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*"»CAU O* NBS TECHNICAL NOTE 696

(Revised September 1977)

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

Near-Field Antenna Measurements

On A Cylindrical Surface:

A Source Scattering-Matrix Formulation

.*>"-* V:

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ril -
NATIONAL BUREAU OF STANDARDS

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1 ' -77 rvdV a<LL - Re£ i lOO Near-Field Antenna Measurements

^

On A Cylindrical Surface:

mi

I A Source Scattering-Matrix Formulation

Arthur D. Yaghjian

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

/

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

! . NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued September 1977
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 696

Nat. Bur. Stand. (U.S.), Tech. Note 696, (Revised September 1977) 40 pages

CODEN: NBTNAE

(Supersedes Technical Note 696 issued July 1977)

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1977

For sale by the Superintendent nf Documents, U.S. Government Printing Office, Washington, D.C. 20402
CONTENTS

Page

1. Introduction 1

2. The Radiating Fields of an Antenna in Cylindrical Waves — 2

2.1. Expressions for the Far-Field, Power Gain Function, and

Complex Polarization Ratio 4

2.2 Total Power Radiated 5

3. Derivation of the Transmission Formula 5

3.1. The Source Scattering Matrix Equations for the Test Antenna 6

3.2. The Source Scattering Matrix Equations for the Probe Antenna 8

3.3. Relationship Between the Probe and Test Antenna Coefficients

in the Common-Center Systems 9

3.4. The Transmission Formula 10

3.5. Inversion of the Transmission Formula 11

4. The Receiving Characteristics of the Probe Antenna 12

4.1. Measuring the Probe Receiving Functions Using a Known Test Antenna 13

4.2. Transformation of the Receiving Functions of a Known Probe 13

4.2.1. Receiving Functions Directly in Terms of the Far-Field

of the Probe 16

5Applications of the Sampling Theorem and Fast Fourier Transform 17

6

Summary 19

Appendix A. Derivation of the Far-Fields in Terms of the Cylindrical

Wave Coefficients 22

Appendix B. Derivation of the Source Scattering Matrix Equations and

Reciprocity Relations for the Probe and Test Antennas 24

Glossary of Symbols 28

Figures 30

References 34

iii

NEAR-FIELD ANTENNA MEASUREMENTS ON A CYLINDRICAL SURFACE:

A SOURCE SCATTERING-MATRIX FORMULATION 1

Arthur D. Yaghjian

The theory for probe-corrected measurement of antennas by scanning on a circular cylindrical surface enclosing the test antenna in the near-field is

formulated from a source scattering matrix description of the test and probe

antennas. The basic transmission formula is derived without recourse to reciprocity, and from a common center approach which separates as an isolated

problem the probe characterization and transformation. Both an experimental

technique and an approximate analytical technique are presented to determine

the required transformed probe coefficients without the use of addition theorems. The approximate technique, which is developed from the exact addition

theorem transformation, yields the probe coefficients directly in terms of

the far-field of the probe, provided the rotation axis of the test antenna

lies in the far-field of the probe. Computer inversion of the transmission

formula is accomplished accurately and efficiently with the aid of the

sampling theorem and FFT algorithm.

Key words: Cylindrical scanning; near-field measurements; scattering matrix.

1 . INTRODUCTION

The ease with which many antennas can be scanned on a cylindrical surface by moving a probe along a single axis for different azimuthal orientations of the test antenna about a parallel axis makes cylindrical scanning a very attractive near-field measurement technique.

In addition to its operational simplicity, cylindrical near-field-far-field deconvolution

utilizes the FFT in a straightforward fashion without the requirement of special measurement

probes, and the far-field emerges in conventional elevation and azimuth angles, the latter

of which covers a full 360°. And, in fact, both the 2-dimensional and 3-dimensional formulation of the probe-corrected technique for measuring antennas on a cylindrical surface

have been derived and verified through experiment by Brown and Jull [l] 2 (2-D case) and

Leach and Paris [2] (3-D case). In both papers the EM fields of the test antenna were

expressed in terms of cylindrical waves and the Lorentz reciprocity theorem was used

(assuming reciprocal probes) to derive the basic "transmission formula" needed to apply the

cylindrical scanning technique. Brown and Jull expressed the receiving characteristics of

the probe in terms of a plane-wave spectrum, while Leach and Paris used cylindrical waves

to express the receiving characteristics of the probe. In the present work, the basic

transmission formula (25) is derived from the "source scattering matrix" equations for the

test and probe antennas referred to a common-center cylindrical coordinate system. This

cylindrical scattering matrix approach, which is analogous to the plane-wave scattering

matrix approach used by Kerns [3], proves simple and straightforward, and does not require

the probe or test antenna to be reciprocal, although reciprocity is expressed conveniently

as a relationship between elements of the scattering matrix (see eqs. (15 and 19)). The

parallels between the cylindrical source scattering matrix formulation used throughout these

notes and the plane-wave scattering matrix formulation introduced by Kerns is striking and

is emphasized to facilitate the reader's understanding of the cylindrical formulation.

Another distinct advantage of the source scattering matrix approach is that it

clearly and completely separates the problem of determining the receiving characteristics

i This technical Note was originally written as a tutorial set of notes for the NBS short courses (July 7-11, 1975 and Aug. 29-Sept. 2, 1977 - Boulder, Colorado).

Figures in brackets indicate the literature references at the end of this paper.
of the probe from the derivation of the transmission formula. The characterization of the

probe can be dealt with as an isolated topic. It is irrelevant to the basic formulation

whether the required receiving functions for the probe are, e.g., measured directly, transformed from its known far-field or plane-wave spectrum [1,11], or transformed from its known

cylindrical mode expansion [2], Of course, the accurate characterization of the probe is a requirement of any reliable probe-corrected theory and is discussed as a separate topic in

section 4 below.

2. THE RADIATING FIELDS OF AN ANTENNA IN CYLINDRICAL WAVES

This work addresses the problem of measuring the radiating fields of a test antenna by

scanning with a probe antenna on an imaginary cylinder surrounding the test antenna. The

problem of measuring the receiving characteristics of the test antenna is very closely

related, but for the sake of brevity it will be left as an exercise for the reader. Of course, most antennas are reciprocal so once the radiating or transmitting characteristics

are determined the receiving properties follow immediately from the reciprocity relationship

(15a).

As a first step in formulating the cylindrical scanning technique, the EM fields outside

an arbitrary test antenna radiating into free space will be expanded in a complete set of

cylindrical waves (eigenfunctions or modes) . The test antenna and its coordinate system

fixed to the test antenna are shown schematically in figure 1. In cylindrical coordinates

the point P is designated by (p, <f>, z), in spherical coordinates by (r, $, 6). It has been shown [4] that in cylindrical coordinates the (E, H) fields in free space

(in this case outside the radius (a) of the smallest cylinder circumscribing the test antenna) can be divided into a TE and TM part with respect to the z-direction. Specifically,

E = V x e

z ^1 + ^V x (v x e^^) (la)

iwy H = kV x g j, + v x (V x 6 to •) (lb) o z 2 z 1

where \\> and if/„, which give rise to the TE and TM parts of the field respectively,

satisfy the scalar wave equations

V

2

(Jj + k

2

ijj = 0, s = 1,2. (2)

All quantities with hats (~) over them denote unit vectors, e (w>o) time dependence

and the rationalized mks system are used throughout, and the free-space wave number k is

defined as usual byJ k = u/u e = — = —— . o o c A The solution to the scalar wave equation (2) in cylindrical coordinates outside the

antenna [ I is written in terms of the well-known scalar cylindrical waves

00 oo K - I I B>) H^ } (K P )e

in

*e

iYZ

d Y . (3)

K =

(k2 _

y 2)l/2

s

u J n ' n
S (l ) The B (y) are arbitrary constant coefficients and H

1 (kd) are the Hankel functions of the n n first kind. The Hankel functions of the first kind are the only cylindrical functions which

possess the proper behavior as p-*» for k chosen positive real or imaginary. (Later, when

the probe enters the picture giving rise to sources outside p, Bessel functions J (<p) are also included in the complete set of cylindrical modes.) When the solutions (3) for ty are substituted into eqs. (1) and the various curls are taken in cylindrical coordinates, the

(E, H) fields can be written

00 oo E(p,<j),z) = I J [bJ(Y) M^ + Bj(y) N^Jdy (4a) n=—°° —°°

H(p,<(.,z) = -±- I j [B 1

( Y ) N

(l) + B

2 (y) M

(l)

K ' T '' ]d Y , (4b) iZ

o n=—

u

°° —°°J n ny nny

(Z = /u /e " 377 ohms)

o o o

where the M and N functions are given by

M^Cp,*,.) = V x (H ( l ) (K P )e

la

*e

1Yze ) = [^H^OcpJe - K H^' 6jein

*eiYZ

(5a)

ny n z p n p n <p

H^Cp,*..) = h

V x M

( l> = £[iYK H( l> V - "Y H

(1)

e

A + k*H< 1 >£ ]e

in

*e lYZ

. (5b)

ny r k ny k

L 'n p p n <f> nz

If the B (y) coefficients were known for the test antenna, eqs. (4) and (5) would give the

electromagnetic field of the test antenna everywhere outside the radius of the smallest

cylinder circumscribing the test antenna. Essentially the main purpose of this work is to

s formulate a procedure for finding the B (y) for an unknown test antenna, by scanning on a cylindrical surface with an arbitrary but known probe antenna. The only quantities in

eqs. (4) which change from antenna to antenna are the coefficients B (y)

If one is fortunate enough to have a probe at his disposal which approximates an ideal

electric dipole, i.e., a probe which measures the transverse components of the electric

field, then the B (y) can be found directly and almost trivially from eq. (4a) alone. To show this, the following orthogonality relationships for the M and N functions are required

J J (M^} x M<}> )-e d* dz = (6a)

' ' ny n y p -co o

CO 2*17 / / (N (l) ''nYnyp x nV',)'« d* dz = (6b)

/ /Vj* xMn

( }> )-e d* dz =il^i H^

) (KP )H ( l ),

(K P ) 5_n .6(y+y') • (6c)

These relationships derive from the definitions of M and N in eqs. (5).

Now cross N into eq. (4a) and use the orthogonalities (6) to yield the B^y),

B* (y) p-r . ,, / /{(N<l) x f(p <j>,z)l-e d* dz , n

4tt 2 k

3 H(i) (k P )H (i) (kP ) -4 o -n,-y °

P
or written out in component form,

.1/ v -1

oo 2ir Bi(Y) „ ~; w / / ^E tp.*,z) + k

2 E.(p.4>,z)

n /._2.-3u (l)/ 4tt 2 k

3 H U; (kp ) -»

p Z O ffi o o

e

T e d<))d2 . ( 7a)

p is the radius of the imaginary scanning cylinder on which the transverse electric field

is measured. Similarly cross M into eq. (4a) to yield the B

2 (y),

B

2

( Y ) - Vl 7 T E (p ,*,z)ein ViYZ d*dz . (7b) n

4Tr 2 K

2 H

n

(l) (K P ) - o

Z °

Equations (7) give the required modal coefficients in terms of the measured transverse

electric field on a cylinder. Moreover, the sampling theorem and fast Fourier transform

(see section 5) can be utilized for the efficient computation of the double integrals in

eqs. (7). Thus for an ideal electric dipole probe, (an ideal magnetic dipole could be used

in a similar manner with eq. (4b)), the deconvoluticn problem reduces to a trivial application

of the orthogonality relationships for the M and N functions. This is true for planar and

spherical scanning as well—and as a matter of fact, for any of the six separable coordinate

systems for which M and N vector wave solutions exist [4].

2.1. Expressions for the Far-Field, Power Gain

Function, and Complex Polarization Ratio

Equations (4) and (5) give the electromagnetic fields everywhere outside the test antenna radiating into free space. Often, however, only the far-fields of the test antenna

are of interest. Thus, it is helpful to derive explicit expressions for the far-field from

eqs. (4) and (5). This has been done in appendix A by a straightforward manipulation of the

Fourier transforms without using asymptotic methods such as the method of stationary phase

or steepest descent. The final results written in the spherical coordinates of figure 1 are:

ikr

y E(r,4>,9) = - 2 k sin 6 £ (-i) n

[B 2 (k cos 6)6. - iB 2 (k cos 6)e.]e n<p

r-x» (8a)

e *E „ , . . ikr r -i j. H(r,<t>,6) = — = ^

x

( -i ) [ Bn( k cos 9 ^ §

e + iB

n^ COS B ^ e J

e ' ( 8b ) r-x» o o n=-oo

The far field equations (8) do not involve an integration, only a summation, and the values

of B (y) for ly^k do not affect the far-fields. In fact in section 5 it is seen that only

|Y|<k are needed to compute the fields beyond a wavelength or so from antennas which do not

s have extraordinarily high reactive fields. Note also that at least some of the B (y) must

behave as the 1/sin 9 as 6-K) if the far-field patterns are finite there.

The power gain function is defined as _ 2 rt* A \ 2irr 2

|e[ r-*»

G(6,<fr) = 2~p ' (9a)

o input
where P. is the net power input to the antenna. The input power can be written in terms input

of the ingoing (a ) and outgoing (b ) modes of the waveguide which feeds the antenna (see

figure 1 and Kerns [3]),

p. = \ n (|a | 2 -|b | 2

) = \ n |a | 2 (i - |r | 2

) , input zo'o' 'o' zo'o' ' o

'

where n is the characteristic admittance of the waveguide and r is the input reflection

o o coefficient at the reference plane r S oo(b = r a ). Substitution of P. _ and the faro o input

electric-field from eq. (8a) into eq. (9a) yields the final expression for the gain

function,

,-, o

oo 00 _,- ,, loir k^sin^ 6 ,

i r / .\ nul/, i a\ ln(f)

1 2 _i_ v / ^^2^1 Q \ in<l>l2\

G(6,cf>) =

z

i

\ 2n_ \ T

ri \U I ( _1 ) Bn

(k cos e)e I + I Z ( -1 ) Bn

( k cos e ) e I * -(9b)

o o ' o

'

' o

'

n=- ° n=-°°

The complex polarization ratio defined here as E./E may be written directly from eq. (8a),

-i I (-i)V(k cos e)e

±n ^ E

9 . n=-°°

m (10)

E()> 7 (-i) n

Bl(k cos e)e

ln<()

L n

n=_oo

2.2 Total Power Radiated

Also, from the far-field equations (8) it is a fairly simple matter to find the total

power P radiated by the test antenna,

- •, t 2tt _

P

r

= ^ j> (ExH*)-e

r da = j— j j

|E| 2 r

2 sin 6 d<^d6 , (lla)

r-x» o o o

where * denotes the complex conjugate.

After substituting the electric field from eq. (8a) into eq. (11a) and making use of the

orthogonality of the e functions, P can be written

, , 2

°° f P

r = -~- I j [|Bl(k cos 0)| 2 + |B 2 (k cos 6)| 2 ]sin 3 6 dO, (lib)

o n=-«° o

A change of the variable of integration from k cos 9 to y transforms eq. (lib) into the final

expression for the total radiated power,

. °° k

P

r =^ I /K2 [|Bl( Y )|

2 + |B 2

( Y )|

2 ]d Y . (12)

o n=-oo -k

Observe again that no radiated power is carried in the waves for |y|>k.

3. DERIVATION OF THE TRANSMISSION FORMULA

In section 2 we derived the E and H fields everywhere outside the test antenna in terms

g of its cylindrical wave coefficients B (y) . Explicit expressions were written for the farfields, power gain function, complex polarization ratio, and total power radiated.

5
In this section we will derive the basic transmission formula needed to compute the

coefficients B ( Y ) for an unknown test antenna by scanning on a cylinder with a probe of

known receiving characteristics. The approach that is taken is to first write "source

scattering matrix equations" for the test antenna with respect to the coordinate system

(previously defined in figure 1) fixed in the test antenna. Secondly, the source scattering

matrix equations for the probe are written for a cylindrical coordinate system (p'.^'.z')

fixed to the probe but with its z'-axis lying along the z-axis of the test antenna coordinate

system. Thirdly, the relationship is derived between the modal coefficients defined in the

coordinate system of the probe and the modal coefficients defined in the coordinate system

of the test antenna. This relationship is entered into the source scattering matrix equations and the resulting equations are combined, neglecting multiple reflections, to give the

desired transmission formula. Finally, the transmission formula is inverted to yield the

B*(Y).

3.1- The Source Scattering Matrix Equations for the Test Antenna

Consider the cylindrical coordinate system fixed in the test antenna as shown in

figure 1. While measurements are being taken with the probe antenna, it will act as a source of EM fields outside the test antenna. Thus the cylindrical mode expansion in the

coordinate system of the test antenna for radii between the test antenna and probe must

include two linearly independent radial solutions—not just the H

1 (<p) solutions as was

the case for the test antenna in free space. As we shall see, the present derivation is

greatly facilitated by choosing J (icp) as the second linearly independent solution.

The E and H fields between the test and probe antennas may be written as in eqs. (4),

but with the J (icp) solutions included as well. Specifically,

GO 00

n=—co -co E(p,*,z) = I /{[bl(Y)M^; + b2< Y )iry] + [a*<Y)M + a*( Y )N ]}dY (13a)

H(p,$,z) = j^- VxE , (13b)

o

where M

l

, N

l have the first Hankel functions H ^(icp) for their radial dependence, and

_ nY nY n M , N have the Bessel functions J (<p) for their radial dependence. The modes with

nY nY n coefficients b ( Y ) represent the fields from sources (induced as well as applied) inside

or on the surface of the test antenna. The modes with coefficients a ( Y ) represent the

fields from sources (induced as well as applied) inside the probe antenna, i.e., outside

the test antenna.

Also, the waveguide modes with coefficients a and b represent fields from sources inside and outside the generator (assumed shielded) of the test antenna, or equivalently,

outside and inside the test antenna proper.
s s In brief then, the modal coefficients (a , a (y)) and (b , b (y)) represent fields from sources outside and inside the test antenna respectively. The inside coefficients

s s (b on, b (y)) are related to the outside coefficients (av on, a (y)) by a linear matrix transformation which will be called the source scattering matrix for the test antenna, to distinguish it from the scattering matrix for the more common H , H representation of

the fields [14] instead of the H , J representation used here. Specifically, these source scattering matrix equations are written

2 b=Ta+y V f R

S (y)aS (y) dy (14a)

o oo s=l

,''

n=-°° -oo

' nn

2 oo 00 b!(Y) = <(y)a„ +11 / S*!(y,B)a>)de . (14b) n no q=l m=-°o -°

S S SQ The functions Rnn(y) , T (y) and S (y,$) embody the receiving, transmitting, and nm

scattering properties of the test antenna. r is the input waveguide reflection coefficient

defined at the surface S . For a given cylindrical coordinate system fixed in the test antenna, these four quantities depend only on the character of the test antenna. In Kern's

notation [3] S„, , S-.„, S.

1

, and S n

_ correspond to R, T, S, and r , respectively.

The source scattering matrix equations (14) can be taken as a definition of linearity

for the test antenna or they can be derived from Maxwell's equations through a

" source adjoint reciprocity lemma ." The derivation of this source adjoint lemma and the scattering

matrix equations (14) may be found in appendix B. It is also proven in appendix B that the

receiving and transmitting functions of a reciprocal test antenna (or between a nonreciprocal

antenna and its adjoint) satisfy the relationship

R!n<^> = <" 1 > n VTk Tn

(Y> , d5a)

o o

and the scattering function satisfies the relationship

S q* (-B,-y) = s!2 (y,3). (15b) -m , —

n

nm

When the probe antenna is removed so that the test antenna is radiating into free space,

eq. (14b) becomes

bn

(Y) = Bn(Y) = Tn(Y) V (16)

(s) Thus if we can determine T (y) for the test antenna we can compute the free-space EM fields

of the test antenna by means of eqs. (4). And, in particular, the far-fields, power gain

function, complex polarization ratio, and total power radiated could be computed from

eqs. (8), (9b), (10), and (12), respectively. Thus the problem of measuring the radiating

g properties of the test antenna reduces to that of determining T (y) . The next step toward

this end is to write the source scattering matrix equations for the probe antenna.
3.2. The Source Scattering Matrix Equations for the Probe Antenna

Consider a cylindrical coordinate system fixed in the probe antenna but with its

z'-axis lying outside the probe antenna along the z-axis of the cylindrical coordinate —system fixed in the test antenna (as shown in figure 2). a' and b' are the coefficients of

"

o o the ingoing and outgoing waveguide mode of the probe antenna, and, in particular, the primes

on the cylindrical coordinate system emphasize the fact that this coordinate system is fixed

in the probe. Of course, the probe coordinate system is nothing more than the test coordinate

system rotated about the z-axis through an angle $ and translated along the z-axis through

a displacement z . (<f> and z vary while scanning.)

The EM fields on a cylinder between the test and probe antennas can be written in terms

of the prime system fixed in the probe:

CO oo E'(p',*\z») = I /{[a^y)!^ + a;

2 ^)N

n^] + [b^CYjM + b;2 (y)N ]}dY (17a) n=—oo -co IT IT

H'Cp'^'.z') = j±- V'xE'. (17b)

o

These equations are identical to eqs. (13) except for the primed coordinates (p'.^'jz') and

primed coefficients (a',b') replacing the unprimed. The a' and b' coefficients have

deliberately replaced the b and a coefficients respectively, so that the a' and b' coefficients would now be associated with sources outside and inside the probe antenna

respectively. In other words, the a' now refers to the H

1 (<p') modes, and the b' to the

J (<p') modes. n

g Just as was done for the test antenna, the inside probe coefficients (b' ,b f (y)) can be

related to the outside probe coefficients (a', a' (y)) by a source scattering matrix for the

probe:

b

o

= r o a

o + I I K

8 TM an

S <Y)dY (18a)

s-1 n=-°° -*>

2 oo CO b

,S

( Y ) = T'V + I I JS

,Sq (y,B)a' q dB . (18b) n ' no L

, q=l n=-°°

L -00

; nmm

The functions R'

S (y). T,S

(y), and S ,Sq

(y,3) represent receiving, transmitting, and scattering

properties of the probe antenna, r' is merely the input reflection coefficient for the probe

waveguide referred to the surface S'. For a given cylindrical coordinate system fixed in

the probe antenna, these four quantities depend only on the character of the probe antenna.

Like eqs. (14), the probe eqs. (18) are derived from Maxwell's equations in appendix B. Appendix B also proves that the source scattering matrix for a reciprocal probe (or for a non-reciprocal probe and its adjoint) obeys the following relationships analogous to
eqs. (15) which apply to the test antenna,

O-Y) = (-D^ll'W (19a)

o o ^-n

(-3,-Y) = s;>,e) . (1%)

The probe scattering equations (18) can be linked directly to the corresponding test antenna

s s scattering equations (14) by relating the coefficients (a' , b' ) of the cylindrical modes

s s in the coordinate system fixed in the probe to the coefficients (a , b ) of the cylindrical

modes in the coordinate system fixed in the test antenna. This relationship between

(a' , b' ) and (a , b ) is surprisingly simple and can be found in a surprisingly simple

fashion.

3.3. Relationship Between the Probe and Test Antenna

Coefficients in the Common-Center Systems

We know that, regardless of whether the probe or test antenna coordinate system is used,

the EM fields at every point in space must be the same, i.e.,

E'(p\ <(.', z') = E(p, <j>, z) (20a)

H*(p*, <f>*, z') = H(p, <f>, z) , (20b)

when (p' , <j>', z') and (p, <j>, z) refer to the same point in space. But as the probe scans the test antenna, the probe coordinate system is merely rotated through an angle <\> about the

z-axis and translated a distance z along the z-axis, i.e.,

,» = A _ A ~„a .,< = p = p , <f> = <j> - <j> , and z =7.-2.

and eqs. (20) become

E'(p, <j) - <f) o

, z - zq) = E(p, <j>, z) (21a)

H' (p, <|> - <|> o

, z - z q

) = H(p, 4., z) . (21b)

Now when (E, H) and (E' , H') are written explicitly in terms of the linearly independent

cylindrical waves in eqs. (13) and (17), the orthogonality relations (6) show that the only

way eqs. (21) can be satisfied is if

in* iyz

a' S

( Y ) = b

S

( Y ) e ° e

° (22a) n ' n

in* iyz

b

. s

( Y

) = a

S

( Y ) e

° e

° . (22b) n

s n

'

These are the desired relationships linking the probe and test antenna scattering

equations (18) and (14).

9
3.4. The Transmission Formula

Equations (22a) recast the probe receiving equation (18a) into the form

2 o° °° in<(> iyz ^W = r oa

o + I I / R

nS(Y)bn

(Y) e ° e ° d ^- (")

s=l n=-°° -°°

Moreover, b (y) from eq. (14b) can be substitued in this eq. (23) to yield

2 oo oo 2 °° oo in<}> iyz

bi<*o'"o) = W + I I / ^;S (Y)t<( Y )ao+ [ I / S^(Y ,3)a^(6)d3]e °e °}d Y . 8=1 n=-°° -oo q=l m=-o° -°° (24)

SG The contribution to b'(<f> ,' z ) from the term in eq. (24) containing S (y,B) represents

multiple reflections. That is, it can be described as that part of the output of the probe

caused by fields which have left the test antenna, reflected from the probe, re-reflected

from the test antenna, and returned to excite the probe. This process repeats itself ad

infinitum. Assuming these multiple reflections are negligible , and the input a' to the probe

is kept zero, eq. (24) reduces to

oo °° 2 in<j> iyz

b

i

( *o'

Zo)=a I /t I R;

S (Y)^(Y)]e °e ° d Y . (25)

n=-oo -oo s=l

(If a

1 is not kept zero but the probe is terminated in a load with reflection coefficient

r' , the right side of eq. (25) is merely divided by (1-rT' ).)

(1) (2) °(1) Here, it can be noted that if (H , H ) functions had been used instead of (H , J ) n n n n for the expansion of the fields in cylindrical waves, it would be impossible without going

through an extremely tedious procedure to show the effect of neglecting multiple reflections

since H waves would be excited by the probe as well as test antenna.

This " transmission formula" (25) relating the output of the probe and the " coupling

2 product" ( I R

1 (y) T (y) where R' (y) is the receiving function of the probe and T (y)

s=l

is the transmitting function of the test antenna) is analogous to the "transmission integral"

of the planar formulation [3], and forms the central relationship on which the cylindrical

scanning technique is based. In essence, it is the same formula as eq. (23) in Leach and

Paris [2], and the same formula as the 2-dimensional equation (18) in Brown and Jull [1] if

the polarization index s, z dependence, and y integration are deleted from eq. (25).

The transmission formula (25) has been derived in a simple, straightforward manner from

the source scattering matrix equations for the test and probe antennas under the assumption

of negligible multiple reflections. No other restrictive assumptions were made - the probe

and test antenna may even be non-recriprocal . (The derivations of Leach and Paris [2] and

Brown and Jull [1] inherently assume a reciprocal probe although both their derivations can be modified slightly, using adjoint reciprocity, to apply to non-reciprocal probes as well.)

10
3.5. Inversion of the Transmission Formula

Let's take a closer look at the transmission formula (25). In principle, as the probe

scans on the surface of an imaginary cylinder surrounding the test antenna, the output

b'(d> , z ) of the probe is recorded for

o o o

< d> < 2tt — o

—°° < Z < °° o That is, the amplitude and phase of b'(4 , z ) is measured for all values of <t> and z . (In

' r r o o o o o practice, the z scan is limited to some finite scan length and b' (<t> e , z ) is assumed

o o o o negligible outside this region. Also, data need be sampled and recorded only at a finite

number of measurement points. These topics of data collection and computations will be

discussed in section 5 below.) Having measured b' (<f> , z ), the Fourier series and integral

of eq. (25) can immediately be inverted to yield the solution for the coupling product in terms

of the measured data b'(<t> , z ): o o o

2 °° 2tt -in<}> -iyz

1 R

n

S (Y) <(y) - ^T- J / b;(* ,z o )e ° e ° d^ . (26a)

S = l O -oo O By computing the double integral on the right side of eq. (26a), the value of the coupling

product,

R

n

1(Y)Tn

(y) + R

n

2 (Y)T^(Y) ,

can be determined for as many n and y as desired. However, even if the receiving characteristics (R , R' 2

) of the probe are known, the transmitting characteristics (T 1, T

2

) of n n

r & n n

the test antenna are not determined uniquely (2 unknowns for each equation) . Fortunately,

all that is needed is another scan with a different probe, or the same probe in a different

orientation (say rotated about its baseline axis by 90°). In other words, two linearly

independent scans are necessary to account for the polarization of the EM fields. The

second scan produces a second equation to complement eq. (26a). Specifically,

2 oo 2tt -inij) -iyz

I R

,,S

( Y )T

S (y) = 7-4— / / b"(<|. ,z )e ° e ° d<j> dz , (26b) S n ' n ' 4ir z a ; ; o

T o o

Y o o s=l o -oo o

where R" (y) and b"(<J>, z ) are the receiving characteristics and output of the second or reoriented probe respectively.

Assuming for the moment that the receiving functions (R' , R") of the probe(s) are known,

eqs. (26) can be solved immediately for the transmitting functions T (y) of the test antenna:

Ti(Y) = [R" 2 (Y)I'(Y) " R' 2 (y)I''(y)]/A (Y) (27a)

t

2 (y) = [r; 1 (y)i^(y) - R^(Y)i;(y)]/A

n

(Y), (27b)

11
where I'(y) and I"(y) are the integrals (equal to the coupling products) on the right side

of eqs. (26a) and (26b) respectively; and A (y) is the determinant

My) - R" 2 (y)R' 1 (y) - R' 2 (y)R" 1 (y) . (28) n n n n n

Provided the receiving characteristics of the probe (s) are known , eqs. (26) , (27) , and (28)

determine the complete transmitting characteristics of the test antenna in terms of the

measured *probe outputs cb'(<j> , z ) and b"(d> , z ). o o o o o o

4. THE RECEIVING CHARACTERISTICS OF THE PROBE ANTENNA

Naturally the question arises as to how the receiving characteristics of the probe can

s s be determined. Specifically, how do we find the R' (y) and R" (y) which are required in

s s eqs. (27) and (28). (In the subsequent discussion, I will mention only R' since R" can be found by the same procedure.)

Two methods will be explained for determining R' . The first involves no coordinate

transformations but requires two cylindrical scans and two test antennas (or one test

antenna in two orientations) with known far-fields. The second method, which is closely

related to the approach used by Brown and Jull [1] and Leach and Paris [2] for cylindrical

scanning, and by Jensen [5] and Wacker [6] for spherical scanning, involves a coordinatelike transformation between the known receiving functions of a probe in a coordinate

system centered on the probe and the required receiving characteristics R' (y) which have

been defined with respect to a coordinate system centered outside the probe.

The chief advantage of the first method is that it requires no coordinate transformations, which can be rather cumbersome expressions. Its chief disadvantage, besides the

fact that it requires the a priori knowledge of the far-fields of a test antenna, may be

g that the given antenna must have non-negligible transmitting functions T (y) for all values

of n and y required of the probe. This will be explained specifically below. The second

method consisting of transforming the receiving functions of a known probe does not exhibit

this limitation. Section 4.2.1 shows that the transformation involved in the second method

can be greatly simplified by an approximate expression when the z-axis of rotation of the

test antenna lies in the far-field of the probe - which is often the case. It should be

mentioned that the two-identical-antenna or generalized-three-antenna techniques sometimes

applicable in planar near-field scanning [3] do not appear feasible for cylindrical scanning

because of the more complicated coordinate transformations (such as eqs. (37) and (38) below)

which are involved in the cylindrical case. In any case, it is emphasized that the common-center, source scattering matrix approach

applied in the present paper to the problem of cylindrical scanning results in a transmission formula (25) which embodies the essence of non-planar scanning in a simple form

(analogous to the planar transmission integral) whose derivation was accomplished without

the introduction of cumbersome coordinate transformations. Questions related to the

characterization of the measuring probe and the associated transformations can now be

concentrated on and discussed as a separate topic.

12
4.1. Measuring the Probe Receiving Functions

Using a Known Test Antenna

The first method for determining R' (y) is quite easy to explain. The unknown probe

whose receiving characteristics are wanted scans two test antennas (or one test antenna in two orientations) whose far-fields (amplitude, phase, and polarization), say, are known.

Since the far-fields are given, eqs. (8) can be inverted using the orthogonality relationships (6) between M and N to yield the transmitting functions T (y) of the known test antenna(s) for |y|<_ k. With the T of the test antenna(s) known, eqs. (27) with the T's

g and R's interchanged can be used to determine the receiving functions R' of the probe. To use eqs. (27), however, the determinant A (y) of eq. (28), now containing the T's, should

not be too small. This implies, as mentioned above, that the T's of the known antenna(s)

must not be negligible for a given n and y.

4.2. Transformation of the Receiving Functions of a Known Probe

g The second method for determining R' (y) assumes a probe of known receiving characteristics (or transmitting characteristics, if reciprocal) but known in a coordinate system

centered at the probe, whereas R' (y) was defined with respect to a cylindrical coordinate

system fixed to the probe but centered outside the probe. Here it will be assumed that the

known receiving characteristics are given in terms of a cylindrical wave expansion, cong forming to the work of Leach and Paris [2]. The transformation to R' (y) from known

receiving characteristics given in terms of a plane wave spectrum derives in a straightforward manner but the derivation will not be included in these notes. This latter transformation is similar to that derived by Brown and Jull [1] for the 2-dimensional cylindrical

problem, and recently by Borgiotti [11] for the 3-D problem. As both Brown and Jull, and

Borgiotti point out, the plane-wave transformation involves integrals which lend themselves

to asymptotic evaluation. In section 4.2.1 we show that convenient approximate expressions

for the probe receiving functions emerge from the cylindrical transformation as well.

g Consider a probe whose receiving functions R.! (y) are known with respect to a cylindrig cal coordinate system fixed in the probe and centered on itself (as opposed to R' (y) which

is defined with respect to a cylindrical coordinate system fixed in the probe but centered

outside the probe). Let the z-coordinates (but, of course, not the z-axes which are separated

by a distance d) be the same for each system. Also choose the baseline from which the

angles in each system are measured to coincide. The coordinate system centered on the probe

will be called the C, system and will have cylindrical coordinates labelled (p.., <j>- , z.,) . The coordinate system centered outside the probe at the test antenna but still fixed in the

probe will be called the C system, and as before, its coordinates will be labelled

(p ',<)>' ,z' ) . Refer to figure 3 for a schematic of the two systems.

13
The relationship between the output of the probe (for a' = 0) and the receiving

functions R' (y) in the C. system is by definition (compare with eqs, (14)),

2 oo 00 b

o

=

? I / R

in (Y)aln (Y)dY ' < 29 > s=l n=-°° -°°

g where a' are the modal coefficients of the J (<p n ) modes which are excited by sources

In n 1

J existing outside the probes, i.e., by sources on the test antenna. The values of these

g a' coefficients are independent of sources within p , i.e., sources on the probe.

The output of the probe b' for a' = is given alternatively by eq. (18a) in terms of

5 the required probe receiving functions R' (y) in the C system,

2 00 00 b

;

= l l I R;

S (Y)a;S

( Y )dY • (30)

s=l n=-°° -°°

g A comparison of eqs. (30) and (29) shows that the required R' (y) can be found in terms of

the given b R' (y) if the a' can be written in terms of the a' , i.e., the J (<p,) modal

In In,... n n 1 coefficients in terms of the H (kp') modal coefficients. Fortunately, this can be

accomplished in a straightforward manner by equating the EM fields written in the C. and

C coordinate systems.

In the C. coordinate system, the EM fields can be expanded in its complete set of

cylindrical modes. Likewise, in the C coordinate system the same EM fields can be expanded

in its complete set of cylindrical modes. The fields, regardless of which expansion is

used, are the same. So the E and H fields written in each coordinate system can be equated,

Before doing this, however, a subtle point should be explained.

The C cylindrical mode expansion is valid only up to a radius p' touching the probe

antenna, and the C, cylindrical expansion is valid only up to a radius p touching the test antenna. Thus, at first sight, we might conclude that the EM fields cannot be equated on any complete circle of either the C, or C system. Fortunately, the problem disappears

G g under closer scrutiny. Since both the a

1 ' and a' refer to modes whose sources lie at the

n In test antenna, the part of the EM fields expressed by both expansions are valid for all

(p , (J> , z ) within any cylinders of radius p 1 up to the radius which touches the test

(1) (2) antenna. Again, it is noted that this would not be true if the (H , H ) description

(1) n n had been used instead of the (H , J ) and the derivation would have become unnecessarily

n n complicated, if performed correctly.

g Specifically, the a' field of system C is given from eq. (17a) as

CO CO E'(p\ <(.', z') = I j [a^

1 (Y)M^) (p

, l <(.', z') +a;2 (Y)N£) G>

, » *\ z')]dY • (31a)

tl=—oo —00 '

Similarly the a' field of system C. is given by

il/„w ^ . .m ^ „.2

1

(p 1, 1

, Z]L ) = I j [a^(Y)M (p r $v

z<) + a^(Y )N (p^ $ ±

, z')]d7 . (31b)

ny 1 1 in ny

J^=—CO —oo

' (z

1 = z')

14
Now M is defined in eqs. (5) as ny

M

(1) (p', f, «•) = V'x e H

(1)

(K P ')e

in<!,, ei^' . (32) ny z n

'

From Graf's addition theorem ([7], formula (9.1.79)) for cylindrical waves, we have the identity

00 HyW^10*' = J *i

l l

(Kd) J (KPl )e

lm^ , (33) n

L- m=-°° n-m ml

which allows M in eq . (32) to be written

ny

oo M^V, ', z«) =

I U

(n

l_l (Kd)M (p r * r

z) , (34a)

' m=-°°

since V'x = V x and z' = z. (That V'x = v x follows from the fact that the curl operator

by its integral definition remains invariant under coordinate transformations.) Similarly

oo N^V, ', z«) = I H^ (Kd)N (p r * 1

, z x

) . (34b) m=_oo

Substitution of eqs. (34) into eq. (31a) yields through comparison with eq. (31b) (after

interchanging m and n, and using orthogonality (6)) the relationship between the J (kp..) and

(1)

ml

H (ko ) modal coefficients, m

oo a<

' S -=

V n (D

I H W (Kd)a'

S

. (35) In L m-n m

v ' m=-°°

Further substitution of a' of eq. (35) into eq. (29) yields (after m and n are again

interchanged)

,

2 oo CO oo K=ll I a Ri>)l£>d)]a;S (y). (36)

s=l n=-°° -oo m=-°o

Comparing eq. (36) with eq. (30) while remembering that the a' are linearly independent

g allows us to write the final relationship between the desired receiving functions R' in

the cylindrical coordinate system centered outside the probe and the known receiving

functions R' in the cylindrical coordinate system centered on the probe:

R

n

S ^ = I R

im ( ^Hn-m

(Kd) ' < 37 >

Equation (37) displays a very interesting and useful property. Even though the probe

receiving functions R' in the system centered on itself may be non-negligible for only a

g few values of m, eq. (37) will give the receiving coefficients R' (y) for much larger

values of n. This means that the transformation expressed by eq. (37) can be applied even

to very small probes which are described by a small number of cylindrical modes. Also

note from eq. (37) and (40) below that as n gets larger than Kd, R' (y) may get very large,

since H * (Kd) gets very large. Further discussion of this increase with n > Kd is contained

in section 5 below.

15
Finally, it should be mentioned that most probes used in practice will be reciprocal

(see eq. (15a)) so that eq. (37) may be rewritten

9 °° k

s m - Ht i <-«"*iV^2i<-> • »»

o o m=-°°

s where the T' (y) times a' are merely the modal coefficients of a probe-centered cylindrical

s s wave expansion of the radiated field of the probe. Of course, the T' a

1 = B' can be com- lm o my

puted from the far-electric-field (phase and amplitude) of the probe by inverting eq. (8a)

for |y| ± k using the orthogonality (6) of the M and N functions. (In section 5 it is seen

that only |y| £ k are needed to compute the fields beyond a wavelength or so from antennas

which do not have extraordinarily high reactive fields.)

s As a check on eq. (38), if the T' (-y) for an ideal electric dipole probe are determined and inserted into eq. (38), eqs. (7) emerge immediately from eqs . (26).

4.2.1 Receiving Functions Directly in Terms of the Far-Field of the Probe

The application and evaluation of eq. (38) is quite straight-forward. The complex

far-electric-field pattern of a reciprocal probe is measured. Equation (8a) is used to compute the cylindrical wave coefficients T' (-y) , and finally the summations in eq . (38) are performed computing the necessary Hankel functions . For typical probes such as openended waveguides, the coefficients T' (—y) become negligible for m greater than approximately

Ka 1 , where a' is the radius of the smallest cylinder (centered at the origin of the C system) circumscribing the probe aperture (see section 5). Thus for a' on the order of a wavelength, the summation needs to include only the few modes from about m = -Ka' to + <a'

.

Although the evaluation of eq. (38) is straight-forward, it is also tedious. It would

s be helpful to express the required probe coefficients R' (y) directly in terms of the farfield of the probe, as can be done in planar scanning [3]. One technique for accomplishing

this, suggested recently by Borgiotti [11], is to express the receiving functions R' (y)

of the probe in terms of an integral of plane waves, and then evaluate the integral

asymptotically. Here we will use an alternative approach which involves replacing the Hankel

function in eq. (38) by the following asymptotic approximation [13]

:

H

( l } (x) = / 2 (-i) n

e i(/^-^ + n Sin_1 5

" 1

} (39) n / ~z~zz

x>n / it/ x

z -nz

If the asymptotic form (39) is substituted into eq. (38) and all terms of order m

2 /Kd

and higher are assumed negligible (more precisely, m

2

/i<cd <<2tt) . eq . (38) becomes.

n + Ka

' -1

R

n

S( ^

= ^z^^ H

n

l} ^d > I ^

< C-Y) e lm S±I1" *f . (40) n n<£«c n L

, lm o o m=-Ka

The restriction x>n translates to d>a+a', and the Hankel funtion H'(icd) in eq. (40) can be

computed from eq. (39) if desired. From the far-field eq. (8a) the summations in eq. (40)

reduce to

16
. -1 n ikr

V/. N m „,1 , s lm sin —r -re _ , .-In. Q -1 -y N ... . IW T

i m

(-Y> e Kd = 2*KT-El<l ,(r

l

,(|, l = Sln ^d 7r ' 9 1 = COS Pr +~ mo (41a)

1

„ ..-In . ikr 1 ., V/. N m m ,2 , x lm sin —r -ire „ , , .-In. Q -1 -y. ,,,, x I(i) TJ mo(-y)e Kd = ^-r- V^l = sin

^ + ,, (^ = cos £) (41b)

1

where E (r , § ,8 ) is the electric field of the probe in the spherical coordinates (r, ,<j>-, , 9-,

)

of the C system. Equations (40,41) show the desired result, that the probe receiving

1 2 functions, R' (y) and R' (y) , can be written directly in terms of the $ and 9 components

respectively of the far-electric-field of the probe in the hemisphere of the test antenna.

The only restrictions in using eq. (40) are that d>a+a' , which says essentially that

the probe and test antenna must not touch, and that m

2

/i<d <<2tt. Since the maximum value of m required is about Ka' , this latter restriction is satisfied if

d >>a

,2 /X . (42)

Equation (42) stipulates that the scan axis of rotation of the test antenna must be in the

far-field of the probe in order for the asymptotic expression (40,41) to approximate the

exact formula (38). (And the approximation becomes better as d becomes larger). If this

condition is met, eqs. (40,41) reduce the problem of probe correction for cylindrical

scanning to nearly the simplicity of the probe correction for planar scanning [3].

Probe coefficients have been computed at NBS using both the exact eq. (38) and the

approximate eqs. (40,41) for open-ended rectangular waveguide at 9 GHz with a separation

d of .5 meters or more. The results agreed within differences that were much smaller than

the accuracy with which the complex far-field pattern of the probe could be measured.

5. APPLICATIONS OF THE SAMPLING THEOREM AND FAST FOURIER TRANSFORM

Let's assume the receiving characteristics of the probe (s) have been determined by

either the "direct measurement method" of section 4.1., or the "transformation method"

using eqs. (37), (38). or (40,41). Then after scanning the test antenna with two linearly

independent probes (or one probe in two orientations), eqs. (27) yield immediately the transs s mitting° functions T (y) of the test antenna. From the values of T , the far-field, for n

' n example, of the test antenna can be computed by summing eqs. (8) with

B

S (k cos 9) = TS (k cos 9) a . (43) n n ' o

There still remains an important question to be answered. How closely must the scan points be spaced, at which the amplitude and phase of the probe outputs b'(<j) , z ) and

b"(cf> , z ) are measured, in order for the coupling product integrals I'(y) and I"(y) in

eqs. (27) to be evaluated accurately. This question is answered by looking at the definition

of I'(y) (or I"(y)) given in eqs. (26): n x,/v n" & ^ max

2 o 2tt -in<f) -iyz

r(y)= I*;

S <Y><<Y)=-f- J J* (* ,z

o )e

°

e °

d V2 o • <">

s=l 4ir^a min o

o z o

17
(The ±°° limits of integration for z have been replaced by (z , z ) because in practice

the probe output b'(<fi , z ) must be assumed negligible outside a certain finite scan length.)

Consider the transmitting functions T (y) of the test antenna. They represent the

coefficients of the cylindrical waves emanating from the test antenna (with a = 1)

th ° transmitting into free space. The subscript n refers to the n order Hankel function,

H

(1)

(k P ) k

2 = k2 - y

2

. n

It can be shown [8] that the reactive energy in the cylindrical modes M.N containing

(1) ny ' ny 5 the Hankel function H (<p) grows extraordinarily large as n becomes greater than <p, for

k real (|y|jL k) . This implies that the reactive fields of the antenna are extremely large

unless the T (y) grow extremely small for n > <a, where "a" is the radius of the smallest

cylinder (centered at the rotation axis z of the test antenna) circumscribing the test

s antenna . Ordinary antennas do not have extremely high reactive fields, and thus the T (y)

will become negligible for n > Ka(|y|j< k)

.

There is a reasonable physical explanation accompanying the interesting result that the

number of modes required is proportional to k = /k2 -y 2 (|y|_£ k) . Equations (8) show that

the values of |y| nearer to but less than k correspond to the far-field on a circle (oj<<j><2tt) nearer to the z-axis (9 = cos ^-) . Assuming the number of far-field fluctuations per solid

angle are roughly the same in all directions, the far-field variation with <J> will be lesser

for circles nearer the z-axis (larger |y|). Thus the number of cylindrical modes needed to expand the field will grow less and less as |y| approaches k, since the higher order modes,

which represent the more rapidly varying parts of the field, will not be needed to expand

the slower variations with <Ji. Quantitatively, this intuited result manifests itself in the

fact that the maximum number of modes needed to represent the far-field at the angle 9_ is

approximately ka s in 8 . Of course, in eq. (44) one is more interested in the behavior of the coupling product

I' (y) than the T (y) alone. Section 4.2.1 showed that, for typical probes, the receiving

11 s n

(i ) functions R' (y) behaved essentially as H

] (icd) . This means that for n > Kd the receiving

functions grow extremely large. However, as long as the distance between probe and test antenna is larger than a wavelength or so, the increase in R' (y) with n is well outweighed

s s by the decrease with n of T (y) . In other words, the coupling product I'(y) as well as T (y)

becomes negligible for n > <a (k real)

.

It can also be shown that as |y| becomes larger than k (k imaginary), the coupling

product expressed by eq. (44) becomes negligible for all n, provided the test antenna does

not possess unreasonably high reactive fields and the probe is further than a wavelength or

so from the test antenna. A comparatively simple demonstration of this fact begins by

writing b'(<f> ,z ) in eq. (44) in terms of a plane-wave spectrum, then using the corresponding

results [12] from the plane-wave description of antennas.

In brief, under the assumptions stated above, the I'(y) will become negligible for all n and y not far outside the circle shown in figure 4 and defined by

18
n

2 =( Ka )2 = ( k

2_

Y

2 )a2

f or n

2 +( Ya) 2 = (ka) 2 . (45)

Applying this information about I'(y) to eq. (44) reveals that b' (<t> ,z ) can be

considered the double Fourier transform of a band-limited function - provided, as explained

above, the separation distance d between test and probe antenna is more than a wavelength

or so, and the reactive fields of the test antenna are not extraordinarily large.

Consequently, the sampling theorem [9] can be utilized to convert the integration on the

right side of eq. (44) to the following summation:

i;(Y) - —J l h' tf . z*)e ° e ° , (46a)

4kn k„aa i,£ 12 o

J

i £ where i> J and z refer to points taken at increments of A, /2a and A„/2 radians, respectively,

Similarly,

o I"(y)o may be written

12

Vy) = 7~ J,

b

o

( V Z o )e

4kn k„aa i,£

1 2 o

J

The values of k.. = 2tt/X

1 and k„ = 2tt/A„ need not be chosen much larger than the free-space wave number k = 2tt/A. It is emphasized that the angular data increments of approximately

V2a radians remains independent of the distance to the probe. However, the minimum radius

"a" circumscribing the test antenna is measured for a cylinder centered at the z-axis, i.e.,

the rotation axis of the scanning system. If for any reason the test antenna had to be

mounted with a large offset distance from the axis of rotation, the minimum circumscribing

radius "a" would be much larger than the overall radius of the test antenna itself.

In summary, eqs. (46) show that the amplitude and phase of the output of the probe (s)

need be sampled only at discrete points on the surface of the scan cylinder. As long as the probe does not get within a wavelength or so of ordinary test antennas, the data point

spacing need be no closer than about A/2a radians for the <f> angular separation (irrespective

of the distance to the probe) and A/2 for the z increments. After the output of the probe(s)

is recorded for the entire cylindrical grid of data points, the "fast Fourier transform" (FFT) can be utilized to compute I

1 (yf and I"(y) from eqs. (46). From the values of (I'(y), I"(y))

and the receiving functions of the probe, the transmitting function T (y) of the test antenna

can be evaluated immediately from eqs. (27). Finally the entire electromagnetic field outside

s s the test antenna is determined by substituting B (y) = T (y) a into eqs. (4) or into

eqs. (8) for just the far-fields.

6. SUMMARY

The basic theory and formulas needed to implement the probe-corrected measurement of

antennas by scanning on a cylinder have been derived by a systematic approach based on the

19
cylindrical-wave source scattering-matrix description of antennas. This approach, which

parallels the plane-wave scattering matrix approach used by Kerns [3], proves simple and

straightforward and does not require the probe or test antenna to be reciprocal.

The derivation began by expressing the EM fields radiating from the test antenna in

a complete set of cylindrical vector wave solutions (M, N) with first Hankel functions

(H ) for the radial dependence. The radiation characteristics of the test antenna are determined by the weighting coefficients of these cylindrical waves, which in themselves

remain independent of the particular antenna. Expressions were derived for the far-fields

(a technique was introduced in appendix A for deriving the far-fields by direct manipulation

of the Fourier transforms rather than by the more involved method of stationary phase)

,

power gain function, complex polarization ratio, and total radiating power of the test antenna in terms of the cylindrical waves and the weighting coefficients B (y) . Thus

the problem of determining the radiation characteristics of antennas by cylindrical scanning

s reduces to that of determining the modal coefficients B (y)

.

If an ideal dipole antenna were available to measure the transverse electric field on

a cylinder about the test antenna, the coefficients B (y) were shown to emerge immediately

through a trivial application of the orthogonality relationships for the vector wave solutions

(M, N) . When an arbitrary probe antenna was inserted into the picture, a second set of

cylindrical waves identical to the first, except for Bessel functions (J ) replacing H , was required to express the EM fields excited by the sources on the probe. The coefficients

of this second set of J cylindrical modes along with the input to the test antenna were

n

(1) related to the coefficients of the first set of H modes and the output of the test

n

r antenna through the "source scattering matrix" for the test antenna.

The source scattering matrix contains a complete and convenient description of the

transmitting, receiving, scattering, and reflection properties of the test antenna. The

transmitting elements of the matrix essentially equals the required modal coefficients

B (y) . Reciprocity was also shown to display itself conveniently as a relationship

between the elements of the scattering matrix.

A similar source scattering matrix was written for the probe antenna but with reference

to the origin or center of the coordinate system situated in the test antenna. This

"common-center" approach along with the use of (H , J ) scattering matrices greatly

facilitated the subsequent derivation of the transmission formula, which contains the

required coupling product between the transmitting coefficients of the test antenna and

receiving coefficients of the probe antenna. Another advantage of the common-center source scattering-matrix approach was that it clearly and completely separated the derivation of

the transmission formula from the problem of determining the receiving characteristics of

the probe.

The characterization of the probe was dealt with as an isolated topic in section 4.

It was found that the receiving coefficients of the probe could be either measured directly,

or transformed from its measured far-field. The specific transformation involved depends

on the choice of coordinate system (not necessarily cylindrical) in which the far-field

20
of the probe is expressed. Conforming to the work of Leach and Paris [2], however,

cylindrical coordinates were chosen to express the far-field of the probe and the

transformation was accomplished with the help of Graf's addition theorem. When the

axis of rotation of the test antenna lies in the far-field of the probe, Graf's addition

theorem simplifies to an asymptotic approximation which expresses the receiving coefficients directly in terms of the far-field of the probe.

Finally, it was shown that the transmission formula could be inverted to yield the

transmitting coefficients of the test antenna, and that the inversion could be accomplished

efficiently through the use of the sampling theorem and FFT algorithm. For ordinary

antennas, i.e., antennas which do not exhibit extraordinarily high reactive fields, the

number of cylindrical modes needed to expand the field beyond a wavelength or so from

the antenna was shown to depend on the elevation angle (90-6) , and was given approximately

as ka sin 6 where "a" is the radius of the smallest cylinder circumscribing the test antenna.

21
APPENDIX A. DERIVATION OF THE FAR-FIELDS IN TERMS

OF THE CYLINDRICAL WAVE COEFFICIENTS

In this appendix the far-fields radiated by a test antenna into free space (eqs. (8)

of the main text) are derived in terms of the coefficients of the cylindrical wave expansion

(eqs. (4) of the main text). Although Leach and Paris [2] have derived the results using

the method of steepest descent, the alternative derivation shown here is interesting enough

in itself to warrant a separate appendix.

The starting point for the derivation is eq. (4a) for the E field outside the test

antenna expressed in a cylindrical wave expansion. If M and N from eqs. (5) are substituted

into eq. (4a), and the large argument value of the Hankel function,

. 7T H (Kp) % (-3.) /-=— e 4

7

, n

K v /TTKp '

is used in M and N, the far-electric-field can be written

°° _ i<p

E(p,<|>,z) = / *£

y

(P,<t>) *-g- e 1YZdy , (Al)

where

TT.

. _ —j. °° VP - W "^ e

4 I

( - 1)n[B

n

(Y)M

nY + H1 ' ' (A2>

nwith

M = i(— e - <e.)

ny P P <j>

M E h-xKe - ^ e. + K

2 e ) . (A3)

ny k

v p p <J) z'

Symbolically, eq. (Al) may be written

-l r ,-. e 1KP

E = F

x [(Ke ) ^— ] , (A4)

r->oo Y K

where F and F will be used to denote the Fourier transform and its inverse, respectively.

The convolution theorem transforms eq. (A4) to

i<p E w=^ F^siFi[V^ • (a5)

The second inverse Fourier transform in eq. (A5) is nothing more than an integral representation

of the zeroth order Hankel function of the first kind, i.e.

TT. -1 e ±Kp

r°°e iKp Iy(E-z) (1), , lr ,i zs /H 4 1 ikr -ik? cos 9 FfUV1 = / ^T e dY = fflo (k/(?_z)+p) r- 7

k7 e e e

^ —oo 22
Thus, the far-electric-field may be written

TT. —,—. / 1 4 ikr r r -± r — , -xKt, i E(r) = LIT e e J F r

[K£: ]e ^ . (A6)

^-KX3

But this last integral is the Fourier transform of an inverse Fourier transform. Thus, the

result is the original function (multiplied by 2tt) evaluated at k cos 0, i.e.,

TT. —,—. /2tF 4 ikr, . — , , E(r) = J-^ e e k sin 9 e (y=k cos 9)

(k = A^-Y Z = k sin when y=k cos 0) . (A7)

Substitution of e (y=k cos 0)from its definition in eq. (A2) gives the required far-field

expression for E shown in eqs. (8) of the main text,

ikr o° . . . E(r,<j>,0) = "2k sin ~~ I (-i) n

[B^(k cos 9)e,-iB*(k cos 0)e

Q ]e

in9 (A8a)

e xl

H(r,d>,6) = -JL-^°° . (A8b)

r-*» o

Finally it is mentioned that the same technique can be utilized to find the far-fields of a radiator in terms of its plane-wave spectrum, i.e., to derive eqs. (1.2-16) of reference

[3b]

.

23
APPENDIX B. DERIVATION OF THE SOURCE SCATTERING MATRIX EQUATIONS

AND RECIPROCITY RELATIONS FOR THE PROBE AND TEST ANTENNAS

Refer first to the test antenna of figure 1. The reference surface S is a dividing

plane between the "test antenna proper" and the shielded "generator" of the test antenna.

Let S be the surface of the cylinder of radius p surrounding the test antenna.

In the region between the generator and the surface S assume that the EM fields

satisfy the following very general form of Maxwell's equations

VxE - icoB =

VxH + ioiD - O'E =

D = e-E + T«H

B = V-E + y-H . ( B1 )

Consider the "adjoint fields" which satisfy the adjoint set of equations,

VxE

a

- icoB3

=

VxH

3

- itoDa

- O'E3

=

D

S

= e -E

a

- v -H3

t t

B3

= -7

t -E^ + ^-"S3

, (B2)

where the subscript t designates the transposed dyadic. A straightforward manipulation of

eqs. (Bl) and (B2) yields an "adjoint reciprocity lemma" [3b]

(E

3

xH-ExH

3 )-nda = , ( B3 )

S +S

o p

where (E,H) are any fields satisfying eqs. (Bl) and (E

3

, H

3

) are any fields satisfying

eqs. (B2).

On the waveguide reference surface S

E = (a +b )e , H. = n (a -b )h

t ooo t oooo

-zA , a , a.~ —a / a , a s , E = (a +b )e , H„ = n (a -b )h

t ooo t oooo

[ e xh *nda = 1 , S ° °

o

\ and thus eq. (B3) becomes

2n (b 3a -b a

3

) = / (FxH-ExH3

) -e da . (B4)

o o o o o

' P

p 24
Divide the fields into fields emanating from sources at the test antenna (denoted by

subscript "o") and fields emanating from the probe (denoted by subscript "1"). Then

/ (F/ScH-ExH^-e da = / (E^xTL-E xH?+E?xH -ELxlfVe da . (B5) ; p *olollolop p p

The integrals over the "o" fields alone and over the "1" fields alone vanish because they

satisfy the adjoint reciprocity lemma separately. Substitution of eq. (B5) into eq. (B4)

gives an expression which will be called the source adjoint reciprocity lemma ,

2n (b a

a -b a

a

) = / (EaxlL-E xH

a+EaxH -ElXHa )-e da . (B6) ooooo •'olollolo p

p

If the E, H fields in the coordinate system of the test antenna given by eqs. (13) are substituted into the right side of eq. (B6), it becomes

2 oo oo 2 VboVVo> " TT- I / ("DM ^(Y)a-(-y)-b-(-y)a^(Y )]dY . (B7)

o n=-°° -oo s=l

To get eq. (B7) in the form shown, the orthogonality relations (6) between the M and N functions and the Wronskian for the Bessel functions were used ([7], Formulas 9.1.15-9.1.17),

Recall that a (y) and a (y) are coefficients of modes with arbitrary sources outside

the test antenna and its adjoint, respectively. Thus, a (y) and a (y) may be chosen independently and arbitrarily. Similarly, a and a , the inputs to the test antenna and its

adjoint, may be chosen independently and arbitrarily.

In particular, if we choose

a 1 as _ a = 1 a =

o n

eq. (B7) yields

2 2 °° oo b

Q = ba

a. +z^r I I [ (-l) n

bas

(-y)a

S (y)dY . (B8) a , tiiK r v f r ,\u,as, . s, ., Vo + 7TYT I I / (-D b_n

(-y)an(y)dy . o o s=l n=-«> -°°

But this is nothing more than the source scattering matrix equation (14b) for the test

antenna with

r = ba

o o

R

S (y) =^|L (.l) n

b

aS

(-y) . n ' nZk ' -n ' o o

In a similar manner the second source scattering matrix equation (14b) emerges if we choose

a

a

= , aaS= 6 6 6(y-g)

o n sq run '

25
In other words , the source scattering matrix equations (14) derived from Maxwell' s equations

through the source adjoint reciprocity lemma (B6) . Equation (B6) also reveals the reciprocity relationships between the source scattering

matrix of the test antenna and it adjoint. When eqs. (14) are inserted into eq. (B7) the

necessary and sufficient conditions for eq. (B7) to be satisfied for all values of the "a" coefficients are

r a = r (a)

o o

o o 0"» - <"!>" *£ «> M

C%<-e,-Y> - <>•« • (d) <B9)

By definition, a reciprocal antenna is one in which the antenna and its adjoint are identical.

Thus for a reciprocal antenna the adjoint superscript "a" in eqs. (B9) are removed and eq. (B9b) becomes identical to eq. (B9c) . (Also, eq. (B9a) becomes a trivial identity.)

We have just completed the derivation of the source scattering matrix equations (14)

and their reciprocity relationships (15) for the test antenna directly from Maxwell's

equations through the source adjoint reciprocity lemma (B6). For the probe antenna we can proceed in the same fashion as for the test antenna, but this time the region of interest

lies between S and the probe generator which is separated from the probe antenna proper

p by the waveguide reference surface S'. Applying the adjoint reciprocity lemma to this

external region yields a surface integral similar to eq. (B3),

(E axH - E xH

a )-n'da' = , (BIO)

S'+S +S

o p o

where the primes have been used to emphasize the fact that we are dealing with the probe

antenna. Immediately, the radiation condition at infinity eradicates the integral over

S , and eq. (BIO) becomes identical in form to eq. (B3).

Substitution of the EM fields of the probe waveguide on S' into eq. (BIO), and division

of the fields on S into those emanating from the test and probe antennas yield an equation

identical in form (except for a negative sign) to eq. (B6) , i.e., a source adjoint reciprocity

lemma for the probe antenna is derived. Moreover, substitution of the E' H' fields from eq.

(17) gives an expression identical in form to eq. (B7) but with the primed coefficients of

the probe system. Continuing the derivation as for the test antenna gives rise to the source scattering matrix eqs. (18) for the probe, and reciprocity relationships for the probe

identical in form to eqs. (B9) . That is, the reciprocity relationships for a reciprocal

probe are 25
R-n^> = ^U Th K^ (Blla)

o o

s'^S n (-3,-Y) = s'^q (Y,3) . (Bllb)

—m

, —

n

nm

Finally it is mentioned that the derivations for the plane-wave scattering matrices and

their reciprocity relationships for antennas have been extended to electroacoustic transducers [10], The analysis in reference [10] can be utilized immediately to extend the

results of this appendix to the cylindrical scanning of electroacoustic transducers.

27
GLOSSARY OF SYMBOLS

Radius of smallest cylinder (centered at the rotation axis z of the

test antenna) circumscribing the test antenna.

Radius of smallest cylinder (centered at the origin of the C. coordinate

system) circumscribing the probe aperture.

a ,a' or a"

o o o

b ,b' or b" v o o o

J

Ingoing

Outgoing

wave-coefficients in the test and probe antenna feeds, respectively.

n n In

ib s ,b'

s ,b;

s

*- j n' n ln}

Outside

Inside

source cylindrical mode coefficients for the test antenna, for

the probe with respect to the C coordinate system and for the

probe with respect to the C. coordinate system, respectively.

B Radiated cylindrical mode coefficients for the test antenna,

C(p' ,<f>' ,z' ), C. (p 1

, <(>.. , z

1 ) : The two cylindrical coordinate systems fixed in the probe;

the first (C) centered on the test antenna and the second (C ) centered

on the probe (see fig. 3)

.

d : Distance between the z-axes of the C and C. system (see fig. 3)

(E,H) , (E' ,H' ) , (E

1 ,H1

) : Complex electric and magnetic fields referred to in the test antenna,

the probe-C, and the probe-C.. coordinate systems, respectively.

§ : a Unit vector for coordinate a.

r ,r

o o

Fourier transform parameter for the z-part of the cylindrical modes

(-co < y < co )

.

The test and probe antenna input reflection coefficients defined at the

i surface S and S , respectively,

o o

r

h

(1 \h<2) n n Cylindrical Hankel functions of the first and second kind.

,(!>' Derivative with respect to argument of the Hankel function of the first kind.

I', I" : n n Coupling product integrals of the transmission formula (defined by the right

side of eq. (26a) and (26b), respectively).

28
J : Cylindrical Bessel functions.

k : Free-space wave number = 2tt/A.

2 2 1/2

k : (k -y )

"

, taken positive real when y < k, and positive imaginary when

Y

> k.

A : Free-space wavelength.

(M , N ), (M , N ) : Cylindrical vector wave solutions with J and H radial

ny ny ny ny n n dependence, respectively.

D : Characteristic admittance of the propagated mode in the test antenna feed,

o

(<|> , z ) : Cylindrical coordinates describing the position of the probe as it scans

the test antenna (see fig. 2).

(p,<J>,z), (r,<J>,6) : Cylindrical and spherical coordinates, respectively, fixed in the

test antenna (see fig. 1).

s 's "s 's R , R or R , L : Cylindrical receiving functions for the test antenna, for the probe

with respect to the C coordinate system, and for the probe with respect to the C. coordinate system, respectively.

T ,T or T ,T n: Cylindrical transmitting functions for the test antenna, for the

n' n n ' In J 6 probe with respect to the C coordinate system, and for the probe with

respect to the C, coordinate system, respectively.

SQ SQ

S H

, S : Scattering functions for the test and probe antenna, respectively, nm nm

t S , S : The waveguide reference surfaces for the test and probe antenna, respectively.

a) : Angular velocity in the suppressed time factor e (u > 0)

.

29
Figure 1. Schematic of test antenna and its

coordinate system.

30
p. p, *'

<£

Probe Antenna

z.z axes

Test Antenna

Figure 2. Schematic of probe antenna and its

coordinate system.

31
Figure 3. Schematic of the two cylindr

coordinate systems fixed in

probe

32
n (order of Hankel function)

Figure 4. Coupling product becomes

negligible not far outside

the circle of radius ka.

33
REFERENCES

[I] Brown, J., and Jull, E. V., The prediction of aerial radiation patterns from near-field

measurements, Proc. of the Institute of Electrical Engineers, 108-B , 42, 635-644

(Nov. 1961).

[2] Leach, W. M. , Jr., and Paris, D. T. , Probe compensated near-field measurements on a cylinder, IEEE Trans, on Antennas and Propagation, AP-21 , 4, 435-445 (July 1973).

[3] a) Kerns, D. M. , Correction of near-field antenna measurements made with an arbitrary

but known measuring antenna, Electronics Letters, 6^, 11, 346-347 (25th May 1970).

b) Kerns, D. M. , Plane-wave scattering-matrix theory of antennas and antenna-antenna

interactions: formulation and applications, J. Res. Nat. Bur. Stand. (U.S.), 80B, 1,

5-51 (January-March 1976).

[4] Morse, P. M. and Feshbach, H. F. , Methods of Theoretical Physics, Part II, pp. 1764-1767

(McGraw-Hill, New York, N.Y., 1953).

[5] Jensen, Frank, Electromagnetic near-field-far-field correlations, Ph.D. Dissertation,

Technical University of Denmark, Lyngby, Denmark (July 1970).

[6] Wacker, P. F. , Non-planar near-field measurements: spherical scanning, Nat. Bur. Stand. (U.S.) Internal Report 75-809, Boulder, Colorado (June 1975).

[7] Abramowitz, M. , and Stegun, A., Handbook of Mathematical Functions (National Bureau of

Standards, Washington, D.C., 1964).

[8] Collin, R. E. , and Rothschild, S., Evaluation of antenna Q, IEEE Trans, on Antennas

and Propagation, AP-12 , 1, 23-27 (January 1964).

[9] Petersen, D. P., and Middleton, D., Sampling and reconstruction of wave-number-limited

functions in N-dimensional euclidean spaces, Information and Control, 5_, 4, 279-323

(December 1962)

.

[10] Yaghjian, A. D. , Generalized or adjoint reciprocity relations for electroacoustic

transducers, J. Res. Nat. Bur. Stand. (U.S.), 79B (Jan. -June 1975).

[II] Borgiotti, G. V., An integral equation formulation for probe corrected far-field

reconstruction from measurements on a cylinder, International Symposium Digest of the

IEEE Antennas and Propagation Society, Stanford University (June 1977).

[12] Yaghjian, A. D., The reactive and far-field boundaries for arbitrary antennas derived

from their quality factor (to be published)

.

[13] Jeffreys, Harold, Asymptotic Approximations, Sec. 4.3, eq. (7) (Clarendon Press,

Oxford, Great Britian, 1962).

[14] Montgomery, C. G. , Dicke, R. H., and Purcell, E. M. , Principles of Microwave Circuits,

Sec. 9.19 (McGraw-Hill, New York, New York, 1948).

34
U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS TN 696 (Revised Sep 77)

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Near-Field Antenna Measurements on a Cylindrical Surface:

A Source Scattering-Matrix Formulation

5. Publication Date September 1977

6. Performing Organization Code

276.05

7. AUTHOR(S)

Arthur D. Yaghjian

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2765276

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as Item 9

13. Type of Report & Period

Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

The theory for probe-corrected measurement of antennas by scanning on a circular cylindrical surface enclosing the test antenna in the near-field is

formulated from a source scattering matrix description of the test and probe

antennas. The basic transmission formula is derived without recourse to

reciprocity, and from a common center approach which separates as an isolated

problem the probe characterization and transformation. Both an experimental

technique and an approximate analytical technique are presented to determine

the required transformed probe coefficients without the use of addition theo- rems. The approximate technique, which is developed from the exact addition

theorem transformation, yields the probe coefficients directly in terms of

the far-field of the probe, provided the rotation axis of the test antenna

lies in the far-field of the probe. Computer inversion of the transmission

formula is accomplished accurately and efficiently with the aid of the

sampling theorem and FFT algorithm.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Cylindrical scanning; near-field measurements; scattering matrix.

18. AVAILABILITY LT] Unlimited

2 For Official Distribution. Do Not Release to NTIS

|v I Order From Sup. of Doc, U.S. Government Printing Office Washington, D.C. 20402, SD Cat. No. C13

i J Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

40

22. Pr

USCOMM-DC 290

•tr U.S. GOVERNMENT PRINTING OFFICE: 1977-777-067-/1201

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of the standards is to establish nationally recognized require- ments for products, and to provide all concerned interests with a basis for common understanding of the characteristics of the products. NBS administers this program as a supple- ment to the activities of the private sector standardizing

organizations.

Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest to the consumer. Easily understandable language and

illustrations provide useful background knowledge for shopping in today's technological marketplace.

Order above NBS publications from: Superintendent of Documents, Government Printing Office, Washington, D.C.

20402. Order following NBS publications—NBSIR's and FIPS from

the National Technical Information Services, Springfield, Va. 22161.

Federal Information Processing Standards Publications (FIPS PUB)—Publications in this series collectively consti- tute the Federal Information Processing Standards Register.

Register serves as the official source of information in the Federal Government regarding standards issued by NBS

pursuant to the Federal Property and Administrative Services Act of 1949 as amended, Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717

(38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulations).

NBS Interagency Reports (NBSIR)—A special series of

interim or final reports on work performed by NBS for outside sponsors (both government and non-government).

In general, initial distribution is handled by the sponsor;

public distribution is by the National Technical Information

Services (Springfield, Va. 22161) in paper copy or microfiche form.

BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey bibli- ographies are issued periodically by the Bureau: Cryogenic Data Center Current Awareness Service. A litera- ture survey issued biweekly. Annual subscription: Domestic, $25.00; Foreign, $30.00.

Liquified Natural Gas. A literature survey issued quarterly.

Annual subscription: $20.00.

Superconducting Devices and Materials. A literature survey

issued quarterly. Annual subscription: $30.00. Send subscription orders and remittances for the preceding bibliographic

services to National Bureau of Standards, Cryogenic Data

Center (275.02) Boulder, Colorado 80302.
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