Dual-beam parabolic antennae in radio astronomy: Theory and application of a method to decrease tropospheric influence on centimetre and miUimetre wavelength observations

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Dual-beam parabolic antennae

in radio astronomy

theory and application of a method to decrease tropospheric

influence on centimetre and miUimetre wavelength observations

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR.IR.C.J.D.M.VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE N A T U U R K U N D E , VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OP WOENSDAG 18 MAART 1970 TE 14 UUR

DOOR

Jacob Wilhelm Martin Baars

N A T U U R K U N D I G INGENIEUR, GEBOREN TE LIENDEN

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A C K N O W L E D G E M E N T

The main part of this work was done during the author's stay at the National Radio Astronomy Observatory (NRAO) in Green Bank, West Virginia, U.S.A. The NRAO is operated by Associated Universities, Inc. under contract with the National Science Foundation. Thanks are extended to Dr.David Heeschen, director of NRAO, for the opportunity to work in Green Bank, and to Dr. Peter Mezger for guidance in the initial stages.

Het proefschrift werd voltooid, terwijl de schrijver in dienst was van de Stichting Radiostraling van Zon en Melkweg, welke wordt gesubsidieerd door de Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (ZWO).

De schrijver richt zijn dank tot de Stichting Radiostraling van Zon en Melkweg voor een tegemoetkoming in de drukkosten van het proefschrift.

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Introduction

This thesis deals with problems in the technique of radio-astronomical observations with a single paraboloidal reflector antenna. In particular we are concerned with the radiation pattern of an antenna at finite distances from the aperture (the Fresnel region). Moreover, we shall allow the feeding point to be located away from the axis of the paraboloid. This problem resiilted from the conception of a novel technique of observation, in which two feeding horns are placed close together near the focal point of the reflector. The radiometer is arranged to switch rapidly between the two feed horns and to produce at the output a signal, which is proportional to the difference in the signals at the two feeds. This technique, originally devised by Dicke, eliminates the effects of instabilities in the gain of the receiver. In the standard radio-astronomical observation the switching is performed between a single feed and a matched resistor at constant temperature. In our dual-beam method we use two feeds, each of which projects a beajn on the sky. Thus we record the difference of the power entering the feeds.

Chapter I contains a short discussion of the paraboloidal reflector antenna as an observing tool in radio astronomy. After a historical introduction we give the terminology and definitions which will be used in the course of the text. Also we establish the formulae representing the response of an antenna to radiation.

Some attention is given to the standard techniques employed by radio astronomers in observations.

An interesting aspect of very large radio telescopes is the impossibility of calibrating their characteristics with the aid of an earthbound test transmitter. Reliable measurements require the trans-mitter to be located in the farfield region of the antenna. This distance

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is usually defined as 2D /X, where D is the diameter of the reflector aperture and X the wavelength. For telescopes of several tens of metres diameter operating at centimetre wavelengths this distance is several tens of kilometres. Radio astronomers have taken a natural step and use cosmic radio sources as test transmitters. Some problems connected with this method will be outlined.

The chapter closes with a description of some unusual methods of observation. The dual-beam method around which most problems in the thesis originate, is outlined in a qualitative fashion.

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The dual-beam arrangement provides, as noted earlier, two

adjacent beams. These will be separated by a certain angle on the sky. We are now able to observe the strength of a cosmic radio source with respect to its surroundings by pointing one beam at the source.

Unfortunately there are several phenomena which may cause unwanted signals in the telescope beam apart from the signal of the radio source. The radio astronomer tries to design an observing method in which he will be free from these interfering signals.

In recent years an increasing number of observations has been caxried out at frequencies from 5 GHz upwards to 100 GHz, in the centi-metre and millicenti-metre wavelength domain. At these high frequencies an

important source of unwanted radiation is the troposphere. The fluctuating thermal emission of tropospheric constituents, mainly oxygen and water vapour, may significantly reduce the sensitivity of the radio telescope. The dual-beam method was devised to improve this situation. It is

based on the notion that there will be a considerable beam overlap in the Fresnel region, where the fluctuating atmosphere is present, while the beams are separated in the farfield region. Thus cancellation of atmospheric noise may be achieved.

The theory of the paraboloidal reflector is the subject of chapter II. Ko rigorous treatment is attempted. We limit ourselves to

so-called "high frequency approximations", where the diameter and curvature of the reflector are large compared to the wavelength. A qualitative description of several approximative methods is presented.

A detailed analysis is given of the vectorial current distribution integration and scalar aperture field integration methods. The effects of the finite distance to the point of observation (Fresnel region) and the off-axis position of the feed are incorporated. In all cases it is possible to reduce the problem to a single integral, provided the amplitude term of the illumination pattern has rotational symmetry.

The final integrals of chapter II have been calculated by numerical integration on a digital computer. The results are given in chapter III. A comparison is made between the results of the current distribution and aperture field methods. The effect of the off-axis feed is similar for both methods. Only at small distances from the reflector, within the Rayleigh distance (D / 2 X ) , are the differences significant.

This chapter also contains a calculation of the degree of beam overlap in the Fresnel region.

In chapter IV we present a review of the structure of atmospheric turbulence. Based on existing theories we describe the characteristics of fluctuations in atmospheric radiation.

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To provide quantitative data on the strength of atmospheric fluctuations and the efficacy of the dual-beam method,observations have been made at wavelengths of about 1, 2 and 6 centimetre, both with a single-beam and dual-beam telescope. In chapter V we

describe the observations and discuss the results. A significant improvement with the dual-beam method is demonstrated. An analysis of the data in terms of autocorrelation functions and power spectral density functions yields the possibility of drawing some conclusions with regard to the structure of the troposphere.

A survey of the main results contained in the thesis and some concluding remarks form chapter VI. References are given in alpha-betical order of author's name and year of publication. The

reference list is not exhaustive. We mention books and articles on which we base part of our work, articles from which we quote specific results ajid literature which has historic value or which gives additional information on the subject.

We use the rationalized international system of units (SI-units) and we note that the unit of temperature is called "kelvin", with symbol K.

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Chapter I

The paraboloidal reflector as an observing

tool in radio astronomy

1. Historical review.

Radio astronomy is one of the young sciences. At the turn of the century some efforts to detect radio radiation from celestial bodies were made. Thomas A. Edison had suggested the possibility, and in Britain Oliver Lodge went so far as to perform the first experi-ment in radio astronomy. He used an open ended circular waveguide in

an attempt to detect radiation from the sun. The observation had a negative result due to lack of sensitivity of the "coherer" detector.

The birth of radio astronomy as a successful branch of science can be placed in 1932. Karl G. Jansky, a radio engineer at the Bell Telephone Laboratories in Holmdel, New Jersey, was studying problems of shortwave interference ("static") by thunderstorms. He used a

directional antenna at a frequency of 20.5 MHz and rotated the antenna in azimuth once per 20 minutes. With every revolution he noted a strong increase in output on his penrecorder. It caused "a hiss in the phones that can hardly be distinguished from set noise" (Jansky 1932).

The direction of maximum radiation changed with time of day and after sufficient measurements were made, a small daily movement was also apparent. This led Jansky to conclude that the radiation came from a fixed region of the firmament. As a matter of fact, the centre of our galaxy, which is a powerful radio source, lies within the boundaries deduced by Jansky from his measurements.

Jansky, althoxigh not an astronomer by education, understood the astronomical importance of his discovery. He also saw the need of a radio telescope with a narrow beam, which could be directed easily at any point of the sky. Thus, in 1935, he proposed to build a para-boloidal reflector, 30 metre in diameter and equiped with a receiver for metre wavelengths. His proposal did not find support. Jansky's discovery went unnoticed by astronomers, but another radio engineer, Grote Reber, in 1937 built a parabolic reflector with a diameter of almost 10 metre in his backyard. It could be moved about one hori-zontal axis, thus one could position it in declination. The dayly rotation of the eaxth moved the antenna beam in right ascension. I f was in effect a meridiantransit instrument. Reber (19^0, I9UU) made a contour map of the sky brightness at I60 MHz and became the first radio astronomer. He is still active in radio astronomy. He was also

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keen in studying the instrumental limitations and characteristics. It is not surprising that Reber too recognized the necessity of larger radio telescopes. In 19^8 he proposed a self-designed steer-able paraboloid of 67 metre diameter. Again no financial support could be found at that time.

A fortxmate circumstance for the development of radio astronomy was the tremendous effort in electronic research brought about by the second world war. Radar receivers and antennae often could be used as radio telescopes after the war. The 7-5 m diameter German "Würzburg" parabolic radar reflectors became very popular. For instance Muller and Oort's (1951) detection of the spectral line of interstellar neutral hydrogen at 21 cm wavelength was made with a "würzburg" antenna, and even today they are being used from Poland to England and from Norway to Italy. Soon, however, the need for larger telescopes became pressing.

We may make a remark at this point about the type of radiation most commonly received in radio astronomy. The strong radiation detected by Jansky, and later by Reber, is emitted by electrons traveling at relativistic speeds through the magnetic field of the Galaxy. We are dealing with a continuum radiation the main

characteristic of which can be written as

where S is the flux emitted by the source, v is the frequency and a is called the spectral index. A typical value of a = -0.7. It is obvious that the strength of the received radiation decreases quite rapidly with increasing frequency. Observations at higher frequencies will thus require larger telescopes. Both high frequency and a large diameter of the antenna increase the resolving power of the radio telescope, a need urgently felt by radio astronomers from the earliest days (Jansky 1935).

The first large paraboloid (D=66 m) was made in England in I9i^6. It was a fixed antenna in the form of a wire net on poles, the

reflector looking straight up. Only a narrow region of about 10 on the sky around declination 55 could be observed. An important

result of the instrument was the detection of weak radiation from our nearest galaxy, the Andromeda Nebula. It is evident that by doing away with the possibility of mechanical movement of the reflector one is able to construct very large antennae. The ultimate result of this ap-proach is the Arecibo 300 metre diameter dish (Gordon and La Londe I96U). By use of a spherical rather than a parabolic reflector one has achieved in this case the possibility of a sky coverage without severe coma effects

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of about 20 around the zenith by movement of the feed.

Generally a fully steerable telescope is greatly preferred, even with a smaller diameter. Many parabolic reflectors were built in the late fifties. Among them is the Dwingeloo 25 metre telescope, which for a short time was the world's largest antenna until the Jodrell Bank 76 metre reflector came into operation in 1957» That is still the largest, fully steerable radio telescope in existence. The rather poor quality of its reflecting surface renders it useless for observations at wavelength below 20 centimetre. Hence its importance is not as great as more modern, but smaller telescopes as the Austra-lian 63 metre telescope at Parkes (Bowen and Minnett, 1903) or the American k3 metre antenna at Green Bank (Small I965, Baars and Mezger,

1966), which incidentally is the largest equatorially mounted telescope. Several telescopes axe built as partially steerable antennae. Generally they axe meridian-transit instruments, steerable only in declination. The largest telescope of this type is the 92m antenna at Green Bank

(Findlay, I963).

The need for larger telescopes at relatively moderate cost has resulted in special types of antennae, which however we shall omit from our discussion. A more complete discussion of radio telescopes can be found in an excellent recent book by Christiansen and Högbom (1969). The advent of the principle of "aperture synthesis" (Ryle I962), in which a large telescope is synthesized by combining many

obser-vations made over a long time with an interferometer consisting of two or more small antennae, brought renewed interest in the fully steerable paraboloidal reflector. In the very young branch of milli-metre wavelength radio astronomy no other than parabolic dishes are being employed at present. The parabolic reflector antenna is no doubt the most straightforward and versatile instrument for radio astrono-mical observations. Its capabilities have not been fully explored as we shall see. But its simplicity is clearly sufficiently attractive to make it the most widely used observing tool in radio astronomy. In this thesis we intend to study some of the less familiar aspects of the theory. The theoretical results will indicate the possibility of \inusual techniques of observation. Experimental verification of the possibility has been obtained and fozms an integral part of our work.

2. Radio Telescope Characteristics

Radio telescopes form a special kind of antennae. Many new results in antenna theory and practice have been obtained by radio astronomers, rather than antenna engineers. The task of observing the

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distribution of sky brightness created unfamiliar problems connected with the antenna. The calibration of large radio telescopes by means of cosmic radio sources has almost entirely been developed by radio astronomers.

This section contains the basic formulae of the interaction be-tween the celestial radiation and the antenna. The terminology to be used in the thesis will be outlined. It is hoped that after reading the present section both astronomers and antenna engineers will feel

sufficiently at ease with the jargon of radio telescopes to read the rest of the thesis or other literature concerned with the subject.

2.1 The "Radio Sky"

The 'i:adio sky" looks very much like the "optical sky". There are numerous discrete sources (in the early days they were called radio stars), both belonging to our galaxy (Milky Way) and outside. The bright band of the milky way, as we see it along the sky, is also present at radio wavelengths. Actually radio radiation comes from the whole of our galaxy, not only from the optically bright plane. However, stars are not radio emitters of any importance. We do observe strong radio radiation from the sun, but only because of its extreme proximity. At the average distance of a star- its flux density at the earth would be below our present detection limit. It would , moreover, completely be obscured by the general "background radiation" of the galaxy. This radiation is being emitted by electrons traveling at relativistic speed in the magnetic field of the galaxy. It is called "synchrotron radiation" or also"Magnetobremsstrahlung", or simply non-thermal radiation. The last name distinguishes it from thermal radiation which is characterized by an equilibrium temperature (blackbody radiation). The free-free emission of clouds of ionized hydrogen (H-II regions), of which many are present in the galaxy, also is called thermal radiation, A well-know specimen is the great nebula in Orion. Discrete sources

(i.e. sources confined to a small and sharply boimded region) of the non-thermal type can also be found in the galaxy, the remnants of super-nova explosions. A famous example is the Crab Nebula.

The many extragalactic radio sources axe all of the non-thermal type. It are in fact galaxies themselves. About 9000 are at present catalogued and many have been identified with objects on optical photo-graphs,such as galaxies or quasi-stellar sources.

The foregoing very short summary of what we "see" in the sky at radio wavelengths may suffice to indicate the primary requirements we demand of a radio telescope. We may wish to make a map of the

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bright-ness distribution over the sky to study the structure of our galaxy. We shall want to measure the strength of the discrete sources above the surrounding background, where the background itself in many cases will not be free of structure. If the telescope has sufficient

resolving power, we shall also want to make pictures of the bright-ness distribution over the extra-galactic sources.

In order to be able to draw meaningful conclusions about the celestial so;u:ce, we need to establish the relationship between para-meters which are characteristic of the physical process in the source and those of the receiving antenna. In other words, we must develop a mathematical formulation for the interaction between the transmitting cosmical radio source and the receiving radio telescope on earth.

2.2 Basic antenna parameters

The fundamental characteristic of any receiving or transmitting antenna is the spatial response to radiation (incoming or outgoing respectively) in different directions. It has the name reception- or radiation-pattern, and by virtue of a reciprocity relation these are identical. Often only the term antenna pattern is used and terms like "radiated power" occur, although the discussion is limited to receiving antennae.

Because we shall use the pattern characteristics throughout the thesis, we introduce their definitions at this point. In analogy to the well-known diffraction pattern of a circular hole in a black screen the pattern of a paraboloidal reflector antenna at a very large distance exhibits a strong lobe in the direction of the telescope axis, the main beam, and weak sidelobes, concentric around the main beam. We denote the

angular dependence of the pattern about the direction of the axis by f(e, 4)) and normalize, so that f(0,0) = 1. Here 9 and <}> denote the eingalax variables of a spherical coordinate system about the axis of the antenna.

The directivity D(e, (j> ) is defined as the power emitted (or received) per unit solid angle in the direction (6,4 ) divided by the average power over a unit of solid angle:

D(6,<>)= ^1-1)

-|^JJf(6,*)dfi

In the direction of the main beam axis we have the maximum directivity, often called "the directivity",

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D = (1.1a) //f(6,<|))dn

The integral in the denominator has the name effective antenna solid angle (or pattern solid angle):

" A " JJf(6,<^)dfi, (1.2)

and thus we write

«A = ^^/Dj^ ^^-3^

The antenna solid angle is difficult to determine because the radiation pattern f (9, (j) ) must be known over the entire sphere surrounding the antenna. Formally we can separate ft. into several parts, e.g. one over the main beam, ü , the other over the sidelobes or stray region, ft . Thus

ft. = ƒ ƒ f(e,(}))dn+ ƒ ƒ f ( 9 , (j) )dfi = fi^ + ftg . (I.it) main stray

beam region

An important antenna parameter is the main beam efficiency

ng = ftj^/fi^ <1 . (1.5) It is the percentage of all received power that enters the main beam.

It will be appreciated that this parameter is a direct indication of the efficiency of the antenna when it receives radiation from a source with an angular diameter equal to the angle subtended by the entire main beam. The concept of beam efficiency is not necessarily restricted to the main beam. Sometimes it has been found convenient to define a beam efficiency, where the "beam" is taken up to a certain angle from the axis and may contain one or more sidelobes. Observations of the moon

(angular diameter 30 minutes of arc) with a narrower beam form an example. Most authors introduce a parameter G, the gain of the antenna, which is written as

G = TigD. (1.6) Here HT, is the radiation efficiency, which represents the ohmic losses

in the antenna reflector and feed. In practice n^ is very close to one. We shall ignore it in the further discussion and only use D as the

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significant parameter. If so desired, n^ can always be considered to be incorporated into the beam efficiency ng. There is another way to characterize the antenna, viz. by the effective absorption area

A{6, <\>). It is defined as the power available at the antenna tenninals divided by the power crossing a unit area of wavefront. Just as the pattern f(9,ij) ) and the directivity D(9,(f> ) , the effective area is a f-unction of the angle from the axis. Generally only A ( 0 , 0 ) is used, the effective absorption area in the direction of the main beam axis, denoted by A. In case the antenna exhibits a physically present capture area to the incoming wavefront, as is always true for reflector antennae, we can define the aperture efficiency n. as

n^ = A/Ag , (1.7)

where A is the geometrical area of the reflector aperture. The S

absorption area is typically a reception parameter, while the directivity pertains to the transmitting mode of the antenna. Through the reciprocity relation there must be a linear relation between A and D. In fact the relation is

D = luT A/X^ _ ^ (1.8)

where X denotes the wavelength. We shall give a proof of the last relation later in this chapter. From (1.3), (l.5)j (1.7) and (1.8) we readily find a relation between n„ and r\^:

Use of this relation is sometimes made in antenna measurements.

2.3.Response of an antenna to radiation from sources distributed in space

We proceed with an important aspect of the general theory. Let us assume a radiating body at temperature T, the thermal radiation of which is partially intercepted by a receiving antenna. The source and the medium between the source and antenna are taken to be homo-geneous and isotropic, linear and without losses.

Our starting point is Rayleigh's reciprocity relation, which may be stated and proved generally as follows. Let there be in space two sources of monochromatic electromagnetic power. From the foregoing assumption the intervening medium will have such properties that D. = e. E, and B. = U-,H, with e. and y. symmetric tensors of permittivity

Ik K 1 IK K IK IK ,

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vmderstood and suppressed t h r o u g h o u t . Maxwell's e q u a t i o n s a r e

Vx E = + iü)B ( 1 . 1 0 ) Vx H = - iuD + j ^ .- ( 1 . 1 1 )

where j is the current density in the source, the other symbols having their usual meaning. Multiply (l.io) and (1.11) for source 1 with Hp, E_ respectively (index 2 means: field of source 2) and (I.IO) and (1.11) for source 2 with -H.. and - E ^ . Then add all h equations to obtain

[H2.(VxE^ )-E^ .(VxHg)] + [ 4 . (Vj^., )-H^(VxEg)J =ia) [(B^ .H^-H^ .B^)+ (II -Dg-D^ .Eg)] +11^ .Eg-j^ .E^ ] .

The first two terms of the right hand side vanish because of the form of e and u, while the left hand part can be written as

div[E^ X Hg - Eg X H^] - [ii.l2 " ^-^^^ ' (l.12) We integrate (1.12) over all space and use Gauss' theorem:

ƒ div [E^xHg - EgXH^] dv = ƒ [E^xHg - E^xH^] .n dS = ƒ [ J.TE2-J_2.E^] d v V S V The surface integral tends to zero if S-x», while the volume integral

at the right side is confined to the sources where j ?^ 0. Thus we find

J

i^.Eg dv = y j^.E^ dv, (1.13)

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where the integration is restricted to the volumes containing a non-zero current density. This is a formulation of the reciprocity

relation. Clearly the field produced at 2 multiplied by its generating ciirrent at 1 equals the field at 1 times its generating current at 2 in case of "delta function" sources in points 1 and 2.

We apply this relation to the antenna and a black body. Thermal fluctuation ciurrents j distributed in the source will give rise to a field E in space. Ciorrent densities j assumed as "delta fixnction"

O Or pulses in a volume element dv of the antenna generate a field E . We have now

v v

a s

where V^ and V^ denote the volume occupied by the antenna and the source respectively. We multiply each side of (I.IU) with the complex

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ƒ .n^-^^^ja'-^)^^^^'= ƒ ƒ

(4-V(jJ"^)^^^^'-V , (4-V(jJ"^)^^^^'-V (4-V(jJ"^)^^^^'-V (4-V(jJ"^)^^^^'-V^, a' a s s'

Since we have taken the current sources j as independent pulse functions in space, the integral becomes, after taking the average over the current sources,

ƒ ƒ 4(ri).j^.(rg) 6(rg-r^) E^.E^ dvdV ,

v_. v.

a

where ö(rp-r^^) is the 6-function. One integration can immediately be performed with the result

V a

In the source we have assumed thermal fluctuation currents j . From the theory of fluctuations we use the following important result

(e.g. Landau-Lifshitz, (I960) vol.8, §87, 88): the crosscorrelation of the fluctuation currents at two points with radius vectors r^ and r^ from the origin is given by

J ^ i ^ ï T l ^ = ^ kT e" 6(rg-r^) , (l.l5) where e" is the imaginary part of the permittivity, k is Boltzmann's

constant and T the absolute temperature of the body. With (1.15) the righthand side of the double integral transforms into

f — kT e" |E I dv.

J IT ' a '

V s

Thus we have now obtained a relation indicating the mutual interaction between the power emitted and received by the antenna and the distant source:

V v a s

The voltage U at the antenna output due to the currents j is equal to E , the field strength, times the "effective antenna length", while the total current I , driving the antenna, is the product of the current density j and the "effective antenna cross-section". Thus we can change the left side of the equation to obtain

^ ,2^2 ^2a>c::k ƒ ^ 1^ |2 ^^

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By definition of the radiation resistance R of an antenna, the total radiated power P is given by

Also from Poynting's theorem for complex harmonic fields the divergence of the Poynting vector is

div S = - ^ E.E*. (1.18) Introduction of (1.17) and (1.18) in ( I . l 6 ) yields

R

| U J ^ = - ^ ^ J T d i v S d v . (1.19)

As first introduced by Nyquist (1928) we can assign to the antenna, more precisely to the radiation resistance R , a temperature which

^ n

is commensurate with the mean squared voltage |u | at the antenna terminals. Since any resistor R at physical temperature T generates a mean squared noise voltage (over the whole spectrum u)

"~2 2 U = - RkT,

U IT

we can define a quantity, the antenna temperature T., due to received radiation as

lu 1^ = - R k T.. /, „»N

' s' IT r A 11.20; IN.B. The familiar form U = Uk TR is based on a spectrum in v

2 2 T

w i t h V = a)/2Tr f o r which U = 2TfU . S u b s t i t u t i o n of ( l . 2 0 ) i n ( l . 1 9 ) g i v e s

^A ^ " P ƒ '^ <ii^ ^ ^'^- ( 1 . 2 1 ) V

s

From vector analysis we know that

T div S_ = div (TS) - S.grad T,

and making use of Gauss' theorem we change (1.21) into

T^ = ^ { ƒ T (S.n)ds + ƒ S grad T dv |. (1.22) S V

s s

where S is the surface of the source body and n the unit vector normal to S and directed inwards.

s

From (1.22) we conclude that in the general case of an antenna and a body at temperature T, separated by an arbitrary distance, we find the antenna temperature due to the radiation from the source as an

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integral over the surface and volume of the body. This is

physicaJJLy reasonable as the radiation pattern of the antenna, i.e. the Poynting vector S., has a three-dimensional structure at finite distances.

The volume integral in (1.22) vanishes in case grad T = 0. This happens if T " constant throughout the radiating soiirce, and also if the cuQgular extent of the source is significantly

greater than the width of the reception pattern. In such circumstances we may taJce to a sufficient approximation that T * constant over the beam.

The last point is important for radio astronomy in general and for our subject in particular. The antenna will receive thermal radiation from the ground and the atmosphere, both of which clearly axe emitters at a finite distance from the antenna. However the temperature of these bodies is generally sufficiently constant over a large region in space to assume that grad T = 0 over the antenna pattern. Hence in analyzing the atmospheric and terrestrial radiation we may ignore the volume integral of (l.22).

Cosmic radio sources are always at extremely large distances, and the antenna pattern has the structure of a locally plane wave in the Fraunhofer region of the antenna. Now we expect T to be defined only by the integral of the source temperature T over the surface as seen from the antenna. In (1.21) is the fraction of

radiated power that is absorbed in the volume element dv. If we consider a volume element dv = dadr with temperature T at a large distance r from the antenna in the direction (9,(t>), the fraction of absorbed power can also be written as

where da „„ = a da is the effective absorption cross section

perpendicular to the Poynting vector and a the absorption coefficient. With the directivity D(9,(()) as defined in (l.l) it is clear that the Poynting vector has a magnitude

S = - ^ — D(9,4)). , 2

Thus we can now write for the absorption in dv at a large distance divSdv S , D(6,(|))

- - ^ — = p d a ^ ^ ^ = ^ - - f a d a .

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t e m p e r a t u r e

dTA = —^ T D(9,<(') a d o . ( 1 . 2 3 ) Uirr

2

Now da/r = dfi, the solid angle under which the source element is seen. Integrating (1.23) over the total extent of the source U we obtain

ƒ a ï M e ^

^,

(I.2U)

A J kv

s

At this point it is convenient to introduce the concept of brightness temperature T , a quantity widely used in radio astronomy. It is defined as the temperature of a black body radiator the surface brightness of which is equal to that of the radio source under

investigation. The brightness B is the power AP per frequency interval V to V + Av which crosses an area AA normal to the waves travelling in a solid angle Aft. Thus

B = '^

AA Av Afi

Application of the Rayleigh-Jeans law gives the brightness temperature as 2

^B = Ik

^-Returning to our discussion we note that a body at temperature T and with absorption coefficient a, according to the above definition, will have a brightness temperature T^ = aT (Kirchhoff's l a w ) .

B

Substituting (l.l) for D(9,(1>), we can change (l.2l+) into T^ = ƒ Tg(9,<fr)f(e,<t>)dd / ƒ f(9,<j))dfl,

or with (1.2):

"s ^^

^A = ^ ƒ V ^ ' * ) ^ ^ö'*)^- (1.25)

Thus the antenna temperatxure is the brightness temperature weighted by the reception pattern of the antenna. We see that the concept of brightness temperature is strictly speaking only valid if the source is in the Fraunhofer region (far-field zone) of the antenna. However, in view of the discussion following (1.22) we may apply the same quantity to characterize the radiation from the atmosphere and the ground in the Fresnel region.

In case the brightness temperature T^, is constant over the solid angle of the source, we can modify (1.25) to obtain

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T^.^T,. (1.26) where U = f f(9,<t>)dfl. sm J Ü s

If the source is larger than or equal to the main beam in angular extent, (1.26) can be written as

^A = ^'B"B • ^^-2^)

The beam efficiency n'^ is calculated up to the solid angle of the source. If the antenna pattern consisted of the main beam only without sidelobes, we would have T. = T\^'I^ for all source where n > fi . Then,

' A B B s m

as soon as the source "fills" the beam, a further increase in source dimensions would leave T. unaltered.

If n «Ü , so that we may assume f(9,(f>) = 1 over Ü , (1.26) changes to o o

^ T = -2 B n

A m

''A = t''^ = f%%'

(^-28)

where HT, is as defined in (I.5). For vanishingly small values of Q.

a s

(1.28) breaks down, while moreover the concept of brightness temperature becomes unrealistic. We take recourse to the concept of flux density S in this case and we call the source a point source.

The flux density at frequency v is defined as the integral of the brightness B over the solid angle of the source:

ƒ Bdn = ^ ƒ T^dft, (1.29) S

^ • X ft fi

s s

and is a measure for the total radiated power. To find the antenna

temperature of a point source with flux density S located in the direction of the beam axis, we make use of the effective absorption area A. The power entering the antenna obviously is S A. From the discussion following (1.19) it is seen that the power delivered to the radiation resistance of the antenna is U /UR = kl

I s '

a spectrum in v) and thus we conclude that

resistance of the antenna is U I /UR = kT. (we are now considering

' S ' A

T. = S A/k , ' (1.30) In (1.30) it is tacitly understood that the polarization state of the

antenna matches that of the received radiation. The antenna radiation is generally linearly polarized. Radio sources, however, emit in general randomly polarized radiation (noise-like signals). Thus upon

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reception by a linearly polarized antenna, only half of the incident power will be in a matched polarization state. Consequently half of the total incident power will be transferred to the transmission line from antenna feed to receiver. Now (l.30) is modified to

T^ = S^A/2k . (1.31) This is the usual form found in text books of the relation between

flux density and antenna temperature.

We conclude this section with a proof of the relation (1.8)

f^ D = UTTA/X^. N^

Clearly this formula exhibits tie "reciprocity between the antenna radiation pattern D(è7$T^nd the reception pattern, expressed as the effective absorption area A(8,<fi). It follows directly from (l.2U),

(1.29) and (1.31). An interesting proof of general validity can be based on thermodynamic considerations. Consider an antenna with a matched resistor at temperature T connected to its terminals. Let there be a voltage generator in series with the resistor. From the reciprocity theorem we know that there will be power transfer from a received wave to the resistor and from the generator to the antenna as well. If we place a black body, also at temperature T, in space, subtending a solid angle fl at the antenna in a direction in which the directivity is D, we can write for the power radiated by the antenna towards the black body in a frequency interval Av

P^ = kT Av Dfi/l+ir.

On the other hand, from the radiating black body a power P will be received by the antenna and delivered to the matched load, which equals

P^ 4 ( ^ )^.

A

The factor — results from the fact that only half the power is matched to the polarization state accepted by the antenna. Because everything is at the same temperature, a condition of thermal equilibrium exists and the principle of detailed balancing requires that P, = P ,

Thus we obtain

D = UTTA/X^.

It should be noted that no use has been made of any detailed specification or special type of antenna. The result applies to any antenna, as long as A can be defined in a meaningful way. In fact the formula may be used to assign a value A to any antenna for which D is known, even without a physical capture area (e.g. a single dipole).

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2.k The calibration of large reflector antennae

The advent of large reflector antennae has brought about hitherto unknown problems in the calibration of the pattern parameters. It is clear from our discussion in section 2.2 that a complete

quantitative knowledge of the radiation pattern enables us to determine all relevant parameters. Unfortunately there are a number of reasons which prevent both the calculation and accurate measurement of the complete pattern of a large antenna. In the theoretical calculations approximations have to be introduced to make numerical evaluation possible. Also influences of the mechanical structure, as for instance aperture blocking by focus box and feed support leg, have to be

incorporated. We shall have a closer look at these theoretical problems in the following chapter. On the experimental side one obvious difficulty is the large dynamic range required to meas\ire the pattern of a large

h 7

antenna. Values of D,, between 10 and 10 are normal. Thus dynamic M

ranges of the order of 50-80 dB are required; a tremendous instrumental task indeed.

A more fundamental problem arises because a calibration transmitter must be located in the "far-field zone" (Fraunhofer region) of the antenna to make a reliable analysis of the measurements possible. The division of space into several "zones" will be described in later chapters. For our present purpose it is sufficient to note that we

2

define the farfield zone as starting at a distance R„ = 2D /X from the antenna, where D is the aperture diameter and X the wavelength. For D = 75 m and X = 20 cm we find R - 56 km and for D = 25 m, X = 6 cm we obtain R - 20 km. Generally it will not be possible to locate an earthbound transmitter at that distance and still maintain a reason-able angle of elevation of the, necessarily unobstructed, line of sight. Radio astronomers have overcome the difficulty by using celestial radio sources as calibration transmitters. The problem of the distance is clearly overcome this way. The dynajnic range, however, may be decreased because of lack of sufficiently strong radio sources. Moreover we have introduced another grave unknown, viz. the strength of the cali-bration source.

One of the main purposes of observational radio astronomy is the determination of the flux density of radio sources over a large range in frequency. Obviously, a radio astronomer wanting to calibrate the directivity qf his telescope with a radio source in order to determine flux densities finds himself in a vicious circle. The difficulty is overcome by establishing the flux densities of a few, strong sources on an absolute basis. Thus a small antenna is applied, the directivity of which can be calculated theoretically with confidence or is determined

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experimentally with the aid of earthbound calibration sources.

In the former category fall horn antennae, where agreement to 1 percent between theoretically calculated and measured directivity has been obtained. Experimental determination of the directivity for a small antenna can be performed with a test transmitter of accurately known power output at a so-called "test-range". Accuracies of 1-2^ are feasible. An interesting approach was developed in Russia, in which a black disk of angular size comparable to the main beam of the

antenna is placed in the fax field zone. Probable errors in the measiured directivity of 5^ are claimed with this method. These antennae with a well-established value of the directivity provide us with accurate and absolute flux densities of a small number of sources, sufficiently strong to yield a good signal to noise ratio in spite of the smallness of the antenna. The celestial calibration sources now enable us to determine the directivity of a large radio telescope and hence to find the flux of weak sources. As a matter of general procedure fLux densities are determined by measuring ratios with reference to a "calibration source" and using the absolute flux density of the calibrator.

Flux density determinations on an absolute base are difficult experimentally and time-consuming. The antenna collecting area generally is calibrated only at a small number of frequencies. Using absolute flux density measurements at several frequencies, one can determine the "absolute spectrum" of the source, i.e. the relation between the flux density and the frequency. By interpolation one finds the expected flux density at a chosen frequency. This method can be used because the spectra of radio sources are very smooth and lack structure over relatively large frequency intervals. For the purpose of calibrating radio telescopes at centimetre wavelengths the

absolute spectra of a few strong sources were calculated entirely from available absolute observations by Baars, Mezger and Wendker (1965). Based on error estimates in the obsejrvations the flux density at any frequency (between 300 MHz and 15 GHz) can be computed with a standard error of approximately 2%. The spectra are displayed in figure 1.1.

The availability of a set of calibration sources enables observers with different instruments to assemble their observations at many

frequencies into a consistent system of flux densities. This, of coiirse, is of importance for radio source spectra studies. The absolute flux density scale of a reasonable sample of radio sources, and hence the directivity calibration of large radio telescopes, can now be determined to an accuracy of a few per cent. Thus with care flux densities should be measured generally to that accuracy. The best catalogues estimate their absolute accuracy to about 5 per cent.

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10" 3«10 10* U) ] 2 3x10' UJ o X 3 10' -3M10* — 10' 1 -1 1 1 1 1 1 (1f.u.=1C""Wm-'Hi"^) \ ^ V CASSIOPEIA-A \ ^ ^ \ S|GH,»3110f.u.(1964,<) \ ^ ^ \ O - - 0 . 7 6 8 CYGNUS-A \ . \ ^ ^ S,sH..2200f.u. \ \ 0 . - 1 . i e 5 ( 1 . 5 < V < 1 5 G H z ) \ \ ^ ^ 1 1 1 1 1 1 -V 1 — " 10 30 10' 3x10^ 10' 3x10' 10' 3x10' 10' FREQUENCY(MHz) 10' 3x10 -10' z 3x10' UI o 10' -30 10 1 1 1 1 1 1 1 VIRGO-A \ S,sH,x277f.u. a < - 0 . 8 1 5 1 1 1 1 TAURUS-A S,sH.-973f.u. a . - 0 . 2 5 5 \ 1 -V 1 10 30 10' 3x10' 10 3x10 FREQUENCY(MHi) 10" 3x10 10

Figure 1.1. The spectra of four strong discrete radio sources, derived from observations with absolutely calibrated antennae. Thus not only the slope a of the spectrum, but also the absolute values of S axe known with high precision (2-3^ for Cassiopeia-A and Taurus-A; 3-5^ for Cygnus-A and Virgo-A). The flux density of Cassiopeia-A has a secular decrease of approximately

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The directivity D„, of course, is only one antenna parameter among several important characteristics. For accurate measurements of the brightness temperature as complete as possible a knowledge of the pattern is necessary. Some attempts have been made to measure the pattern with the aid of a monochromatic transmitter at a distant elevated point (TV tower or hill top) (see e.g. Higgs, 1967)- The results are generally not too reliable, and levels of -U5 to -50 dB with respect to the main beam form the sensitivity limit. A severe disadvantage of a transmitter at a low elevation angle above the horizon is the occurence of reflections at the ground. It is difficult to calculate reliable correction factors for this effect. Much better results are obtained with the following method. Again we use as transmitter in the far-field a radio source with well-determined flux density. Because its strength generally will not enable us to make a good measurement at levels more than 25-30 dB below D^^^^ with a straight-forward method, we can use a second telescope to form an interfero-meter with the antenna under investigation.

It is well known from general theory that the response of an inter-ferometer to received power is proportional to the product of the far-field (voltage) patterns of the individual elements. Let us now change the orientation of the antenna under test with respect to the source and keep the main beam of the auxiliary antenna directed at the source. The variation in output of the interferometer is proportional to the field-pattern of the test antenna. Since in general we are interested in the power pattern of the antenna, we see that a -UO dB sidelobe will still give a 1 percent output signal in the interferometer. Because the second antenna is always receiving maximum power, it can often be of rather small size.

There is another advantage in the interferometric method. Unwanted radiation (e.g. from ground or man-made interference) entering the antenna beam will generally not give any output signal because of difference in propagation time to the two interferometer elements. The method has been applied by several workers, notably by Hartsuijker, Baars, Drenth and Gelato (1970), who made a detailed measurement of the complete reception pattern of the Dwingeloo telescope at a frequency of IU1U MHz. They determined the pattern strength with good reliability to a level of about 60 dB below the main beam. Unfortimately the method can only be used at observatories where a second, small, antenna is available. We have dwelled at some length on the problem of radio telescope calibrations in a qualitative way. To conclude this section we direct the reader's attention to the excellent review article by Findlay (I965): "Absolute Intensity Calibrations in Radio Astronomy". Two other works

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on measurements of antenna characteristics deserve mention: a review article by Tseytlin (I965) and a book by Kuz'min and Salomonovitch (1966).

2.5 Imaging properties of paraboloidal reflector

Compared to an optical telescope the usual radio telescope exhibits remarkably poor imaging properties. The reason is twofold. First, spherical reflectors are commonly used for optical telescopes. The image is sharp along a circle through the centre of the sphere. However, the radiation is not concentrated at a point-focus, but rather along a part of the caustic. The parabolic reflector on the other hand focusses the radiation from the direction of the axis at one point. But as soon as the direction deviates from the main axis, the concentration of the radiation rapidly declines and no sharp image is formed.

A second difference between optical and radio telescopes is the ratio of focal length to diameter of reflector (the F/D ratio). Optical telescopes typically have a "prime focus" F/D ratio of 3, while F/D = O.U is common with radio telescopes. Thus the effect of coma is much more serious with radio telescopes than with optical ones. This means that the quality of the image for off-axis radiation is poor in a radio telescope for very small displacements from the focal point. In other words, the beam of a radio telescope with "out of focus feed" has a strong coma side-lobe for small feed deviations. These circumstances give the optical telescope a "field of view", i.e. a region of good image, of the order of a few minutes of arc with a resolution of the order of 1 second of axe, thus providing a large number of independent image elements. On the other hand, the radio telescope has a field of view equal to the size of its beam; it yields one image element. Adjacent points on the sky are observed by movement of the telescope.

In a few instances attempts have been made to increase the size of the image in a parabolic antenna by putting several feeds, each with its own radiometer, near the focal point. Due to the strong coma

effect not more than one or two points at either side of the central focus can be used with confidence. A large improvement can be obtained with radio telescopes of the Cassegrainian type. A second, small reflector of hyperbolic shape near the prime focus of the paraboloid concentrates the radiation at a point near the vertex of the para-bolic reflector. The focal point of the paraboloid coincides with one focus of the hyperboloid; hence the radiation is focussed at the

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other focus of the hyperboloid. Such a system behaves as a prime focus antenna of much larger focal length, and consequently the coma effect is an order of magnitude less serious. The possibilities for image extension in parabolic antennae have not yet been fully explored and deserve more attention.

3- Observational Procedures

3.1 Standard methods of observation

We proceed with a short description of some methods of

observation which axe more or less standard in radio astronomy practice. Normally the telescope is equiped with a single receiver and feedhorn. Thus a pencil beam with its side-lobe pattern is directed towards the sky.

The simplest way of observation is with a stationary antenna. The sky drifts through the antenna beam and we record a "drift curve". This method is suitable for purposes of mapping extended areas of the sky. One advantage is that no varying radiation is received in side and back lobes from the ground. A severe drawback is the very slow operation due to the small drift velocity of the sky. Observations can be speeded up by scanning the antenna through the region of interest with some known speed. Now the antenna moves also with respect to the earth. If the source is very weak, we may need a long integration time in order to bring the signal above the noise fluctuations of the

receiver. This can be achieved by tracking the source. Thus we move the telescope exactly opposite to the earth's rotation. If we observe a small, discrete source, we need its level above the general back-ground. So after tracking it for some time, we track a point in the "blank" sky near the source (2 beamwidths away, say). The difference in received power is a measure for the source strength. This procedure is often called an "on-off" observation. It can be of advantage to repeat an "on-off" cycle several times to increase the effective

"integration time" of the observation.

3.2 Other, more special, procedures; the dual-beam technique An unusual type of radio telescope or a particular observation may lead to special methods. Their purpose can be brought under one or more of the following:

a. increase in the speed of observation, c.q. observation of a larger area of sky, if the sensitivity of the radio telescope is ample for the purpose of the programme;

b. increase of the sensitivity of the telescope with a given receiver; c. removal, or reduction, of unwanted environmental influences, as e.g.

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from ground and atmosphere.

In category a we mention a method applied with meridian-transit instruments. By rocking the telescope up and down in elevation ("wobbling") a larger area is observed in declination. Of course, the time spent at each point is less and hence the sensitivity of the observation decreases. By stacking several feeds vertically near the focus, each with its own receiver, the sky coverage of a telescope can be increased without motion in elevation. With a transit telescope a given source can only be observed once a day for the short time it takes the soiurce to "drift" through the beam. In one experiment this time was increased from Uo seconds to about 5 minutes by moving the feed about the focal point at the correct rate. It is worth-while to note that this rate is not the sidereal rate of the source but somewhat faster. The reason for this is the fact that the angular displacement of the antenna beam on the sky is less than that of the feed by a factor ("beam deviation factor") which depends on the F/D ratio of the antenna. This example falls under category b.

Clearly both a and b can be achieved simply by using more observation time. The problems of influence from ground and atmos-phere are more basic and hence examples which fall in category c^ have greater intrinsic importance. It is obvious that the variations in antenna temperature due to atmospheric and ground radiation will depend strongly on the elevation angle of the antenna. The azimuth dependence will be considerably less. To observe the brightness distribution of the whole visible sky in a reasonable time one will scan the telescope. With an instrument on an elevation-azimuth mounting we can scan a large part of the sky by moving the antenna

in azimuth only at a fixed elevation. With this method the effects of ground and atmospheric radiation are much less, and easier to correct for, than with a telescope scanning in right-ascension or declination (Davis, 1967).

A method devised originally to increase the sensitivity for the detection of point sources (Conway, Daintree and Long 1965» Davis

1967) (category b) has proved to be of great value in category c,. In a search for sources the observer is troubled by two kinds of spurious signals: the noise generated by the receiver and the confusion noise due to the unknown, weak and unresolved sources radiating in the antenna beam. Optimum use of the telescope is made if both "noises" are at the same level. Without going into the details of this interesting problem we note that the following method in some cases provides a situation closer to optimum. Two feeds are placed in the focal region, projecting two beams on the sky separated by a few beamwidths.

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The receiver is arranged to record the difference between the signals from the two feeds. A source moving through the beams will produce a signal with twice the amplitude of that of a single feed.

Radiation with an angular structure (significantly) exceeding the beam separation does not contribute to the receiver output signal. Also Conway (I963) made the suggestion on qualitative grounds that the

"dual-beam" method might reduce the effect of tropospheric fluctuations. The argument is as follows.

At short wavelengths in the centimetre and millimetre range (X< 6 cm, say) serious problems can arise in radio-astronomical observations due to fluctuations in the thermal radiation from the tropospheric

constituents, notably water vapour. Even in cloudless conditions the amount of water vapour, and hence the intensity of its thermal radiation, is subject to variations. The structure and intensity of the ttirbulence field in the atmosphere determines the fluctuations. It will be

appreciated that a "turbulon" (name for a region in which the turbulence is correlated) moving through the beam of a radio telescope will give a signal which is difficult or impossible to distinguish from the passage of a radio source. Thus the sensitivity of the radio telescope may well be determined by atmospheric influences at the very short wavelengths.

The dual-beam method just described is expected to improve the situation. To make this plausible let us first note that the density of water vapour in the troposphere decreases rapidly with altitude, the main fraction being located in the first 2-3 km. Besides we argue that the two beams , though separated by a few beamwidths in the fax-field zone, will have a considerable overlap in the Fresnel region, close to the antenna. For in this region the Fraunhofer diffraction pattern has not yet formed itself and the power is more or less flowing in a tube.

We reach the conclusion that a significant amount of the tropo-spheric fluctuations might be cancelled by dual-beam observations in which the difference signal is recorded. It was this suggestion that prompted the work to be described presently. At the time of publication of Conway's suggestion preparations for observations at millimetre waves were being made. It was decided to carry out a more detailed, quantitative analysis of the method in the light of antenna theory and structure of the atmosphere. Experimental results were to form an integral part of the work. The following chapters contain the results of the

investigations. Apart from any possible merit of the present work for its own sake it may perhaps be noted that the dual-beam technique has

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already found wide application in point source observations at wave-length of 6 cm and less. It has significantly increased the useful observation time for work in the region of very short wavelengths.

With this encouraging perspective we conclude the chapter on the general theory of the paraboloidal reflector in radio astronomy.

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Chapter II

Theory of the paraboloidal reflector antenna

1. Qualitative description of several approximations.

The present chapter deals with aspects of the theory of the paraboloidal reflector antenna. It may be stated from the outset that we shall not attempt to give a complete and rigorous electro-magnetic treatment of the theory. The problem of the diffraction of a plane wave by a perfectly conducting paraboloid of finite dimensions has not been solved in its generality by an exact method. However several approximative procedures have yielded sufficiently accurate results under certain circumstances. Since our main concern is to obtain reliable numerical results in cases which concern us in practice, we have concentrated on the development of a suitable approximation, valid in our region of interest.

Qualitative properties of some approximations are described in the following sections. They all belong to the class of high frequency approximations. in which we assume the wavelength to be much smaller than the dimensions of the reflector. This applies to radio astronomy antennae, where the usual wavelength range is 1-100 cm and reflector diameters vary between 10 and 300 m.

1.1 Geometrical optics method.

This is the simplest method, which may be expected to yield fair results in cases were the primary source has a broad radiation pattern and the scattering occurs at a non-focusing reflector

with-out sharp edges, located at a laxge distance from the primary source (feed). The reflection on the reflector is assumed to obey Snell's law at the tangent plane in the point of reflection. Application of Snell's law is valid if both the incoming wavefront and the reflector are locally plane. Therefore we require the radii of curvature of the wavefront and of the reflector to be much larger than the wavelength. Clearly, the higher the frequency of the waves, the better will be the approximation.

The scattered intensity in a certain direction is found from considerations of conservation of energy in the incident cone of rays and the reflected ray tube. The final field is a superposition of the broad radiation pattern of the feed and the broad scattering pattern from the reflector. The results of this method are independent of the wavelength. The quality of the approximation is best for X->0.

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For reflection on the concave side of a paraboloidal reflector no reliable results are expected since we assumed a non-focusing, i.e. convex reflector surface. The approximation implies that no

discontinuities occur in the phase and amplitude of the rays and their derivatives. Hence foci and boundaries are excluded.

There is however one interesting feature of the geometrical optics approximation in the case of reflection of a plane electromagnetic wave at the exterior (the convex side) of an infinite paraboloid of revolution. An asymptotic expansion of the electromagnetic field shows that all terms but the one of lowest order vanish (Kline and Kay, 1965, Ch. ^k). Thus it appears that in this case the geometrical optics method provides an exact solution of the electromagnetic reflection problem. It is remarkable that the case of a scalar (e.g. acoustical) wave does not provide such a simple solution. This is an unusual situation.

1.2 Current distribution method.

It is well-known that the field at a point can be expressed in terms of the field values over a surface enclosing all sources. This can be shown e.g. by using a vector analogue of Green's theorem and applying it to the Maxwell equations. Thus the radiation field of a radiator is obtained by performing an integration over the current and charge distribution on the surface of the body, which we assimie perfectly conducting. The resulting field satisfies Maxwell's equations, provided that the source distributions over the surface of the radiator satisfy the equation of continuity.

In the current distribution method we approximate the current distribution over the surface of the reflector from the known radiation field of the primary source. The primary waves and the reflector axe assum.ed locally plane; thus on the surface we have

geometrical optics reflection. The surface current density distribution is simply found from the plane wave boundary condition at the reflector surface. On the illuminated area of the reflector a surface current density

io= 2(n X H^) (II.1) is required to satisfy the boundary condition. The magnetic field

strength of the incident wave at the surface is denoted by H , while n is the unit vector normal to the reflector surface. On the

geometrical shadow area of the reflector the current is assumed to vanish. The discontinuity at the edge of the reflector must be removed

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achieved by introducing a line charge along the boundary euxve be-tween the illuminated and dark parts of the surface. It can be shown

(Silver 19^9» Ch.5) that the line charge contribution removes the longitudinal field component of the radiation field at infinite

distance (the farfield) from the reflector. This is physically obvious, because the field at a large distance from the source behaves as an outward traveling wave in free space and is necessarily transverse.

The present method differs from the geometrical optics method in several aspects. First the requirement that the reflector be non-focusing can be dropped. This is of crucial importance for the treat-ment of a paraboloid with a primary source at the focal point. Second the radiation field at a point of observation is obtained from an

integration of the surface current over the whole area of the reflector. Moreover, because the field satisfies Maxwell's equations, it exhibits a frequency dependence. We can summarise the characteristics of the method by stating that diffraction effects are taken into account in the calculation of the radiation field. The geometrical optics method completely ignores diffraction.

Thus in the current distribution method we pass from the surface current to the radiation field in an exact way. But the current density on the reflector is still approximative, since we use a local geometrical optics relation for its calculation.

In the general case the surface current density at a point Q on the reflector will be influenced by the current at all other points of the surface. Actually, it can be shown (Fock 1957» Franz 1957) that the surface current j (p) at a point p on the reflector surface S satisfies the following integral equation:

j. (P) = j ( p ) + ^ n x ƒ [j_ (p')x(^-£')] ^^^i|^e^™dS . (II.2) ~^ S R-^

Here R denotes the distance between the fixed point £ and the variable point £_' , while n is the outward unit normal on the surface at the position £_; j is given by (ll.l). The integral term in (II.2) results from the interaction between the currents at points £_ and £_'. In the usual approximation the integral is ignored. It can be shown (Cull en

1958) that its contribution tends to zero for k-x», provided the point £ is not very near the shadow boiindary.

Generally the omission of the integral term causes an error because the following effects are neglected:

(i) ciirvature of the reflector surface,

(ii) a possible near field contribution in the current j , caused by the finite distance to the primary source,

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(iii) the edge effects which cause perturbation currents at the edge and on the shadow region. As already stated, these are needed to satisfy the condition of continuity.

The magnitude of these errors has been investigated among others by Tartakovskii and Tandit (I960) and by Kinber (I961).

ad (i). The integral equation (II.2) can be solved by successive

approximation of the current j(p'), starting with j_(p') = 2 n x H ( p ' ) . After one such iteration the resulting formula is valid to within terms

_2

of the order (kR ) , where R is the radius of curvature of the ^ o ' o

reflector. Tartakovskii and Tandit show that the first correction term vanishes for reflectors with a quadric surface of revolution and the primary source at the focus. They give the correction for a paraboloid, with an elementary plane wave source at the focal point, as

|(i-io)/J| = c o s ^ d ) / (kf)2. (II.3)

where f is the focal length and i(/ the angle between the vector p and the reflector axis. For kf >10, hence f è 1.7X, the correction current is less than ^%. It should be noted that the correction current is 90 out of phase with respect to j . For all practical purposes the influence of curvature appears completely negligible.

ad (ii). The electric field of a Hertzian dipole has the form

(1 • 7^'^ h — F(e.'t') + term in p"^.

It is natural to neglect the term with 1/ikp in case k p » 1 . Thus, if we write p=f, we require that 2TTf/X>>1, which gives <'\% error if f>l6X. It appears that the near-field effect is stronger than the influence of reflector curvature. Tartakovskii and Tandit find the

near field current to be 90 out of phase with respect to the geometrical optics current and with a relative amplitude proportional to the ratio of feed and reflector diameter. Typically one has D„ == 2X, while generally D >100X. Since the spatial distribution of this current component deviates significantly from the main current, some care must be taken e.g. in cases, where the geometrical optics illumination decreases towards the reflector edge.

ad (iii). Tartakovskii and Tandit have estimated the edge currents; the 2

magnitude is of the order 1/k f relative to the geometrical optics field, and the region of influence is only of the order of a wave-length along the rim. Kinber (I96I), however, does not agree and argues that the edge currents can be noticed over the whole surface. Still for a large reflector (f»X) the edge effects have negligible