1492 IEEE TRANSACTIONS ON AhTENNAS AND PROPAGATION, VOL. Ap-35, NO. 12, DECEMBER 1987

= 0. Thus, when the poles of G(s) are far from s = 0, the magnitude jQs: is large and the asymptotic expansion of F( jnsf) is given by

Substituting (24) into (23) yields

which is the correct expression for the case when g ( z ) does not have any poles near zs.

IV . CONCLUSION Two complete uniform asymptotic expansions (for large 0) of the

integral shown in ( l ) , obtained by two different methods, were compared. It was shown that both expansions are exactly the same (term by term). It was also observed that the uniform asymptotic solutions given in [6] (multiple pole singularities) and in [4, eq. (4.4.16)] (one pole singularity) are applicable for the general case where the pole@) of g(z ) cross the SDP path anywhere in the z-plane. Obviously, the generalization of Felsens result [4] to multiple pole singularities is still valid for the general case described above. Furthermore, for the special case when all the poles of g(z ) are far from the saddle point, the transition function F(x) was replaced by its own asymptotic series for large 1x1. As expected, the asymptotic series in (25) is the same as the one obtained by Felsen [4] when g ( z ) is regular near the saddle point.

REFERENCES

N. Bleistein, Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities, J. Marh. Mech., vol. 17, pp. 533-559, 1967. A. Erdelyi, Asymptotic Expansions. New York: Dover, 1956, ch. 2. E. L. Van der Waerden, On the method of saddle points, Appl. Sci. Res., B2, pp. 3345, 1951. L. E. Felsen and N. Marcuvitz, Radiation and Scattering of Wava. Englewood Cliffs, NJ: Prentice-Hall, 1973, ch. 4. P. C. Clemmow, Some extensions to the method of integration by steepest descents, Quart. J. Mech. Appl. Math.. vol. 3, pp. 241- 256, 1950. C. Gennarelli and L. Pdumbo, A uniform asymptotic expansion of a typical diffraction integral with many coalescing simple pole singulari- ties and a first-order saddle point, IEEE Trans. Antennas Propa- gat., vol. AP-32, pp. 1122-1124, Oct. 1984. D. L. Hutchins and R. G . Kouyoumjian, Asymptotic series describing the diffraction of a plane wave by a wedge, Ohio State Univ. ElectroSci. Lab., Techn. Rep. 2183-3, Dec. 1969. J. Boersma and Y. Rahmat-Samii, Comparison of two leading uniform theories of edge diffraction with the exact uniform asymptotic solution, Radio Sci., vol. 15, pp. 1179-1194, Nov.-Dec. 1980. E. L. Yip and R. J . Chiavetta, Comparison of uniform asymptotic expansions of diffraction integrals, IEEE Trans. Antennas Propa- gat., to appear. J. L. Volakis and M. I. Herman, A uniform asymptotic evaluation of integrals, Proc. IEEE, vol. 74, pp. 1043-1@44, July 1986. R. G. Kouyoumjian and P. H. Pathak, A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, Proc.

R. Tiberio and G. Pelosi, High-frequency scattering from the edges of impedance dscontinuities on a flat plane, IEEE Trans. Antennas Propagut., vol. 31, pp. 590-596, July 1983. R. G. Rojas, A uniform GTD analysis of the EM diffraction by a thin dielectric/ferrite half-plane and related configurations, Ph.D. disser- tation, Ohio State Univ., 1985.

IEEE, V O ~ . 62, pp. 148-1441, 1974.

An Optimized Polarization Sensitive Salisbury Screen

FRANK E. GROSS AND ERIC J. KUSTER

Abstruct-A transmission line equivalent circuit model for strip gratings is used to determine the performance of both conducting and resistive gratings. A polarization sensitive Salisbury screen is designed in which the space cloth is replaced by a resistive grating and the ground plane is replaced by a parallel conducting grating. The new Salisbury screen performs like the traditional screen for an incident E-field parallel to the strips, but is transparent for the perpendicular polarization.

I. INTRODUCTION

Salisbury screens are well-known devices for absorbing plane waves at a design frequency. The traditional Salisbury screen is polarization insensitive at normal incidence. By replacing the 377 $2 space cloth by a resistive strip grating and replacing the ground plane by a conducting strip grating, an optimized screen can be made transparent for (E perpendicular to the strips) and still perform as a Salisbury screen for (E parallel to the strips). The optimization and solutions will be accomplished by treating the Salisbury screen components as shunt elements in a transmission line model.

n. THEORY The strip grating geometry is depicted in Fig. 1. The general

subject of strip gratings is adequately treated by Larsen [I] and Lamb [2] and thus will not be expounded on here. The gratings can be modeled as transmission line shunt elements and the model can be used to optimize grating performance. The transverse electric (E) mode is when E is polarized parallel to the strips, the transverse magnetic (TM) mode is when E is polarized perpendicular to the strips.

A . Strip Grating Equivalent Circuits The strip grating has an effective impedance for each mode (TE or

TM). Thus a grating in free space can be modelled by a transmission line with a shunt impedance as seen in Fig. 2. The equivalent impedance for the grating at oblique incidence can be derived from the works of Marcuvitz [3], Redheffer [4], and Ram0 [5 ] . Marcuvitz derived the shunt reactance for a grating at normal incidence and for a limited range of parameters at oblique incidence. Redheffer demon- strated the angular dependance of the shunt reactance, and Ram0 showed the form of the shunt resistance for round wires when the skin depth is greater than the radius. The shunt impedance is given below for the TM and TE cases.

TM Case Z , = ( R + jX , ) sec 0

where the reactance is given by [3]

($) -I=? I n sec *+ 1 + Q Q sin4 cos4 \k \k

Manuscript received Februav 6, 1987; revised June 24, 1987.

MA 01730. F. B. Gross is with The MITRE Corporation, Burlington Road, Bedford,

E. J. Kuster is with Georgia Tech Research Institute, Georgia Institute of Technology, Atlanta, GA 30332.

IEEE Log Number 8717286.

0018-926X187/1200-1492$01.00 0 1988 IEEE

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-35, NO. 12, DECEMBER 1987 1493

The result is

4 wl+

0 a 0 0 0- r=-. - 2 0 2 0 + 22, (6) When 2, is substituted into (6) for the TE case with arbitrary R,, the results agree well with Hall and Mittra [6] at various angles of incidence.

B. Optimizing Strip Gratings Side View

TE I

Top View

Fig. 1. Strip grating geometry; w = strip width, d = grating constant.

I Fig. 2. Strip grating equivalent circuit.

with

Q= 1 41 -(d/X)2

- 1

Z = impedance of medium.

The resistance is given by

with

t thickness of strip S skin depth of strip u conductivity of strip R, impedance of strip.

The strips will be optimized for the application of a polarization sensitive Salisbury screen. The resistive grating will replace the traditional 377 Ci space cloth and the conducting grating will replace the traditional ground plane. The optimization will be performed at normal incidence.

Conducting Grating: To optimize the strip parameters, we first consider the conducting grating alone. The power reflection coeffi- cient at normal incidence is calculated by substituting the first-order approximation for (1) and (4) into (6). The first-order approximation retains only the first term in (2) and (5). (This results in an error of less than 10 percent when d/X 5 0.5. This greatly simplifies the optimization and yields very satisfactory results.)

,w\\\ 2

(2) =t [In csc *+ Q ws4 f 1 + Q sin4 f

with

RII reflection coefficient for E parallel to the strips (TE case) , Rl reflection coefficient for E perpendicular to the strips (TM

case).

The above reflection coefficients are plotted for various values of w/ d i n Figs. 3 and 4. It can be seen that for a proper selection of w/d, the grating can be a good conductor to the TE mode while being

The conducting grating can be optimized by defining the following: (3) nearly transparent to the TM mode.

Rn= 1 - A (9)

and

RL = A (10)

+ ( $)2 (1 - 3 sin2 9) cos4 f . ( 5 ) 1 Also, R is the same as for the TM case.

When the grid or strip grating is in free space, the reflection coefficient for TE or TM incidence, can be calculated by calculating the reflection coefficient for the equivalent transmission line model.

where

A is an arbitrarily small parameter.

This condition, after some algebraic manipulation, can be satisfied when w/d = 0.5. For a given d/X, w/d = 0.5 insures the maximum RII and the minimum R I . Substituting this result into (4), we derive a condition on d/ X such that

-,

We thus can determine the grid performance by the parameter A. As an example let R L = -20 dB then A = 0.01 and d/ X = 0.145 yielding RII = - 0.04 dB. RH and Tl for the optimized conducting

. -- I - . , . . . .

. '1 494 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-35, NO. 12, DECEMBER 1987

8 .2 .4 .6 .8 1.0 W D

Fig. 3. Reflection coefficient for E pal le l to the conducting strips.

0

-10

R(dBB) -28

-30

-48

5 8 8 .2 .4 .6 .8 1 .e

w/D

Fig. 4. Reflection coefficient for E perpendicular to the conducting saips.

grating versus frequency are plotted in Fig. 5. RII = TI when w/d = 0.5.

Resistive Grating: The resistive grating will replace the traditional space cloth and should be 377 Q for the TE case, but should look like an open circuit to the TM case. Thus, to optimize the resistive grating, the criteria are

1) Zg (TM)=jm

2) Zg (TE) = Zo. (12)

Using the first-order approximation for (1) and (4) at normd incidence, we have

Z , ( T " ) = R + J X , = R - j AZO (131 ?TW

4d In sec - 2d

dZ0 T W h 2 d

Z E ( T E ) = R + J X L = R + j - l n c s c - . (14)

Both conditions in (12) cannot be satisfied siinultaneously (unless d/ X = 0). The best compromise is when R = Zo and the imaginary terms for (13) and (14) are at least 10 Zo and 0.1 Z,, respectively. This is accomplished when d/X I 0.155 and when

Equation (15) results in w/d = 0.352. RU and TL for the optimized resistive grating are plotted in Fig. 6. The optimized strip grating coefficients are given in Table I.

Optimized Salisbury Screen: The Salisbury screen composed of the optimized polarizing strips is depicted in Fig. 7. The optimization

0.00 --r

-0.20 - m a 2 -0.40 E

-

u k4 w a

v

2 -0.60 -

E -0.80 -

-1.00 " ' " " " " ' ~ " " " 0 5 10 15 20

FREQUENCY (GHz) Fig. 5. A, and T , for conducting ships: w/d = 0.5, d = 7.5 mm.

-9.00;.~0.00 ----__ ----__ ----_ ---_ -.-_

h -9.20 9

---- -0.20 z a v

-lo.ooo - 5 10 15 20 -1 .oo FREQUENCY (GHz)

Fig. 6. A, and T, for'resistive strips: w/d = 0.352, d = 4.65 mm.

TABLE I OPTIMIZED STRIP GRATING FOR POLARIZATION SENSITIVE SALISBURY

SCREEN

~ ~ Type 1 .:< 1 ,5wid Conducting grating Resistive grating .155 .352

RESISTIVE GRATING

Fig. 7 . Optimized polarization sensitive Salisbury screen: conducting grat- ing-w/d = 0.5, d = 7.5 mm, resistive grating-w/d = 0.352, d = 4.65.

assumes that the coupling between the conducting and resistive gratings is negligible. It can be shown that for a separation between gratings of xo/4 the lowest order Floquet modes are evanescent and small. Thus, the no coupling assumption is valid for this application. The reflection and transmission coefficients for the optimized screen

1

-25 -

FREQUENCY (GHz) Fig. 8. Reflection and transmission for the optimized polarization sensitive

Salisbury screen.

are plotted in Fig. 8 with the ideal Salisbury screen results superimposed. The absorption for the TE mode at the design frequency is - 45.2 dB while the insertion loss for the TM mode is merely - 0.23 dB. Thus, the Salisbury screen is very effective for the purpose for which it is designed.

El. CONCLUSION

The Salisbury screen can be made polarization sensitive by replacing the 377 Q space cloth by a resistive strip grating and by replacing the ground plane by a parallel conducting grating. The gratings can be optimized to absorb TE incident waves but be nearly transparent to TM incident waves. The overall performance is limited by the grating constant d/X.

REFERENCES

T. Larsen, "A survey of the theory of wire grids," IRE Trans. Microwave Theory Tech., vol. MTT-12, pp. 191-201, May 1962. H. Lamb, "On the reflection and transmission of electric waves by a metallic grating," Proc. London Math. Soc., vol. 29, p p . 523-544, 1898. N. Marcuvitz, Waveguide Handbook. New York: McGraw-Hill, 1951. R. Redheffer, "The dependence of reflection on incidence angle," IRE Trans. Microwave Theory Tech., pp. 423429, Oct. 1959. S. Ram0 et ai., Fields and Waves in Communication Electronics, 2nd ed. New York: Wiley, 1984, pp. 181-182. R. Hall and R. Mittra, "Scattering from a periodic array of resistive strips,'' IEEE Trans. Antennas Propagat., vol. AP-33, no. 9, pp. 1009-101 1, Sept. 1985.

Low Angle Signal Fading at 38 GHz in the High Arctic

W. I. LAM

Abstract-In 1984 a microwave propagation experiment was conducted at 83'N latitude in Alert, Canada, to study the characteristics of low angle fading at a frequency of 38 GHz. By monitoring the continuous wave (CW) signal transmitted from the orbiting LIB-8 satellite, propaga-

Manuscript received January 23, 1987; revised May 19, 1987. The author is with the Department of Communications, Communications

Research Centre. Government of Canada, 3701 Carling Avenue, P.O. Box 11490, Station H, Ottawa, ON, K2H 8S2 Canada.

IEEE Log Number 8717291.

tion data were gathered over a range of elevation angles from 1" to 21". A total of three sets of measurments were made in the spring, summer, and winter. These allowed comparisons to be made of the seasonal character- istics of low angle fading in the arctic. The experimental data were examined with respect to the atmospheric conditions observed at Alert. The results presented include the variation of the median signal level with the elevation angle, cumulative distributions of the received signal level and fade rate statistics. The amount of signal fading increased rapidly as the elevation angle decreased. Fading was most severe in the summer which also had the highest fade rates. Very little fading was observed in the winter.

INTRODUCTION

As the elevation angle of a satellite-earth path decreases, the effects of tropospheric fading become increasingly severe. The use of low angle propagation paths is necessary, however, if coverage of high latitude locations via geostationary satellites is to be achieved.

The low angle fading of 4 and 6 GHz signals has been studied at Eureka (latitude 80"N, elevation angle 1") using the Anik A satellite [ 11. Measurements have also been made at Resolute (75"N) and Ottawa (45%) for a frequency of 7.3 GHz [2]-[4]. These measure- ments indicate that there is less fading in the arctic than at midlatitude locations, and that the fading is more severe in the summer than in the winter. An experiment performed at Spitzbergen (78"N) for 11.8 GHz also gave similar results [5]. At millimeter wave frequencies measurements of low angle fading have been made in Newfoundland [6], Virginia [7], and Texas [8] using the ATS-6 satellite but none have been reported for arctic locations.

During 1984 a series of measurements was made of the strength of a 38 GHz signal that was transmitted from the Lincoln Experimental Satellite (LES-8) and received at Alert, NWT. This communication summarizes the results of that experiment.

a. DE~CRIPTION OF EXPERIMENT A circularly polarized 38 GHz continuous wave (CW) signal is

transmitted from LES-8. The satellite follows an orbit which is inclined at 23" relative to the earth's equatorial plane and the orbital period is 24 h. When observed from earth, the path of the satellite forms a figure eight pattern in the sky.

The measurements were made with a receiver at Alert (82"30'N, 62"20'W) on the northern tip of Ellesmere Island, Canada, approxi- mately 850 km from the north pole. At a latitude of 82"N the satellite is accessible for approximately 11 h per day since only the upper half of its figure eight path is visible. During each pass the satellite rises from below the horizon to a maximum elevation angle of 21" and then sets back below the horizon. While the satellite is above the horizon its azimuth angle varies over a range of 6". The physical horizon for the receiver is at an elevation angle of about 1 because of blockage by hills along the path.

The receiving system was equipped with a 0.45 m diameter Cassegrain antenna which had a gain of 42 dB and a 3 dB beamwidth of 1.2". The antenna was mounted on a steerable pedestal which had a pointing resolution of better than 0.05". A preprogrammed tracking method was used to point the receiving antenna toward the satellite. For each day the elevation and azimuth angles at 10 min intervals were calculated from the satellite orbital elements and stored on magnetic tape. To control the antenna pointing, these values were used every few seconds by a computer to calculate the look angles by linear interpolation. Finally, accurate pointing was ensured by entering fine adjustments into the computer to maximize the received signal.

A steerable spot-beam antenna was used on the satellite to transmit the 38 GHz signal toward earth. Although pointing errors of this