1965 Bickel and Ormsby: Polarization Scattering Matrix 1087

Error Analysis, Calibration, and the Polarization Scattering Matrix

S. H. BICKEL AXD J. F. A. ORMSBY

Abstracf-In conjunction with the polarization scattering matrix, an error analysis is developed for first-order errors. A set of con- straints on the various error quantities is established for a passive antenna. The effects for dissipative and nondissipative errors, to- gether with perturbations due to deviation of the scatterer from the main beam axis and due to thermal noise, is included. The influence or lack of it, of these errors on polarization invariant quantities, rota- tion about the line of sight, and polarization axial ratios is explicitly developed.

Emphasis is also given to calibration procedures for taking ac- count of the various errors as well as Faraday rotation, if present. The essential dependencies are exposed, and the features of both a two-body and a one-body technique are given.

INTRODUCTION

Considerable effort has been directed recently toward an analysis of the polarization scattering matrix for determining characteristics of the scatterer [1]-[3]. This paper considers the errors to be ex- pected in certain useful parameters derived from the scattering matrix. These errors can be caused by improper antenna polariza- tion, deviation of the scattering body from the main beam axis, and thermal noise.

The effect of these errors on a measured scattering matrix can be described analytically by a congruent transformation in which the parameters in the transformation characterize the errors. Polariza- tion invariant quantities and rotation about the antenna line of sight can be measured in the presence of certain of these errors.

In addition, a calibration procedure offers a convenient method to evaluate the overall error problem. Alternative procedures are presented using either one or two calibrating scatterers.

DEFINITION OF THE ERROR MATRIX

The various errors can be represented by an error matrix E , which perturbs the ideal antenna polarization vectors. Thus, if a represents the ideal antenna polarization vector, then the vector for the actual polarization transmitted is given by

a' = Ea. (1)

For small errors, E is defined by

with

The eij denote error in the antenna polarization and will be a function of position in the antenna beam. As one deviates from the axis of the beam, these errors will tend to increase [l] . In any event, for an ideal antenna with no error (Le., q j = O for all i, j ) , E becomes the identity matrix I, resulting in the desired polarization being transmitted.

For a passive antenna, it is required that the power contained in a' be less than, or equal to, that contained in 4. Allowing for com- plex polarization vectors, this can be expressed as

a'*(a')* = Ea.E*a* 5 a.a*, (4) where the asterisk denotes the complex conjugate.

following restrictions on the elements of E : The condition given by (4), where a is arbitrary, leads to the

in this paper was sponsored by the Air Force Electronics Systems Dw., Air Force Manuscript received April 5 , 1965; revised May 10, 1965. The research reported

Systems Command, under Contract AF 19(628)2390. This paper is.identi6ed as ESD Technical Documentary Report 65-116. Further reproduction IS authorized to satisfy needs of the U. S. Government.

The authors are with The MITRE Corporation, Bedford. Mass.

Re €11 I 0, (5) Re €22 I 0, (6)

1 €21 + EIZ* 1 ) I 4 Re €11 Re €22) (7)

If the two polarizations of the antenna remain orthogonal after

€21 + el** = 0. (8)

where Re stands for real part.

perturbations, then we require that

Three distinct possibilities exist for orthogonality. The first is the lossless case, in which Re €11 and Re e?? equal zero and E is unitary (Le. E*TE =I). The next is for loss that is independent of polariza- tion. Here loss is said to be isotropic, in which case E equals a real attenuation factor times a unitary matrix. The last case is for asym- metric loss, where either Re €11 or Re €29 are zero.

ceiver terminals as given by the bilinear form [4] The scattering matrix S specifies the voltage developed at the re-

I i a b = a S b , (9)

where a and b are the received and transmitted polarizations, respec- tively, and Sb accounts for the effect of the scattering.

The bilinear form in (9) is preserved if the scattering matrix, as measured in the presence of errors, is given by [SI

3 = ETSE, (10)

where ET indicates the transpose of E. Thus, the perturbed scattering matrix, neglecting second-order effects, can be written as

eeS1l + C21S3 S I I + E ~ H + t d h

s21 + €12Sll + € ? S T 2 ect2Sp. + el& = eel ( ), (11)

where the Sij; i, j = 1, 2 are the elements of the scattering matrix S which would be measured in the absence of antenna error, and

so = Sl? + s21, (12) €1 = €11 + €22, (13) €2 = €11 - €22. (14)

In the absence of nonreciprocal effects, SI^ equals S?I, so that S3 becomes 2 S l ~ or 2&.

A model for random errors can include the effects both of un- known body motion in the beam and thermal noise.

Let e represent rotation about the beam axis and 4 represent the angle between the main beam axis and the direction to the body. For moving bodies, 0 and 4 will be functions of time, so that q ( @ , 6) specifies the associated error. The nature of this error is passive so that the inequalities (S), (6), and ( 7 ) hold.

Using eij(") to represent thermal noise, and c,~(O, 0) to specify fixed errors when the body is along the axis of the beam in the ab- sence of noise, the total error for each i, j pair becomes

E" ,, - - C i j ( o , 0) + Eij(e, 4) + e i j ( . ~ ) . (15)

Since the average value of thermal noise is zero and such noise is independent of motion, the average value of the error eij is ei,(O, 0) added to the average value of s i J ( & 4). On the other hand, the vari- ance (Le., squared rms error) of eij becomes the sum of the variances of e& 6) and q , ( ~ ~ ) , considering both errors as random.

EFFECTS OF A LOSSY ANTENKA ON SCATTERING M a T R I X INVARIANTS

Reactive antenna errors produce changes in the axial ratio and rotation of the axes of the polarization ellipses. However, orthogonal- ity of the polarizations, as well as the power contained in each po- larization, is preserved. Hence, the inequality condition of (4) be- comes an equality. This results in E becoming a unitary matrix and, consequently,

Re €11 = Re €22 = €21 + e12* = 0. (16)

This condition, applied to the transformation (lo), results in the invariance of both the determinant of the scattering matrix, det S, and the trace, PI, of the power scattering matrix P, where

P = STS, (17)

1088 Bickel and Ormsby: Polarization Scattering Matrix

and the trace of P is described directly in terms of the element of s by

From these invariants the modulus of the eigenvalues of S can be found [3].

In the general case, with dissipative antenna errors, power is lost. from the main antenna beam either by heat loss or, more probably, by reflections from the support structure or radome. For this situa- tion, the determinant of Sand the trace of P are no longer invariant under the transformation expressed by (10). In particular, the de- terminant becomes, neglecting second-order effects,

det S- = det ET det S det E = e*’ det S, (19)

and the trace of the measured power scattering matrix P becomes

P = 5 exp [Re (ei, + e;i)] I Si; l e + 2 Re (€12 + w*) ReS&* i j-1

+ 2 Im (e12 + enl*) Im S&* (20)

From the inequalities (S), ( 6 ) , and (7) , PI and the modulus of det as measured by a dissipative antenna will be less than these

quantities as measured by an antenna experiencing no loss or purely reactive errors.

If the dissipative part of the error is isotropic, with orthogonality preserved, both Pl and the modulus of det .!? are attenuated by the same amount. Thus they can be used in this case to find the ratio of the moduli of the eigenvalues of S.

In the case of small independent random errors with zero mean values, both det S and P1 are preserved for average values. Thus, ob- servation over a sufficiently long period tends to maintain the in- variance properties of det S and PI in the presence of errors of this type.

where Im stands for imaginary part.

EFFECTS OF ANTENNA ERRORS ON MEASUREMENT OF ROTATION ABOUT THE LINE OF SIGHT

In studying the effects of antenna errors on the relative orienta- tion of the antenna and scatterer about the line of sight, it is con- venient to assume that the scattering matrix measurement is made !sing circularly polarized antennas. For this case the basis vectors L and i? represent left- and right-circular polarization, respectively, so that

R?th the antennas free from error, the measured scattering matrix is given on the circular polarization basis as

where e designates the relative rotation between the antenna and scatterer about the line of sight, and OF designates any Faraday rotation present between the antenna and the scatterer.

\Yith the elements of C’ in (22) denoted by Cij’, 0 is given by

1 1 4 4

e - eo = - (arg cl1’ - arg C22‘) = - Im In Cd/Cz?’, (23)

where arg stands for argument, and the reference angle 00 is obtained by setting e to zero in ( 2 2 ) as

1 c22

4 Cll eo = -ImIn--. (24)

The effect of the error matrix as defined in ( 2 ) is to produce two slightly elliptically polarized antennas given by

where the axial ratio for the left- and right-circularly polarized an-

tennas become, respectively,

I & = - = 1 + 2 j e 2 1 l major axis mnor axis

August

rR = 1 + 2 I 6 1 2 1 . (27)

From (11), In I?11/I?22 in the presence of small errors is given by

where CI = c12 + c21.

The error in using (23) to determine 0 is given, from (28), by

or

The first error term in (31) is a constant term caused by a mis- alignment of the antenna axes about the line of sight and is indepen- dent of the scatterer. The second term involving e21 and €12 is caused by the transmission being slightly elliptically rather than circularly polarized [see ( 2 6 ) and (27)].

Equation (31) thus shows that transmission of slightly noncircu- lar polarization causes an error in 0, and this error depends on both the scattering body and e itself. For a body that causes little de- polarization, the magnitude of the off-diagonal terms will be much larger than the magnitude of the diagonal terms for circular polariza- tion. In this case, the second term of (31) which is proportional to the ratio of these magnitudes can be quite large.

If we consider small independent random errors, for example thermal noise, having uniform phase distribution and magnitude distribution with variances pij* (i.e., r m s value of 1 ei;] is k;), then the variance of rotation error denoted by w2 is given by

As in the deterministic case, we can associate an uncertainty in alignment with the ~ 2 2 term and polarization uncertainty with the p12* and ~ 2 1 2 terms, which are affected by the properties of the scat- terer but, in this case, are independent of rotation angle e. The effects of axial errors on gL6 are illustrated for rotationally symmetric scat- terers under ‘Application.

For zero mean random errors, observations over an extended time interval will result in an average value of 6 which tends to zero, as does the average of the observed errors for the invariant quantities.

CALIBRATIOX PROBLEMS

In order to calibrate the antenna, it is necessary to determine the error quantities e l , €2 , €12. and €21.

From (1 l), e l acts as a scale factor on all elements of the scatter- ing matrix and is, therefore, important only if absolute measure- ments are required. Referenced to circular polarization, the imag- inary part of e? gives a measure of the tilt of a reference axis on the antenna, while e l ? and €21 measure how much the antenna axial ratio, as given by ( 2 6 ) and ( 2 7 ) , deviates from the unity value associated with pure circular polarization.

One method of determining these effects is to measure the return from test scatterers for which the scattering matrix is known.

For reciprocal scatterers, S 1 2 equals 5’21, so that there are only three independent quantities in (1 1) while there are four .unknowns. Thus, it is impossible to determine all the error terms with a single calibrating body at a given aspect.

One of the most common calibration scatterers is a sphere for which the scattering matrix in circular polarization is given by

(33)

1965 Bickel and Bates: Magneto-Ionic Propagation and the Polarization Scattering Matrix

where US is the square root of the cross section of the sphere. From ( 1 l), the scattering matrix of a sphere measured in the presence of errors is given by

(34)

By taking ratios ? I I / ? ~ and c??/ca, the quantities e12 and €21

Can be determined directly without absolute measurements. Also, Cl?/C?, can identify the presence of Faradayrotation [6]. U’hen prac- tical, the antenna feed structure shou!d be adjusted until a null is obtained for the diagonal elements of C so as to insure the transmis- sion of purely circular polarization. If absolute cross section is known, then a determination of 61 can be made. Imperfections in the calibrat- ing sphere can be neglected only if I Cii/Cal <<eij; i # j , i = 1, 2.

To locate a reference axis for the antenna it is necessary to mea- sure e?. As indicated by (34), no information is obtained about this difference quantity using returns from a sphere. This is to be ex- pected since a sphere is a nondepolarizing body and, as such, scatters independent of rotation. Thus, to specify a reference axis, it is neces- sary to measure the return from a second scatterer, such as a dilpolel

The scattering matrix for a dipole whose axis lies along the L+R direction is given by

C d 2 ( ( ‘ I ) , 2 1 1

where Ud is the square root of the cross section of the dipole. Assum- ing that calibration has been performed on the sphere so as to make €12 and e21 zero in (34), then the measured scattering matrix return from the dipole becomes

Then CI~/??Z in (36) gives €2, and hence the antenna axes are readily determined for known body orientation.

The calibration also offers a convenient method to evaluate the behavior of the errors as a function of line-of-sight direction.

Another calibration procedure is possible using a single body of revolution at different orientations. For bodies of revolution in the absence of errors, C1l is equal to C2?. Therefore, the real part of (28) can be written as

All the elements of (37) are either ratios of observables or error terms. Since there are five error quantities associated with the three real and two imaginary parts of the error terms, it is necessary to make measurements a t five different aspects of the body. Since (37) does not involve the orientation angle explicitly, these error terms (i.e., axial ratios and difference between the power in orthogonal circular channels) can be determined without knowing the orienta- tion of the scatterer. This is analogous to calibration from a sphere. I f the orientation of the projection of the body axis on the radar line of sight is known, then from (31) the imaginary part of e? which locates the antenna axis can be found.

L4PPLICATIOs

Of the various considerations which can be derived from the re- sults of this paper, one of particular interest relates the variation in c(6 to axial errors for rotationally symmetric scatterers.

For this situation, and with ~ ~ 1 1 2 ~ and ~ 9 1 ~ both taken equal to p*, we have from (32),

The depolarization [SI, D, for the case at hand is given by

.- I

0 0 . 2 a4 0.6 , 0.8

BODY DEPOLARIZATION, D __c

Fig. 1. Rotation error vs. body depolarization and axial errors.

[Ye thus obtain the relationship

1089

.A plot of l r g in degrees as a function of the body depolarization D for various axial errors specified in decibels by p is given in Fig. 1.

REFEREKCES [l] P. J. Allen and R. D. Tompkins. “Pobrization techniques and components

for radar and communication systems,” 1963 N E R E M Rec.. PP. 16 and 17. [2] I. D. Olin and F. D. Queen, ‘A multj-band. polarization di,versity system for

dynamic radar cross-section studies, 1964 Radar Rcllectrvdy Measurements Symp.. vol. 11, pp. 166-176; Rome Air Development Center. Gri5ssAFB. N. Y.. Rept. TDR-64-25.

131 S. H. Bickel. ‘Some invariant properties of the polarization scattering matrix,”

141 V. H. Rimsev. “Transmission between elliDticallv wlarized antennas.” Proc.

. . this issue page 1070. . .

[SI C. D. Graves, Radar polarization power scattering matrix.” Proc. IRE, vol.

161 S. H. Bickel and R. H. T. Bates, “Effects of magneto-ionic propagation on the

IRE. vol. 39,Dp; 535-540, May 1951.

4 4 , pp. 248-252. February 1956.

. . .

polarization scattering matrix.” this issue, page 1089

Effects of Magneto-Ionic Propagation on the Polarization Scattering Matrix

S. H. BICKEL AKD R. H. T. BATES, MEMBER, IEEE

AbstractMagneto-ionic propagation effects are considered at radio frequencies that are su5ciently high so that there is negligible physical separation between the ordinary and extraordinary propaga- tion paths. It is recognized that the sensible magneto-ionic propaga- tion effects are birefringence and Faraday rotation. A matrix repre- sentation for magneto-ionic propagation is employed in order to facilitate the determination of the effects of magneto-ionic propaga- tion on the polarization scattering matrix of an arbitrary, monostatic radar target. It is shown that the propagation path must be calibrated with a known, four-way symmetric target when both birefringence

Manuscript received March 26. 1965; re!& May 6. 1965. The reseafch reported in this paper was sponsored by the A n Force Electronics Systems DIV.. Air Force Systems Command, under Contract AF 19(628)2390. Thls paper, is identified as ESD Technical Documentary Rept. 65-105. Further reproduction is authorized to satisfy, needs of the U. S. Govemment.

The authors are wth the MITRE Corwratlon. Bedford. Mass.