′′′′+ = −+ = −+ = −+ = − −−−− −−−−

= −= −= −= − − −− −− −− −

∫∫∫∫

∫ ∫∫ ∫∫ ∫∫ ∫

15. MDS for Singular/Hyper-Singular Integral

Equations

H

(((( ))))

(((( ))))(((( ))))

(((( )))) (((( )))) (((( ))))

21 1

2 2

02

1 10

1

2

0 0 0

01

1 11 ln 1

8

1, 1

v t k Lt dt v t t t t dt

t t

v t N t t t dt L U tn

π π

π

− −− −− −− −

−−−−

− − − − +− − − − +− − − − +− − − − +

−−−−

∂∂∂∂+ − = −+ − = −+ − = −+ − = −

∂∂∂∂

∫ ∫∫ ∫∫ ∫∫ ∫

∫∫∫∫ (((( )))) (((( )))) 21j t v t t= −= −= −= −

Hyper-SIE contour parameterization

SIE contour parameterization x(t),y(t), -1≤t≤1

transformation of SIE into Cauchy-singular IE

note: K, M, N

–are

smooth functions

16. Method of Discrete Singularities

Discretization is done by using the quadrature formulas of interpolation type with the nodes in the nulls of the Chebyshev polynomials of the 1st and the 2nd kind:

2 1( ) 0, cos , 1,2, ...

2

n n

n i i

iT t t i n

nπ

−−−−= = == = == = == = =

1 0( ) 0, cos , 1, 2, ... 1

n n

n oj j

jU t t j n

nπ−−−− = = = −= = = −= = = −= = = −

17. Method of Discrete Singularities

E

H

Resulting matrix equations are uniquely solvable for each polarization case and give us the required current functions, v(t). Accuracy is controlled by the number of nodes "n":

(((( )))) (((( ))))

(((( ))))

0 0 0

1 0

1

0

0 02

1 1 0

1 1 1( , ) ( ) ( ) , 1,2, ... 1

4

1 1( ) ( ) ( ) ,

4 1

nn n n n

q j q jn nq q j

nn n

q q

q

iK t t v t U t j n

n t t

dtiM t v t U t j n

n t

π

π

====

==== −−−−

′′′′+ = − = −+ = − = −+ = − = −+ = − = −

−−−−

= − == − == − == − =−−−−

∑∑∑∑

∑∑∑∑ ∫∫∫∫

(((( )))) (((( ))))(((( ))))

(((( ))))(((( )))) (((( ))))

(((( ))))(((( ))))

(((( )))) (((( ))))(((( )))) (((( )))) (((( ))))

12 2

2 0 0 0 0 0 02

10 0

2

0 2

2 0 0 0 0 0 0

1 111 , ,

1, , , 1, ..., 1

2

j qn

n q q n j q j q

qj qq j

j

n j n j j j j j

v t t S t t N t tn t t

tnv t S t t N t t f t j n

n

λ

λ

++++−−−−

−−−−====≠≠≠≠

−−−−

− −− −− −− −

− + + +− + + +− + + +− + + + −−−−

−−−− + − + + = = −+ − + + = = −+ − + + = = −+ − + + = = −

∑∑∑∑

here (((( )))) (((( )))) (((( ))))(((( ))))

21

2

0 0 0 0

1

11, ln 2 2 ,

2 8

j qn

n j q l j l q

l

k LS t t T t T t

l nλ

++++−−−−

====

−−−−= + + == + + == + + == + + =

∑∑∑∑

18. Convergence and Accuracy in MDS Modelling

of 2-D Parabolic Reflector Antennas

f/d=0.5, kb=2.6

MDS mean-square errors in the

surface current functions of

symmetric parabolic antennas

versus the interpolation order, n

MDS mean-square errors in the far-

field patterns of the same antennas

versus interpolation order, n

19. Complex-Source-Point Beam Field

Normalized radiation patternsof CSP feed for different kb

Near fields of several CSP feeds

Advantages of CSP beam field:• Exact solution of the Helmholtz

equation at any space domain• The greater the kb, the narrower the CSP beam

r0x

y

ββββ2

b x

0

y0

Branch cut plays the role of emitting aperture

20. Front20. Front20. Front20. Front----Fed Parabolic ReflectorFed Parabolic ReflectorFed Parabolic ReflectorFed Parabolic Reflector

d/λλλλ = 10

kb = 2.5

d/λλλλ is fixed, kb is varied:

main beam width – no change

side lobe level - decreasing

kb is fixed, d/λλλλ is varied:

main beam width – decreasing

side lobe level - no change

::: Symmetric dish :::

Focal point

21. Front21. Front21. Front21. Front----Fed Parabolic Reflector Near FieldFed Parabolic Reflector Near FieldFed Parabolic Reflector Near FieldFed Parabolic Reflector Near Field

d/λλλλ = 50

kb = 2.5

d/λλλλ = 5

kb = 2.5

d/λλλλ =10

kb = 2.5

22. Front22. Front22. Front22. Front----Fed Parabolic Reflector Focal ShiftFed Parabolic Reflector Focal ShiftFed Parabolic Reflector Focal ShiftFed Parabolic Reflector Focal Shift

Total radiated power and main beam directivity:

(((( )))) (((( ))))2

22

0 0 0

0

2| | , max

π

πϕ ϕ ϕΦ ΦP d D

P= == == == =∫∫∫∫

= total field Radiation Pattern

= Observation Angle

(((( ))))0ϕΦ

0ϕ

Directivity as a function of kb

grows up with reflector size d/λ λ λ λ , and reaches maximum if kb≈2.5

reaches maximum if the source is slightly shiftedfrom the reflector focus

D as a function of x0 (source location)

grows up with d/λλλλ

kb = 2.5

focal point

::: Shallow dish :::

23. Offset Parabolic Reflector Simulation23. Offset Parabolic Reflector Simulation23. Offset Parabolic Reflector Simulation23. Offset Parabolic Reflector Simulation

kb = 11

d/λλλλ = 47

ββββ = 1400

Phase: main beam is locally close to the plane wave

Near field perfectly illustrates the wave effects and the interference in near zone

The larger the reflector => the

narrower the main beam width

Focal point

24. Offset Parabolic Reflector 24. Offset Parabolic Reflector 24. Offset Parabolic Reflector 24. Offset Parabolic Reflector ““““Quiet ZoneQuiet ZoneQuiet ZoneQuiet Zone””””

Phase Pattern: main beam is locally close to the plane wave in the so-called “quiet zone” of a quasioptical-size offset reflector

d/λλλλ=47, f/d=0.5, kb=11 (-10 dB edge illumination)

::: Deep dish :::

d/λλλλ = 30

kb = 3

ββββ = 1230

d/λλλλ = 5

kb = 3

ββββ = 1230

25. Offset Parabolic Reflector Simulation25. Offset Parabolic Reflector Simulation25. Offset Parabolic Reflector Simulation25. Offset Parabolic Reflector Simulation

26. Elliptic Reflector Focuser Near Field26. Elliptic Reflector Focuser Near Field26. Elliptic Reflector Focuser Near Field26. Elliptic Reflector Focuser Near Field

kb = 9

d/λλλλ = 30

kb = 2

d/λλλλ = 30

kb = 0.5

d/λλλλ = 14

27. Elliptic Cross27. Elliptic Cross27. Elliptic Cross27. Elliptic Cross----Section Focuser Near FieldSection Focuser Near FieldSection Focuser Near FieldSection Focuser Near Field

kb = 2

d/λλλλ = 12

kb = 2

d/λλλλ = 9

kb = 10

d/λλλλ = 46

kb = 10

d/λλλλ = 20

28. 28. 28. 28. CassegrainCassegrainCassegrainCassegrain Antenna SimulationAntenna SimulationAntenna SimulationAntenna Simulation

kb = 9 d1 = 30λ λ λ λ d2 = 7λλλλ

29. Offset 29. Offset 29. Offset 29. Offset CassegrainCassegrainCassegrainCassegrain Antenna SimulationAntenna SimulationAntenna SimulationAntenna Simulation

kb = 25 β β β β = 22= 22= 22= 220000

d1 = 10λ λ λ λ d2 = 4λλλλ

30. 230. 230. 230. 2----D Model of "D Model of "D Model of "D Model of "OmnidirectionalOmnidirectionalOmnidirectionalOmnidirectional" PACO Antenna " PACO Antenna " PACO Antenna " PACO Antenna

single parabola directivity

PACO directivity

Edge Illumination

= 600

single-feed design provides higher directivity if kb is small.

however, if kb gets larger, double source is able to

increase the PACO directivity

31. 231. 231. 231. 2----D PACOD PACOD PACOD PACO----type Antenna Simulationtype Antenna Simulationtype Antenna Simulationtype Antenna Simulation

single source

double source

kb = 9.2 provides maximum to directivity of double-source PACO

in case of 2 CSP the main beams are narrower

single source double source

32. Conclusions32. Conclusions32. Conclusions32. Conclusions

>> Complex-Source-Point field is a very good model of typical small-aperture feed

>> MAR & MDS possess fast convergence and controlled accuracy

>> MDS implementation is simpler than MAR

>> MAR & MDS compute 2-D models of real-life multi-reflector antennas of arbitrary shape mostly in seconds

>> MAR or MDS can be used as a core of the computer optimization and synthesis software