A11 holographie

8/6/2019 A11 holographie

http://slidepdf.com/reader/full/a11-holographie 1/5

PUBLICATION..., VOL. 1, NO. 1, JUNE 2007 1

Reconstructing distortions on reflector antennas

with the iterative-field-matrix method using

near-field observation data

Jose A. Martinez-Lorenzo, Member, IEEE, Borja Gonzalez-Valdes, Student Member, IEEE, CareyRappaport, Fellow, IEEE and Antonio G. Pino, Senior Member, IEEE

Abstract— This work extends the mathematical formulationof the iterative-field-matrix method for observed data from thenear-field region of a Perfect Electric Conductor. The method isused as a diagnosis tool for reflector antennas, to determine thepositions and extent of distortions from their idealized shapes.The new formulation is tested on a reflector antenna with severalsignificant bumps, and excellent results are achieved. This workalso presents an example where the Method of Moments is usedto generate the synthetic data and the inversion is performed

using Physical Optics. Such a configuration ensures that theforward model is unbiased with respect to the inversion model,demonstrating that the new formulation is also robust for theserealistic scenarios.

Index Terms— Diagnosis, reflector antennas, Physical Optics,Method of Moments, distortions.

I. INTRODUCTION

Many diagnosis techniques have been applied to determine

distortions of reflector antennas from their idealized shapes.

Microwave holographic metrology reconstruction, [1], [2] uses

the far-field amplitude and phase patterns of the antenna

to compute the field distribution on the focal plane of thereflector, this information combined with ray theory leads to

the shape distortion determination [3].

This contribution presents a variation of the iterative-field-

matrix method [4], [5] for finding the shape variations of

reflector antennas. The new formulation presented in this paper

is not only valid for using data in the far-field region of

the reflector antenna but also for using data in the near-

field region. The near-field version of the iterative-field-matrix

method avoids intermediate steps, including computing the

field distribution on the focal plane of the reflector [3],

reducing the error introduced to the inversion procedure, and

so it results in reducing computational time.

The results presented in the far-field version on the iterative-field-matrix method published in [4], [5], uses Physical Optics

(PO) for generating the synthetic data and for performing

the inversion (PO-PO configuration). The near-field version

of the iterative-field-matrix presented in this work not only

uses the PO-PO configuration, but also uses the Method of

Moments of generating the synthetic data and Physical Optics

for performing the inversion (MoM-PO configuration). The

Manuscript received Jan 1, 2007; revised June 1, 2007. This work issupported y CenSSIS, the Gordon Center for Subsurface Sensing and ImagingSystems under the ERC Program of the NSF (Award number EEC-9986821)and by Spanish Government grant FEDER-MEC ESP2005-01894

later configuration makes the inversion procedure robust, since

it ensures that the forward model for generating the synthetic

data is unbiased with respect to the inversion procedure.

The structure of this paper is as follows: forward models

used in this work are described in section II., formulation for

the near-field version of the iterative-field-matrix method is

introduced in section III., a numerical example is addressed in

section IV., and conclusions to this work are summarized insection V.

I I . FORWARD MODEL: METHOD OF MOMENTS AND

PHYSICAL OPTICS

The analysis, or forward model, consists of computing the

fields Es scattered by a Perfect Electric Conductor (PEC)

surface due to a known incident field Ei. Once the scattered

field is known, it is added to the incident field in order to

determine the total field Et at any point in the space [6].

The latter procedure can be divided into two steps. First

compute the induced electric currents J on the surface of the

PEC. Second determine the radiation in free space due to the

electric currents which establishes the scattered field at any

point in the space. The two steps are described with more

detail in the following sub-sections.

A. Step one: Calculation of the induced electric currents

The Electric Field Integral Equation (EFIE) in combination

with the Method of Moments (MOM) [7] is used to compute

the electric current J induced on a PEC surface illuminated by

a known incident field Ei. The later procedure can take large

amount of time, since it requires the computation and inversion

of the impedance matrix [7]. In the particular case where

the PEC is electrically large, the computation and inversion

of the impedance matrix can be avoided and the induced

currents can be approximated by using the Physical Optics

(PO) approximation [6], [8].

B. Step two: Calculation of the scattered fields

The scattered electric fields due to the induced electric

currents can be computed by using [6]:

Es(r) = − jη04πk

Ω

[g1J(r) + g2(J(r) · R)R] e−jkRdΩ

(1)




g1 =−1 − jkR + k2R2

R3, g2 =

3 + 3 jkR − k2R2

R3

R = r− r , R = |R| , R = R/R

k = ω√

µ00 , η0 =

µ0/0

where r is an observation point and r is a source point on

the PEC surface. k and η0 denote the wavenumber and the

free-space wave impedance, and a ejωt time dependence is

assumed.The integral equation (1) can be numerically computed by

dividing the PEC domain Ω into a set of triangular subdomains

Ωi placed at ri, for i = 1...N Ω, and then by evaluating the

integral on a set of observation points rl, for l = 1...N obswhere N Ω and N obs are the number of subdomains and

observation points. When the observation point rl is in the far-

field region of each subdomain, the electric field contribution

of subdomain i in observation point l can be expressed as [8],

[9]:

Es(rl, r

i) = Es

l,i =

−j

2λ

e−jkRli

Rli Ω

i

[J(ri

)−Rli(J(r

i

)·Rli)] ejkr

i·RlidΩ (2)

Rli = rl − ri , Rli = |Rli| , Rli = Rli/Rli

where λ is the wavelength. The scattered field, Esl , produced

by the entire PEC surface at the observation point, rl, is

computed by adding the contribution of each subdomain:

Es

l = Es(rl) =

N Ωi=1

Es

l,i (3)

III. INVERSION METHOD: NEA R-FIELD VERSION OF THE

ITERATIVE -FIELD-MATRIX METHOD

The inversion method based on the iterative-field-matrix ap-proach [4], [5] consists of determining the localized unknown

distortions, τ , to be added to a nominal, undistorted PEC

surface, r ∈ Ω, when the scattered fields produced by the

undistorted surface, Es, and by the distorted surface, Es,∆,

are known. The iterative-field-matrix approach is described

in the following subsections, which specifically indicate the

differences introduced to [4] which apply for observed in the

near-field.

A. Formulation for the near-field version of iterative-field-

matrix method

The iterative-field-matrix approach [4] computes the dis-crete distortions τ i, for i = 1...N Ω, introduced to the triangular

facets ri in the unit direction τ i by using a linear system of

equations. In the particular case where the scattered fields are

measured or calculated in the near-field region of the overall

PEC object, but still in the far-field of every triangular facet,

the linear system of equations can be written in a matrix as:

[A][x] = [b] (4)

Al,i = Ψ ·Esl,i jφl,i (5)

φl,i = k (Rli − pinci ) · τ i (6)

bl = Ψ · (Es,∆l −Es

l ) (7)

where [A] is a N obs×N Ω matrix containing the value Al,i in

row l and column i, Ψ is any polarization vector, pinci is the

unit poynting vector of the incident field at the center of the

triangular facet Ωi, the term [b] is a N obs × 1 column vector

containing the value bl in the row l, and the term [x] is a

N Ω

×1 vector containing the unknown value xi = τ i in row

i.The matrix A in (4) is ill-posed, and its inversion is

performed using the Singular Value Decomposition (SVD)

factorization [10], and the solution of the system is regularized

using the Tikhonov regularization [10], [11]. Once the discrete

distortions are calculated, a best fit procedure to a Polynomial

Fourier Series (PFS) [4], [12] is performed in order to obtain

an analytic continuous surface. The matrix inversion and best

fit approximation are iteratively repeated until the computed

distortions reach to an stable solution [4].

B. Differences between the near and far field versions of the

iterative-field-matrix method The far-field version of the iterative-field-matrix method [4]

requires the observation points - where the scattered fields are

measured or simulated - be in the far-field region of the PEC

scatterer. On the other hand, the near-field version, presented in

the previous section based on (2), only requires the observation

points to be in the far-field region of each subdomain. For

instance, the far-field version of the method applied to the

geometry described in Table I requires the scattered field to

be measured at distances Rfar ≥ 2D2/λ = 11.94 m. The

near field version of the method constrains field sampling at

distances Rnear ≥ 2H 2d/λ = 0.0133 m.

The far-field version of the method considers Ψ to be the

vector associated with the copolar polarization componentof the field [13]. The near-field version considers Ψ to be

any arbitrary component (or combination of multiple field

components) measured or computed in the near-field region

of the PEC.

Finally, the near-field version of the method computes the

value of φl,i in (6) by using the unit vector Rli, while the far

field version [4] uses the unit vector rl.

IV. APPLICATION EXAMPLE: DISTORTIONS

RECONSTRUCTION ON A BUMPED REFLECTOR

The numerical example presented in this section consists

of using the near-field formulation of the iterative-field-matrixmethod to determine bump-like distortions on the surface of

an ideal paraboloidal reflector.

The selected geometry is a single centered reflector with

parameters described in Table I and represented in Fig. 1.

Three bump-like distortions are introduced to the reflector

surface. Each distortion is characterized by: 1) its relative

position with respect to the center of the projection of the

reflector into the focal aperture in cylindrical coordinates φand ρ, 2) its diameter Db, and 3) its depth δB (see Table

II). The configuration of the distortions is similar to the one

described in [3].




TABLE I

CHARACTERISTIC PARAMETERS OF THE SIMULATED EXAMPLE

Main Reflector

Reflector diameter D = 0.3982 mFocal length F = 0.3982 m

Maximum patch size H d = 0.0133 m

Feed

Frequency f = 11.3 GHzElectric field polarization p = xf

Subtended angle θs = 28.0725o

Taper edge E-plane tE = 12 dBTaper edge H-plane tH = 12 dB

Fig. 1. Vertical cross section of the centered reflector antenna

The scattered fields were evaluated on N obs = 1800 obser-

vation points uniformly distributed along an on-axis circular

aperture, which is located at zobs = 5.4 m from the origin

of coordinates and has a diameter of Dobs = 3.982 m. The

reflector is discretized into N Ω = 5400 triangular facets,

producing a maximum patch size of H d = 0.0133 m. The

reflector surface is approximated using 534 Polynomial Fourier

Series (PFS) basis function [4], [12].

In the general case, the scattered field produced by the dis-

torted geometry Es,∆l can either be measured in an anechoicchamber or synthesized using a forward model. This data need

only be measured or generated once before starting the inver-

sion algorithm. In this work, the term Es,∆l is generated by

using both the Method of Moments and Physical Optics. The

later technique is referred in this work as “PO for the forward

model”, while the former is referred as “MoM for the forward

model”. The terms Esl,i and Es

l cannot be measured and need

to be regenerated for each iteration of the inversion procedure.

Computing such terms using the Method of Moments for

inversion would be possible but computationally impractical.

As a result, the Physical Optics approximation is always used

in order to compute Esl,i and Es

l

. The later procedure is

referred in this work as using “PO for the inversion”.

Fig. 2(a) presents the nominal reflector with the bump-like

distortions described in Table II. The reconstructed distorted

surface when using a PO-PO configuration for the first, fifth

TABLE II

CHARACTERISTIC PARAMETERS OF THE BUMP DISTORTIONS

φ[deg] ρ[λ] Db[λ] δB [λ]45 5 1.5 0.1667

180 5 1 0.2170 3.75 2 0.2

and tenth iterations are presented in Fig. 2(b), 2(c) and 2(d)

respectively. The equivalent cases are presented in Fig.3 when

a MoM-PO configuration is used. Both configurations predict

the distortions on the surface of the reflector after only a

few iterations. The residual errors, computed as the difference

between the analytical and reconstructed distortions for both

configurations are shown in Fig. 4, Fig. 5 and Fig. 6 for

the first, fifth and tenth iteration respectively. In the case of

MoM-PO configuration the residual error is bigger than in

the case of PO-PO configuration. This is because MoM-PO

configuration uses basis functions for expanding the current in

the forward model, with MoM using the Rao-Wilton-Glison

(RWG) [14] basis functions, which are different from those

used in the inversion model, while PO uses basis functions

which have constant amplitude and linear phase variation

across each triangular facet [8]. The PO-PO configuration

uses the same basis functions for the forward model and the

inversion, leading to a lower residual error. The configuration

using the different forward model from the inversion model

ensures that the former is unbiased with respect to the later,

demonstrating that the new formulation is also robust on theserealistic scenarios.

V. CONCLUSIONS

This work presents a new formulation for the iterative-field-

matrix method, which is valid when the scattered fields are

evaluated in the near-field region of a PEC scatterer but still in

the far-field region of each independent facet used to discretize

the PEC object.

The method has been used as a diagnosis tool to reconstruct

bump-like distortions on a reflector antenna surface. The

proposed formulation does not require computing intermediate

fields on the reflector aperture, as done in other diagnosismethods, reducing error and optimizing the execution time for

the inversion procedure.

Two configurations have been considered in order to eval-

uate the performance of the method. The first uses Physical

Optics for the forward and inversion procedures, while the

second uses the Method of Moments for the forward procedure

and Physical Optics for the inversion. Both configurations

are able to quickly and accurately reconstruct the distortions

on the reflector surface. As expected, the results for the

first configuration are better in terms of residual error due

to the algorithmic similarity between forward and inversion

procedures. The second configuration represents a more real-

istic situation where the forward and inverse procedures are

unbiased relative to each other while still providing accurate

reconstructions.

REFERENCES

[1] J. C. Bennett, A. P. Anderson, P. A. McInness, and J. T. Whitaker,“Microwave holographic metrology of large reflector antennas,” IEEE Transactions on Antennas and Propagation, vol. 24, no. 3, pp. 295 –303, 1976.

[2] C. E. Mayer, J. H. Davis, W. L. Petersw, and W. J. Vogel, “A holographicsurface measurement of the texas 4.9-meter antenna at 86 ghz,” IEEE Trans. Instrum. Meas., vol. IM-32, pp. 102 – 109, 1983.




(a)

(b)

(c)

(d)

Fig. 2. a) Nominal distortions. Reconstructed distortions in millimeters forb) 1 iteration c) 5 iterations d) 10 iterations using PO-PO configuration

(a)

(b)

(c)

(d)

Fig. 3. a) Nominal distortions. Reconstructed distortions in millimeters forb) 1 iteration c) 5 iterations d)10 iterations using MoM-PO configuration




(a)

(b)

Fig. 4. Reconstruction error in 1 iteration a) PO-PO configuration b) MoM-PO configuration

(a)

(b)

Fig. 5. Reconstruction error in 5 iterations a) PO-PO configuration b) MoM-PO configuration

(a)

(b)

Fig. 6. Reconstruction error in 10 iterations a) PO-PO configuration b)MoM-PO configuration

[3] Y. Rahmat-Samii and J. Lemanczyk, “Application of spherical near-fieldmeasurements to microwave holographic diagnosis of antennas,” IEEE Transactions on Antennas and Propagation, vol. 36, no. 6, pp. 869 –878, 1988.

[4] J. A. Martinez-Lorenzo, C. M. Rappaport, and A. G. Pino, “Reflectorantenna distortion: an iterative-field-matrix solution.” Radio Science,43:RS4019, doi:10.1029/2007RS003813, 2008.

[5] C. M. R. B. G. Valdes, J. A. Martinez-Lorenzo and A. G. Pino,“Generating contoured beams with single shaped reflectors using an iter-ative field-matrix approach,” IEEE Antennas and Wireless Propagation

Letters, vol. 7, pp. 697–700, 2008.[6] C. A. Balanis, Engineering Electromagnetics, 1st ed., R. M. Osgood,

Jr., Ed. New York, USA: John Wiley and Sons, 1989.[7] R. F. Harrington, Field Computation by Moment Methods, 1st ed. New

York, USA: The IEEE Press Series on Electromagnetic Wave Theory,1989.

[8] J. A. Martınez-Lorenzo, A. G. Pino, I. Vega, M. Arias, and O. Rubinos,“Icara: Induced-current analysis of reflector antennas,” IEEE Antennasand Propagation Magazine, vol. 47, no. 2, pp. 92–100, 2005.

[9] A. M. Arias, J. O. Rubinos, I. Cuinas, and A. G. Pino, “Electromagneticscattering of reflector antennas by fast physical optics algorithms,”

Recent Res. Devel. Magnetics, no. 1, pp. 43–63, 2000.[10] P. Hansen, “Regularization tools: a matlab package for analysis and

solution of discrete ill-posed problems,” Numer. Algo., no. 6, pp. 1–35,

1994.[11] Hansen, “Perturbation bounds for discrete tickonov regularization,”

Inverse Problems, no. 5, pp. L41–L44, 1989.[12] J. B. Bergman, F. J. V. Hasselmann, F. L. Teixeira, and C. G. Rego,

“A comparison between techniques for global surface interpolationin shaped reflector analysis,” IEEE Transactions on Antennas and

Propagation, vol. 42, no. 1, pp. 47–53, 1994.[13] A. C. Ludwig, “The definition of cross polarization,” IEEE Transactions

on Antennas and Propagation, vol. 21, no. 1, pp. 116–119, 1973.[14] D. W. S. Rao and A. Glisson, “Electromagnetic scattering by surfaces

of arbitrary shape,” IEEE Transactions on Antennas and Propagation,vol. 30, no. 3, pp. 409–418, 1982.

Documents

A11 holographie