IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-33, NO. 7, JULY 1985 749

Phase Retrieval in the Radio Holography of Reflector Antennas and Radio Telescopes

D. MORRIS

Abstract-Methods of phase retrieval from simulated intensity informa- tion have been tested for use in the radio holography of reflector antennas. In nnmerical simulations the Misell algorithm has been used successfully to retrieve the aperture phase distribution from two numerically simulated power polar diagrams, one in focus and the second defocused. The technique uses no anxilliary reference antenna. However, it does need a high signal to noise ratio, typically 50 dB if a 60 x 60 array is to be measured to a precision such that the gain is within 1 percent of ideal. It should be most useful where no direct phase measurements are possible and ground-based or satellite transmitters can be used as sources. The use of astronomical maser sources (22 GHz) can give information on large scale deformations.

I. INTRODUCTION

T HE ACCURACY needed in the setting and measurement of the surface of large radio-telescopes designed for operation

at millimeter and shorter wavelengths is now at the limit of conventional surveying methods. This is particularly true if the surface is to be checked at several angles of tilt. “Holographic“ methods have been used with some success but until now have required additional reference antennas and phase stable receiving systems.

This paper explores the possibility of using measurements of intensity only and relying on “phase retrieval” methods to ob- tain the necessary phase information. As a first step we have made a numerical simulation to test the feasibility of the method and its sensitivity to noise and measurement errors.

The idea of using measurements of the far-field pattern of an antenna to deduce the phase errors present in its aperture is not new (see for example Blum et al. [6] ). However, its practical use dates from the work of Bennet et al. [4] and Scott and Ryle [26]. A “holographic” recording technique was used by Bennet et al. [4], and this name has remained even though recent meas- urements have used direct phase recording (Scott and Ryle [26]), Godwin et al. [16], Anderson et al. [1] , Bennett and Godwin 151, Mayer et ai. [21]). A similar method has been ana- lyzed by Von Hoerner [31]. It was first proposed by Shenton and Hills [ 271 and uses the focal plane fields.

The need for accurate phase measurements makes these methods relatively difficult and expensive if additional equipment such as auxiliary reference antennas and phase sensitive receivers must be found. On the contrary, most reflector antennas have receivers capable of measuring the power polar diagram. What can be deduced about the reflector shape from such power measure- ments alone? So posed, the problem becomes the well-knowv ”phase retrieval problem.”

11. THE PHASE RETRIEVAL PROBLEM

There is extensive literature on calculating the phase of a complex function from knowledge of only its magnitude, or the

Manuscript received November 30, 1982; revised September 2, 1983. The author is with IRAM, Domaine Universitaire de Grenoble, 38406 St.

Martin d’Heres. France.

magnitude of its Fourier transform (its spectrum). For reviews see Taylor [28] and Fenverda [ lo ] .

For the antenna problem the usual positivity constaint (Fienup [l 11 ) cannot be used and we must use additional information to obtain a unique solution. Two possibilities from electron microscopy may be useful. The first uses the magnitude of the function and the magnitude of its Fourier transform (Gerchberg and Saxton [ 131 )> i.e., the intensity distributions over the antenna aperture and in the far-field diffraction pattern. The second method, due to Misell [22], uses the intensity distributions in two defocused images, i.e., two far-field patterns recorded with different axial focus settings. This second method seems a priori more suitable for antenna diagnosis. Both measure- ments (in-focus and defocused radiation patterns) use the same experimental technique. Furthermore the solution for the aper- ture phase distribution has been shown to be unique for the one- dimensional case (Huiser and Ferwerda [ 181 , Hoenders [ 171). This result can probably be extended to two dimensions (Drenth et al. [9]). In contrast, the measurement of aperture and far-field intensities involves different techniques and the calculated solu- tion for the phase is not always unique (Huiser et al. [ 191 , Huiser et al. [20] ).

111. POSSIBLE ALGORITHMS

Three classes of algorithms have been proposed for solving the phase retrieval problem. The direct methods (Van Toorn and Fenverda [30], Cerchbcrg and Saxton [ 131 ) are recursive and are so sensitive to errors and noise in the input data that most authors have dismissed them for practical use. The remain- ing two methods can be regarded as error reduction techniques. They include gradient search methods such as the method of steepest descent, and the iterative Fourier transform method (Gerchberg and Saxton [ 141 , Misell [22]). The relation be- tween the two has been discussed by Fienup [12]. Boucher [7], [8], in one-dimensional tests, has concluded that the Misell algorithm is slightly superior to the gradient search technique in terms of sensitivity to errors and noise. In two-dimensional tests Saxton [25] found that both methods (gradient search and iterative transform) were reasonably insensitive to noise.

Both methods have the disadvantage that they. may find “local” minima instead of the desired “global” minimum. In the interative transform method this has been termed “locking” (Gerchberg and Saxton [ 141 ). It is readily detected in computer simulations but may be a practical limitation since measurement errors produce similar effects on the convergence of the algorithm.

In a survey of tests of the iterative transform algorithm, Taylor 1281 has concluded that success depends on the particular func- tion involved, the number of samples, and the initial trial func- tion. It thus seemed useful to us to make computer trials simulat- ing, as closely as possible, the measurement of a reflector an- tenna. We report here on tests of the iterative transform method

0018-926X/85/0700-0749901.00 0 1985 IEEE

750 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-33, NO. 7, JULY 1985 $f

applied to defocused pairs of far-field power patterns, that is, the After transformation to the aperture plane the total error power Misell algorithm. is

error, it is clear that a penalty in precision must be paid if inten- and is given by Parseval,s theorem as sity (power) measurements alone are used to deduce the phase distribution across the antenna aperture. N - 1 N- 1 N- 1 N- 1

grain noise or photon statistical noise) on the Misell algorithm i=o F 0 i=o j = o

has been studied numerically by Boucher [8] and similar studies were also made by Misell himself [ 2 3 ] ) . In both cases only one- s 2 N P 2 dA2(ui, bi). (2) dimensional tests were made. For radio measurements, the domi- i=o j = o

nant error is likely to be additive receiver noise. We now analyze The root mean square (rms) phase errors dQ in the aperture its contribution to the phase errors in the aperture plane. plane are given by

lengths with an aperture electric field distribution E(x, r); the dQ2 = 0.51v-Z 2 {dE(xi,yi)/E(xi,yj)}z harmonic time variation is suppressed. This can be represented i=o j = O

The influence of multiplicative errors (e.g., photographic dE2(xi,Yj) = N - 2 A(ai, bj) - q a i , bil2

N- 1 N- 1

Consider for simplicity a square antenna of diameter D wave- ?v-1 N - l

by a square array of N X N equally spaced complex numbers. Its far-field pattern is then given over a finite data window, whose diameter is N/D rad, by the discrete Fourier transform

and substituting for dE(x, y ) from ( 2 ) N - I N - 1

OFT) dQ2 K 2 i=n 2 i=o { d A 2 ( q , bj)}/[A2(0, O ) ] (3 )

Thus N X N complex samples of the far-field pattern spaced at the critical interval 1/D rad will define the aperture distribu- tion with a surface resolution of D/N wavelengths.

The final trial function at the end of the iterations of the Misell algorithm is, in the in-focus far-field plane, T(a, b). Inverse Fourier transformation yields the solution for the aperture distribution

S(x, y ) = DFT- { T(a, b)}.

In the ideal case when the algorithm introduces negligible error, T(a, b ) differs in amplitude from the correct valueA(a, b) by the measurement error ~ A ( Q , b) of the in-focus measure- ments. Similarly, the phase of T(a, b) differs from that of d(a, b) by an angle @(a, b) introduced during the previous iteration and depending on both the measurement error in the defocused plane and the history of previous iterations. This is equivalent to the addition of an error vector dT(a, b).

We assume, from the symmetry of the algorithm, that its operation is such that, overall, the error contributions to dT(a, b) from d4(a, b) and #(a, b) are equal. So the total error power when summed over the N X N measured points in the far field is just twice the contribution due to dA (Q, b) alone:

N - 1 N - 1

{ A ( Q ~ , t)i) - bi)12 i=o j=o

. " , -

where A(0, 0) is the maximum value (on boresight) of the far- field amplitude, assumed to be approximately N2E(0, 0) if the phase errors are small. K is a factor to allow for any taper in the aperture distribution and any associated weighting of the phase errors. For a Gaussian amplitude taper t then K2 = ( l / t - 1)/(1 - t ) if amplitude weighting is used (for -14 dB taper K = d,.

In the case of the interferometer analyzed by Scott and Ryle [26], the electric field amplitude in the far field is propor- tional to the output V of the detector whose rms noise fluctua- tions dV are constant during measurements. Then substituting in (3)

dQ2 2 K 2 N 2 d v 2 / v2(o, 0)

dQ KNIR (4)

which differs from the value given by Scott and Ryle [26] by the normalization factor K . (Here R = V/dV is the signal to noise ratio on boresight.)

However, in the present case when only power can be meas- ured, the amplitude is taken as the square root of the detector output, or in the presence of noise, the square root of the abso- lute value of the detector output. For dA/A small we have approximately dA = d V / U ( n , b) which varies ulthin the polar diagram. The aperture plane rms phase errors are, from (3) ,

or dQ KS(N)/2 R .

Since the sum S(N) can become very large in the far side- lobes where A (a, b) is small, the method is increasingly suscepti- ble to the errors introduced by noise at large scan angles. The exact value of S(N) and its variation with N depend on the side- lobe structure which in turn will be a function of taper, blocking, and surface errors. In practice a variation faster then N 2 can be expected. Thus in the regime where the amplitude errors are

c

rrl

4

I

MORRIS: RADIO HOLOGRAPHY OF REFLECTOR ANTENNAS 751

small the Misell algorithm will result in larger phase errors than when phase can be directly measured (4). This is particularly true for large N (high surface resolution).

Another limiting case corresponds to measurements made near nulls where the signal is negligible with respect to receiver noise. In this case

dA(a, b ) = { I dV(a, b)

and

where we have used the result for the mean output of a full wave linear detector when fed with Gaussian noise (Bennett ~ 3 1 )

Then (3) becomes

~ Q ~ I S [ ~ / ~ ] ~ . ~ ~ N / R ~ . ~ . (6)

Hence: in the worst case the Misell algorithm demands a sig- nal-to-noise ratio which is approximately the square of that needed when phase can be directly measured. In fact the nu- merical trials described below show that this regime is more nearly applicable in practice. For small phase errors the fractional loss in antenna gain is

dG = dQ2.

Then the signal-to-noise ratio needed is

R z (2 /n)0 .5K2N2/dG.

V. THE MISELL ALGORITHM

The algorithm is supplied with a first guess at the aperture distribution-the initial trial function-and an in-focus and a defocused far-field-amplitude distribution. For the present tests, the latter were derived from a model antenna.

The trial function is in turn repeatedly compared with the two far-field amplitude patterns. At each iteration the amplitude

tude but its phase is retained, and thus is progressively modified as the iterations proceed. The connection between the two far- field patterns is made via the aperture plane (by fast Fourier transform) where focussing and defocusing is achieved by sub- tracting or adding a phase correction which varies as radius squared. One further constraint, not strictly necessary for the algorithm, has been included. During each change of focus in the aperture plane, the trial function is put to zero beyond the edge of the telescope, Le., a mask is applied. At each comparison, in either focused or defocused far-field plane, the mean square difference, expressed as a fraction, is taken between the trial amplitude and that simulated by the model. This error can at worst remain constant during the iterations (a “locked” solution at a local minimum) or decrease toward zero at the correct solution (Gerchberg and Saxton [14], Boucher [8]). In the presence of noise or measurement error the algorithm converges to or “locks onto” a solution with nonzero error since the two far-field patterns, when corrupted by noise, are no longer consistent with a single aperture distribution.

r of the trial function is replaced by the simulated far-field ampli-

VI. COMPUTER SLMULATIOKS The model antenna consisted of a two-dimensional complex

array of N X N elements simulating the aperture distribution.

The amplitudes were given a Gaussian taper (usually -14 dB) and this distribution could be offset to produce a slight asym- metry. It could be masked to zero within regions “blocked” by a central obstruction supported by four “legs,” and also beyond the edge of the circular aperture (see Fig. 1). The radius of this aperture was usually about 0.45 of the array diameter. Thus the simulated far-field values obtained by DFT correspond to slightly oversampled measurements, to avoid aliasing. The phase distribution could simulate defocus errors, astigmatism, a displaced panel or random surface errors.

Far-field patterns were calculated by the discrete Fourier transform (Arembepola [2]) for the in-focus and defocused cases. For the latter, the phase error in the aperture plane was assumed to vary as radius squared. In practice, when low F/D systems are measured, higher order terms would be necessary. The amplitude changes produced by defocusing are small for high gain antennas and have been neglected. In the tests described below, a maximum defocus amounting to 9.4 rad at the antenna edge was used. However, other tests indicate that this is not critical as long as it can reduce the antenna gain significantly bearing in mind the measurement errors. Thus with a 40 dB signal to noise ratio a defocus of about 1.6 rad was needed but with 50 dB signal to noise ratio even 0.393 rad gave satisfactory convergence. For a typical prime focus telescope these values correspond to motions of the order of a wavelength in the re- ceiver (or the secondary mirror in Cassegrain systems). The magnitude of the far-field patterns were corrupted by simulated measurement errors before being input to the Misell algorithm. The effects of additive noise (simulating receiver noise), multi- plicative noise (simulating random gain variations), and non- linearity were studied. To avoid negative amplitudes, the absolute values of the corrupted amplitudes were taken before being input to the algorithm.

The errors of the solution were estimated by comparing the aperture distribution at the output of the Misell algorithm with that supplied by the model. Phase errors have been expressed as amplitude weighted rms values.

VU. RESULTS An initial trial function with a Gaussian taper (either -7 dB

or -20 dB) and random phase (rms = 0.5 rad) was used but these values were not critical. Usually, after several tens of iterations the algorithm converged to a solution close to the desired function originally supplied by the model. Thereafter convergence was slow (Boucher [SI) and calculations were usually stopped after some hundreds of iterations. The devia- tions between the model and solution (the errors) were apparently random and near to those expected from the simulated measure- ment errors (calculated using (3)). As a measwe of the per- formance of the algorithm, the ratio of the rms phase error in the aperture plane to that expected from the simulated meas- urement errors has been calculated. It was of the order unity (range 0.5 to 2.0) depending on the noise level. For this purpose the actual amplitude errors dA(a, b) were used in (3), and the assumptions of (1) can be tested.

To illustrate the algorithm‘s performance we give the results for a simulated antenna having astigmatism (0.5 rad) and a displaced panel (0.5 rad). The illumination pattern was offset from centre by about 0.15 radii. A central blockage of about one tenth of the antenna diameter has been assumed. Random surface errors made little change in performance.

752 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-33, NO. 7, JULY 1985 @

( C) (dl

Fig. 1. Grey scale plots showing typical results from the Misell algorithm for a signal to noise ratio of 45 dB. Fig. I(a) shows the input phase distribution supplied by the model antenna. It has an astigmatism of + - 0.5 rad and a panel displaced in phase by 0.5 rad (I/ 25 wavelengths surface displacement). Fig. l(b) gives the output solution phase. The errors in amplitude and phase (difference

the extreme amplitude erron are + 9.0 and - 1 .O percent. The amplitude weighted rms phase error is + - 0.094 rad corresponding solution minus model) are displayed in Figs. l(c) and l(d), respectively. The extremes of phase error are +0.82 and - 0.51 rad and

to a gain loss of 0.87 percent.

For most tests a 32 X 32 array was used and the antenna diameter was taken as 30 resolution elements. The results are summarized in Table I for this case and for a limited series of tests on 64 X 64 and 128 X 128 arrays (60 and 120 resolution elements per diameter, respectively). For each signal-to-noise ratio (column 3), the sum-square error (percent) reached after the number of iterations listed in column 2 is given in column 4. Three performance estimators are used. They are the weighted r m s phase error in the aperture plane (column 9 , the ratio of this is that expected from the noise level (column 6), and the antenna gain loss (percent) due to the phase errors (column 7).

The results for a 45 dB signal to noise ratio are dlsplayed in the “grey scale” plots of Fig. 1. The phase distribution across the aperture of the model antenna is shown in Fig. l(a). The

c

solution after 200 iterations is given in Fig. l(b), and the phase and amplitude error distributions in Figs. l(c) and l(d), respec- tively. Tne random nature of the residual phase errors (solution minus model input) is demonstrated in these latter two plots. Note that the phase errors are not distributed with circular symmetry since the illumination pattern has been offset.

The dependence of the amplitude weighted phase errors on the signal to noise ratio and the number of samples used (which determines the surface resolution achieved) is austrated graphi- cally in Fig. 2. In these tests no “blocking” was present in the aperture. Fig. 2 shows that the form of the dependence is in agree- ment with (6) over most of the range considered. (A steepening and separation of the curves at high signal-to-noise ratios indicates the transition to the regime of (5)). However, the magnitude of

I

MORRIS: RADIO HOLOGRAPHY OF REFLECTOR ANTENNAS 753

TABLE I RESPONSE OF MISELL ALGORITHM TO ERRORS DUE TO RECEIVER NOISE

ARRAY ITER S IZE(N) NO.

32 50

a00

200

2OG

200

( 200

63 200

100

100

( 30

12E 200

200

( 200

RAT IO S / N

i n f .

60 d b

50 d b

40 d b

30 d b

20 d b

60 d b

50 d b

40 d b

30 d b

60 d b

50 d b

4 0 d b

SUM SQUARE ERROR ( % )

9. 2x10

$ . & x 1 0

1 .5x10

-5

-4

-2

-1

1. 5x10

8. 6 x 10 -1

2. 4

6. 3 x 10

5. ox 10

3. 9x10

1. 46

-3

-2

-1

2 1 x 1 0

1. 5x10

-2

-1

-1

PHASE ERROR

PHASE ERROR/ GAIN LOSS EXPECTED ( % I

-3 4. 16x10

-1

--- --- - <o. 01

5 . 4x 10 1. 32 a . 1 -?

3. 7x10 1.42 0. 14 -1

1. 7x10 1.26

6.OxlO 1. 04 -1

2. 7

27. 0

1 3 0. 58 79.0 )

3 3 x 1 0 1. 96

l.lxl0 0. 99

3. 7x 10 1. 003

9. 8x10 0. 71

-‘2

-1

-1

-1

-9

7 . 0 x l O 1. 28

2. 3x10 0. 43 -1

-1

0. 1

1. 1

11. 7

57. 1 )

0. 48

5. 0

9. ox 10 7 . 4x10 0. 84 3 7 . 0 )

R ( d b.1

7 0 60 50 LO 30 2 0 Fig. 2. The dependence of the amplitude weighted rms phase errors dQ (rad) on the signal to noise ratio R (dB) for several sizes of the

data window (diameter N pixels). The expected errors calculated using (3) are shown dashed. The dotted line represents the predictions of (6) in the “small signal approximation” (N = 64).

the errors is overestimated by about four times. Thus the de- obtained by averaging several solutions made with different pendence found in these tests can be approximated by (6) with initial trial functions. Table I1 summarizes a limited number of k = 0.6. tests. It shows that a reduction in phase error by a factor of

On the other hand: the errors predicted from (3) (shown about two can be obtained in this way. This suggests that the dashed in Fig. 2) afgee with those observed over most of the ‘‘locked’’ local minima are randomly spaced about the “global” range. This gives some confmation to the assumptions of (1). minimum. Averaging in this way may be useful in reducing the The discrepancies at low signal to noise ratios (e.g., 20 dB) can “computational noise” and arriving at a solution independent be attributed to failure of the small angle approximations. A t of the initial trial function. high signal-to-noise ratio the deviations from the predictions of The influence of random receiver gain fluctuations and of non- (3) may be due to “locking” onto local minima near the “global” linexities in the measuring devices has also been investigated. minimum. It has been noticed that the solutions depend in their As expected from the tests by Misell [23], [24] and Boucher details bn the initial random phase distribution assumed for the [7], [8] the algorithm is not sensitive to random gain errors. initial trial function. Furthermore, smaller phase errors can be For example, an n n s fluctuation of 10 percent in amplitude

754 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-33, NO. 7 , JULY 1985

TABLE II AVERAGES OF SOLUTIONS MADE WITH DIFFERENT INITIAL TRLAL FWNCTIONS

S/t< R A T I O -57 d b -47 d b -37 d b

NO. AVEEAGED PHASE ERROR

PHASE ERROR

PHASE ERROR

-2 10 10

-2 IO

- 1

1 2. b57 8 908 3. 193

4

16

64

1.748

1.408

1. 340

5. 991 ? . 896

5 044 1. 506

4. 612

e x p e c t e d 1 058 6.087 2.753

(0.83 dB) gave an rms phase error of 0.072 rad, corresponding to an antenna gain loss of 0.5 percent. Similarly, a quadratic term in the system response amounting to a maximum of IO percent in amplitude (0.83 dB) on boresight led to an rms phase error of 0.038 rad with a gain loss of 0.34percent. This insensitivity is presumably because a high dynamic range is only needed in a small region in the main lobe of the antenna pattern. Further- more, to a fust approximation, both in-focus q d out-of-focus beam patterns wiU be distorted in similar ways. This suggests that the optimum observing method would use measurements taken with equal but opposite defocus values.

VIII. CONCLUSION

From the present tests the MiselI algorithm appears to be satisfactory for use in antenna diagnostics. No spurious solu- tions or solutions corresponding to local minima (“locked solu- tions”) which differed greatly from the true solution were found. However, in tests with simulated receiver noise, the errors in the solutions were greater than expected on the basis of noise alone. This effect was most pronounced at low noise levels (high signal- to-noise ratios). It may indicate “locked solutions” very close to the correct solution. Their influence can be reduced be averaging several solutions obtained with different initial trial functions.

The main limitation to the method for antenna diagnosis seems to be the high signal-to-noise ratio which is required (about the square of that needed when phase can be directly measured). However, where satellite or ground-based transmitters can be used as sources, the method may prove useful when direct phase measurements are impossible. A peak signal-to-noise ratio of 50 dB allows about 60 X 60 points to be measured (or set in position) such that the antenna gain is within one percent of ideal at the measurement wavelength (Table I). Values of 40 dB should be attainable with large telescopes observing the astro- nomical sources of 22 GHz maser emission. This is sufficient for studies of large scale (e.g., gravitational) deformations. Some practical tests on real antennas are desirable.

Preliminary tests of the Gerchberg-Saxton algorithm to analyze intensity data in the aperture plane and in the far field have given similar performance. In principle this method may fail since the solutions are not always unique (Huiser et al. W I , [ W ) .

ACKNOWLEDGMENT

I thank E. Arambepola for advice on the use of his FFT routine, D. Emerson and U. Schwarz for computing help, and L. Wieliachev for directing me to the literature on speckle inter- ferometry. Thanks are due to D. Downes for improving the text and to S. Halleguen for photography.

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D. Morris, photograph and biography unavailable at time of publication.

Abstract-The radiation characteristics of a parabolic dish with a loaded peripheral flange are examined in detail in order to assess the effectiveness of such a loading in further reducing the backward scattered field. Uniformly valid diffraction coefficients are developed to deal with both isotropic and anisotropic surface impedances. It is shown that substantial improvement of the antenna performance can be obtained in a wide rear angular sector, and the optimal loading conditions are determined.

I. INTRODUCTION

T HE ENHANCEMENT of the front-to-back ratio as well as the reduction of the far out sidelobes are relevant problems

in many applications of reflector antennas, as, for instance, the broad-band microwave communications links.

A (possibly loaded) cylindrical shroud is often used to this end, and its performance has been examined from both the experimental and theoretical points of view [ 11 - [4] .

The more general case of flanged parabolic antennas has been recently considered and thoroughly analyzed [5] ~ showing that the best performance is obtained by using a flange forming a right angle with the reflector surface, and not the shroud con- figuration. In such a way a significant field level reduction can be obtained in a wide angular sector without affecting the field radiated in the forward half-space.

This paper is devoted to the detailed analysis of the radiation properties of a parabolic dish with a loaded right-angled flange,

Manuscript received July 23, 1984; revised March 8, 1985. 0. M. Bucci is with the Dipartimento di Elettronica, Universiti, via Claudio

C. Gennarelli is with the Istituto Universitario Navale, via Acton 38, 80133

L. Palumbo is with the Dipartimento di Energetica, Universiti La Sapienza,

21, 80125 Napoli, Italy.

Napoli, Italy.

via Scarpa 14, 00161 Roma, Italy.

in order to assess the further reduction in the backscattered field level which can be achieved by such a loading.

The loading is described in terms of a surface impedance boundary condition, a model whose validity has been examined both from the theoretical and the experimental points of view

Geometrical theory of diffraction (GTD), properly extended to deal with the surface impedance case, is used in the study, save in the neighborhoods of the forward and rear axial (caustic) directions, wherein physical optics (PO) and equivalent currents method are used, respectively.

In a GTD approach the field radiated by the flanged dish, whose geometry is depicted in Fig. 1, is obtained by summing up the geometrical optics and the diffracted rays. Since the flange is shadowed by the reflector, the relevant diffraction contribu- tions are given by

t6I-POl.

1) rays single diffracted by the paraboloid edge, which are dominant in the angular region not shadowed by the flange

2) rays doubly diffracted by both the paraboloid and the flange edges, which provide the m a order contribution in the rear region.

To deal with these rays we need uniform asymptotic evalua- tions of the solutions of two canonical problems: 1) the scattering of a plane wave by a right-angled loaded wedge and 2) the dif- fraction of such scattered field by a loaded half-plane. These problems are considered in Section 11. Once appropriate diffrac- tion coefficients have been determined, the radiated field can be evaluated in a standard way by using the aforementioned tech- niques. The backward field is analyzed in Section 111, wherein the effect of the loading for both the cases of isotropic and

0018-926X/85/0700-0755$01.00 0 1985 IEEE