Theory of diffraction in microwave interferometry, D. M. Kerns and E. S. Dayhoff x (August 25, 1959)

Full text of “Theory of diffraction in microwave interferometry

JOURNAL OF RESEARCH of the National Bureau of Standards— B. Mathematics and Mathematical Physics 

Vol. 64B, No. 1, January- March 1960 

Theory of Diffraction in Microwave Interferometry 

D. M. Kerns and E. S. Dayhoff x 

(August 25, 1959) 

Microwave Michelson and Fabry-Perot interferometers are respectively considered as 
instances of: (1) A "reflection system", consisting of a radiating-receiving system and a 
reflecting object (e.g., a finite mirror) ; and (2) a "transmission system", consisting of a 
radiating system and a receiving system with an object (e.g., a Fabry-Perot etalon) inter- 
posed. The basic theoretical objective is the calculation of the amplitude and phase of the 
(time-harmonic) received signal in the systems considered. The electromagnetic field in 
space transmission paths is represented in terms of continuous angular spectra of vectorial 
plane waves, and the elements of the systems are described by means of suitable tensor 
scattering matrices (having both discrete and continuous indices). Needed scattering 
matrices are considered known; relationships to experimentally determinable data are 
outlined. The general case of either the reflection or transmission system is soluble 1 formally 
in terms of a series of integrals stemming from the Liouville- Neumann series solution of 
certain integral equations. Formulas are obtained for models of the Michelson and Fabry- 
Perot instruments with arbitrary radiating and receiving characteristics. The theory and 
various features of the instruments considered, including Fresnel-region (or quasi-optical) 
behavior, are illustrated by means of examples obtained by choosing relatively simple and 
rather hypothetical analytical expressions for the radiating and receiving characteristics. 



1. Introduction 

Microwave versions of the Michelson and the 
Fabry-Perot interferometers are being studied and 
developed at the National Bureau of Standards with 
a view to high-accuracy measurements of wave- 
lengths, lengths, and the speed of light [l]. 2 The 
data provided by these instruments are, however, 
subject to corrections for effects which are completely 
negligible in optical interferometry but significant 
in the microwave region. In this work — hi contrast 
to the corresponding optical case — the principal 
such effect results from the wavelength not being 
negligibly small relative to the dimensions of the 
apparatus, and consequently the effects of diffraction 
upon the wavelength as observed by the instrument 
must be carefully considered in order to exploit 
fully the potential accuracy of the method. In 
this paper we consider these effects and present 
analytical tools suitable for making a correction to 
the apparent wavelength [2]. This should make 
possible the use of Fresnel region microwave inter- 
ferometry for work of the highest accuracy. 

For the purposes of our analysis the pertinent 
aspects of the class of microwave Michelson inter- 
ferometers are represented by a "general reflection 
system/' Similarly, the microwave Fabry-Perot 
instruments are represented by a "general trans- 
mission system.." (These basic arrangements are 
described more fully below.) Inasmuch as we wish 
to provide a theory inherently capable of dealing 
with high-accuracy experiments, a considerable 



i Present address: U.S. Naval Ordnance Laboratory, Silver Spring, Md. 

2 Figures in brackets indicate the literature references at the end of this paper. 



degree of generality is required to" avoid over-idealiza- 
tion. It is interesting that the required generality 
is such that the general reflection system could 
equally well represent a system consisting of radar 
and target, for example; similarly, the general 
transmission system could represent a point-to-point 
communication system. In view of the generality 
of the basic arrangements it seems likely that appli- 
cations of the theory will also be found in problems 
other than the ones that motivated this work. 

The specific theoretical objective is the calculation 
of the amplitude and phase of the received signal in 
the systems considered. Theoretical expressions 
for the received signal will contain, at least implicitly, 
all available information regarding diffraction errors, 
intensities, etc. The analytical technique employed 
involves a transducer point-of-view, scattering- 
matrix formalism and constitutes a generalization 
of a part of the theory of waveguides and waveguide 
junctions. 

The arrangements to be considered involve 
radiating, receiving, and radiating-receiving systems 
as elements, which we shall occasionally refer to as 
"terminals" or as "terminal apparatus." Such an 
apparatus is shown schematically in figure 1. The 
left-hand block in the figure may contain sources of 
power, detectors, and any other required auxiliary 
equipment, all of which is assumed to be shielded. 
This equipment is connected by means of waveguide 
to a waveguide-space transducer, represented by the 
triangular symbol in the figure. For simplicity in 
the formulation it will be assumed that the wave- 
guide is of a self-enclosed or shielded type. The 
arrangement and details of the apparatus are 
highly arbitrary, an essential but not very restrictive 



532245—60- 




Figure 1. Schematic radiating-receiving system. 



condition being that the transducer proper possesses 
the property of linearity with respect to electro- 
magnetic fields. (Details of the statement of 
hypotheses are given in sec. 2, A.) 

In one of the basic arrangements considered — the 
general reflection system, illustrated in figure 2— 
radiation from a radiating-receiving system is re- 
flected or scattered by a general reflecting object at 
a variable distance d and is partly received (and 
partly reflected or scattered) by the same system. 
The Michelson interferometer can be considered in 
general terms as a reflection system in which the re- 
ceived signal is observed, as a function of d, by 
causing it to interfere with a suitable reference 
signal and detecting the resultant amplitude. 

In the laboratory version of the Michelson inter- 
ferometer being considered in connection with the 
present w r ork, the radiation forms a well-defined 
beam (within its Fresnel zone), the reflecting object 




= 66 



RAD.-REC. SYSTEM 



SCATTERING OBJECT 



(Z=0)J 



Figure 2. Reflection system. 

If the scattering object is a finite mirror, for example, the arrangement represents 
a microwave Michelson interferometer. 



is a large but finite mirror placed well within the 
Fresnel zone of the beam, and multiple reflections 
between the mirror and the radiating-receiving sys- 
tem are relatively unimportant. It may be said 
that the instrument is operated in the Fresnel or 
quasi-optical region. This mode of operation im- 
plies the inequalities a^>\ and d<Ca 2 /X, where 
a is a measure of the cross section of the beam and 
X is the free-space wavelength corresponding to the 
frequency of operation. Under these conditions the 
radiation in the space between radiator and mirror 
approximates a homogeneous, plane standing-wave 
and the received signal is approximately propor- 
tional to exp (2ikd), where Jc=2w/\. 

The second basic arrangement considered is illus- 
trated in figure 3. Here a structure or object that 
in general may reflect, transmit, and scatter is inter- 
posed between a radiating and a receiving system. 
If the interposed object is some form of Fabry-Perot 
"etalon" [3], the arrangement represents a general 
form of the Fabry-Perot interferometer. In this 
type of instrument the received signal exhibits am- 
plitude variations sharply dependent on the spacing 
of the plates of the etalon, and this is attributable to 
interference among components of the radiation 
having experienced 0, 1, 2, . . . reflections between 
the plates. 



s 




> 



RADIATING 
SYSTEM 



(2 =0) j 



SCATTERING 
OBJECT 



RECEIVING 
SYSTEM 



Figure 3. Transmission system. 

If the scattering object is a suitable form of Fabry-Perot etalon, the arrangement 
represents a microwave version of the Fabry-Perot interferometer. 



In the laboratory version of the Fabry-Perot be- 
ing considered in connection with the present work, 
the instrument is a Fresnel-region instrument in the 
sense explained above, and multiple reflections, ex- 
cept between the plates of the etalon, are again 
relatively unimportant. 

In the theoretical approach used in the present 
paper, the (vectorial) electromagnetic field in a space 
transmission path is represented as the superposition 
of an angular distribution or spectrum of (vectorial) 
plane waves. Such a spectrum is in general deter- 
mined not only by the free-space radiation charac- 
teristics of a radiator, but also by the effects of all 
scattering elements involved (the radiator itself 
would be such an element, for example, in the case 
of a Michelson instrument with appreciable multiple 
reflections). Although the use of the plane-wave 
representation of the fields is suggested by the close 
approximation to a single plane wave that might 
exist in a Fresnel-region instrument, it is hardly 



necessary to remark that in a practical case even the 
unmodified free-space radiation spectrum ls likely to 
be extremely complicated in detail. This is because 
of such things as edge effects, geometrical imperfec- 
tions, and limitations in the design and construction 
of lenses and horns. 

The concept of the plane-wave resolution of the 
Held permits one to form a useful qualitative picture 
of the origin of an effective wavelength. Consider, 
for example, a Michelson instrument. Each ele- 
mentary wave in the spectrum whose normal makes 
an angle with the line along which translation of 
the mirror is measured has an effective wavelength 
X sec 0; the resultant effective wavelength can be 
thought of as a kind of average of the elementary 
contributions over the existing spectrum. Tims one 
might anticipate, for example, that the resultant 
effective wavelength should rather generally tend to 
be greater than the free -space wavelength. 

A main section of this paper, section 2, is devoted 
to establishing a scattering matrix formalism for 
the description of radiating-receiving systems of the 
general type considered. In this connection it is 
helpful — at least to one acquainted with the theory 
of waveguide junctions — to regard the space side 
of the waveguide-space transducer as a waveguide 
of infinite cross section. A scattering matrix having 
both discrete and continuous indices, which corre- 
spond respectively to the mode in the ordinary 
waveguide and the continuum of modes in the wave- 
guide of infinite cross section, is required. The 
expression of reciprocity, which seems to be of in- 
terest in itself in the class of problems considered, 
is established. 

For the purpose of the present work, the scattering 
matrices of the radiating-receiving systems involved 
are considered known; the determination of the nec- 
essary data for a practical application is considered 
to be an independent problem, experimental or pos- 
sibly theoretical. (Fortunately in the practical cases 
under consideration, scattering by the radiating- 
receiving systems involved is of minor importance, 
and it thus appears that the necessary data, can be 
obtained from suitable measurements directly or 
indirectly determining the far-field radiation charac- 
teristics. Such measurements would yield t he neces- 
sary information about the underlying diffraction 
problem, however complicated it might be.) 

In section 3 the scattering matrix formalism is 
extended and applied to the calculation of the re- 
ceived signal in the basic arrangements described 
above. The general problem involving multiple 
reflections is soluble formally (in the sense that the 
problem is reducible to quadratures) in the form of 
a scries of integrals, which stem from the Liouville- 
Neumami series solution of a system of integral 
equations. Formulas are derived for models of the 
microwave Michelson and Fabry-Perot interferom- 
eters in which the radiating and receiving spectra 
are arbitrary. Finally, these formulas are illustrated 
by means of examples obtained by choosing specific 
and relatively simple analytical expressions for the 
spectra involved. 



2. Scattering Matrix Description of Radiat- 
ing-Receiving Systems 

A. Basic hypotheses. To the general description 
of the radiating-receiving systems given in section 1 
we add the following details. 

We choose a terminal surface S Q in the waveguide 
and a supplementary surface Sq (such that Sq+Sq 
forms a closed surface) coinciding with the shielding 
around source, receiver, etc. (See fig. 1.) We 
further choose a rectangular coordinate system 
Oxyz such that coordinate surface 2=0, which we 
denote by Si, will serve as a terminal surface on the 
space side of the waveguide-space transducer. 
As a convenient artifice we employ an infinite 
hemisphere S m lying in z<0 and centered at O. 
The interior — i.e., the domain of the electromagnetic 
field — of the transducer is the region V bounded 
externally by Si+S m and internally by S +Sq. In 
figure 1 the structure of the transducer considered 
lies in V; other structures or objects may also be in 
V but no attempt is made to illustrate this possibility. 

The whole of the space and structure within V is 
counted simply as a (decidedly) non homogeneous 
medium which may also be dissipative and aniso- 
tropic. Linearity is assumed and is essential; 
reciprocity, in the sense that tensors describing the 
medium within V are required to be symmetric, is 
assumed and is useful but not essential. 

In the space transmission paths (i.e., in 2>0) the 
medium is to be homogeneous, isotropic, and non- 
dissipative, as well as linear. For the time being it 
is assumed that these properties hold for arbitrarily 
large 2; the interposition of elements (such as a 
reflecting object) will be considered later. (The 
theory is not actually restricted to transmission 
media having the ideal characteristics listed, since, 
as will be seen, an arbitrary linear transmission 
medium is to be treated as a suitable interposed 
(dement.) 

It is assumed that the electromagnetic field quan- 
tities vary harmonically with time t at frequency 
w/(27r). We employ the usual complex electric and 
magnetic field vectors, E, H, which are functions of 
the position vector r of Oxyz, and omit the time 
dependent factor exp (—iut). 

It should be observed that in taking S m to be 
hemispherical it is tacitly assumed that the problem 
is not two-dimensional (by a "two-dimensional" 
problem is meant one in which all quantities are 
independent of one rectangular coordinate perpen- 
dicular to the ^-direction). To avoid unduly com- 
plicating the discussion, two-dimensional problems 
are not considered in the main part of this paper. 
However, two of the examples at the end of the 
paper are two-dimensional. For convenience in 
discussing these and other two-dimensional cases, 
some of the more important or less obvious of the 
needed formulas are summarized in an appendix 
(sec. 4, B). 

B. Representation of field on S . It, as is assumed, 
only one mode (a propagated mode) is of importance 



in the waveguide in the neighborhood of S , then 
from waveguide theory it is known that the trans- 
verse components E H t of E, H on S may be 
written 



E(r) t =(a +b )e(r), 
n(r) t =(ao-b )h(T), 



■ (r on So) 



(1) 



where e, h are real basis-fields for the mode involved. 
The basis-fields are subject to the impedance normal- 
ization 



h(r) = 77 nXe(r) 
and the power normalization 



JSr 



e(r)Xh(r). ndS=^T 2 r} 



(2) 



(3) 



where n is the unit normal on S drawn into V and 
rjo is the wave-admittance for the mode involved. 
These equations implicitly define the quantities 
a 0} b ; it can be verified that a , b so defined are 
respectively linear measures of the electric field of the 
incident and emergent traveling-wave components 
of the waveguide field at S Q . The net time-average 
power input P at S is given by 



Js Q 



P =Ue ^EX^ndS=2w 2 r lo (\a \ 2 -\b 



(4) 



where Re denotes that the real part is to be taken 
and the superposed bar denotes the complex con- 
jugate. 

C. Representation of the Held in the region z>0. 
As mentioned in the introduction, the electromag- 
netic field in the region z>0 is to be represented as a 
superposition of plane-wave solutions of Maxwell's 
equations. This type of representation is well 
known, at least in the case of solutions of the scalar 
wave equation [4]; the generalization to the vector 
electromagnetic field offers no particular formal 
difficulty. 

The electromagnetic field in the region under 
consideration satisfies Maxwell's equations in the 
form 



VXE=ico M H, vXH=-icoeE, 



(5) 



where /i, e are constant real scalars representing 
respectively the permeability and the permittivity 
of the medium. (Rationalized MKS units are 
employed.) We derive our basis fields from the 
general plane wave 



E=Texp (ik-r), 
H=(co / x)- 1 kXTexp (ik-r), 



(6) 



which is a solution of (5) for any propagation vector 
k such that F=coV and any vector T satisfying 



the transversality relation k-T = 0. (In spite of 
this occurrence of " transversality/ ' in what follows 
the term "transverse" will always mean transverse 
with respect to the s-direction.) 

The propagation vector will be regarded as a 
function of its transverse components k x , k v ; the 
3-component is then 



h=±y, 



(7a) 



where y=(k 2 — k 2 x —k 2 y )^. It will be convenient to 
denote the transverse part of the propagation 
vector by K, so that K=k x e x -\-k v e y and 



y=(k 2 -K 2 )?. 



(7b) 



Since k x , k y must be allowed to vary independently 
in the range (—<», <*>), real and imaginary values 
of 7 will occur. 7 will be taken positive for K 2 <Ck 2 y 
positive imaginary for K 2 ^>k 2 . Superscripts "-f" 
and " — " will be used when it is desired to indicate 
the choice of sign associated with k z . 

In virtue of the relation k-T = 0, (6) yields just 
two linearly independent fields, hence just two basis 
fields, for any given k. The appropriate polariza- 
tions for the basis fields are those with the electric 
vectors parallel or perpendicular to the plane of 
k and e 2 , which is the plane of incidence for a ray 
incident on any plane z= const. This choice of 
polarizations corresponds to the practice in elec- 
tromagnetic theory in deriving the Fresnel equa- 
tions, for example ; it also corresponds to the resolu- 
tion into "transverse magnetic" and "transverse 
electric" modes of waveguide theory. 

In setting up the desired basis fields it is convenient 
to employ the transverse unit vectors 



k 1 = K/K, K 2 =e g X«i, 



(8) 



which are respectively parallel and perpendicular 
to the plane of k and e 2 . (The notation is illustrated 
in fig. 4.) As a temporary abbreviation we put 
u ± =exp (*±-r). For the "Ef (or TM) compo- 
nents we take T=Ki^Ky~ l e z and obtain from (6) 



Ef=[* 1 = F K7" 1 c f ]w ± > 



(9) 



where m=ue/y. For the U E± J (or TE) components 
we take T=k 2 and obtain from (6) 

E?=k 2 u ± "I 

Y do) 

n?=[±7 l2 e 2 XK2+K(a>tx)- l e 2 }u± J 

where rj 2 =y/(o)iJ.). Among other similarities it may 
be observed that 7] b 772 are "wave-admittances" that 
correspond exactly to the wave-admittances en- 
countered in the theory of waveguides with discrete 
modes. The expressions in (9) and (10) are essen- 




Figure 4. Illustrating k, K, k\ } and *2. 

tially the desired basis fields (^-components, in- 
cluded here, will be dropped later). The normaliza- 
tion and orthogonality properties of these fields are 
of course implicit in the expressions themselves. 

We now introduce spectral density functions 
b m =b m (K) and a m =a m (K) for outgoing and incoming 
waves, respectively, and form the general super- 
position 



E(r)=Jl](6 m E;i+a„E-)./K, 

H(r)=j2(ai+a;)^K. 



(11) 



Here and subsequently in expressions of this type 
summation over the two values of the polarization 
index m and integration over the domain of the two 
variables k x , k y will be understood. The E, H given 
by (11) will satisfy Maxwell's equations provided 
merely that the neressary differentiations can be 
taken under the integral sign. However, for our 
purposes, the 0-eomponents of the* fields are redun- 
dant. By discarding the ^-components we obtain 
the mueh more convenient and explicit expressions 
for the transverse field components 



ewfJi](^/ 2 +^-^«,/' k,r («, 

H(r) t =[s(^e i72 -^" m )^e 2 XM iK,R dK, 



Y (12) 



where R denotes the transverse part of r. These 
equations exhibit E r , H, as two-dimensional Fourier 
transforms of the quantities multiplied by exp 
(it'K-R) in the respective integrands; from the inverse 
t ra informations one may obtain 



6 m (K)=^ic w .J(E+^ 1 HXe z )^«- B dR, 



a m (K)=-^K fl 



i 



(E- v - l nxe z )e- iK - R <m. 



(13) 



Here and subsequently integrations with the differ- 
ential symbol dR are to be taken over all values of 
x and y for some fixed 2>0. These equations enable 
one in principle to evaluate a m b m from a given 
distribution of E t , H c on any transverse plane. 

The ease where there is no field incident on Si from 
the '"right" [i.e., where a m (K)=0] is of particular 
interest in what follows, and we shall list several 
relations holding in this case. From (12) and (13) 

E(r)« = fB(Ky k -7/K, (14) 

B(K) = (2t)- V m fE(r),6-'" K - R dR (15) 

wdiere B = B(K) is the transverse vector 6 1 k 1 +6 2 «2. 
The time-average power, P r , radiated into the half- 
space 2>0 will also be of interest. This power is 
the same as the time-average power flux in the 
+ ^-direction across the surface S x . Hence, from 
(12) [witha*(K)-0], 

P r =iRe fEXH.e^R=2T 2 f 2^(K)|6 m (K)| 2 rfK, 

^ J jK 2 <k 2 

(16) 
where, as indicated by the notation below the 
integral sign, evanescent waves are excluded from 
the integration. This equation is of a type, known 
as Parseval's formulas; it may be derived formally 
with the aid of the rule thai 



j: 



e *<*z-*p*dz=2TS(k x —ks), 



(17) 



where <5 denotes the so-called impulse or 6-func- 
tion [5]. 

D. Definition of scattering matrix. Having set up 
representations for the fields on the two terminal 
surfaces of the waveguide-space transducer under 
consideration, we are now in a position to consider 
the transducer as a whole. It may be assumed that 
a. set of out-going wave-amplitudes [b and b m (K)] 
will be determined by a set of incident wave-am- 
plitudes [do and a m (K)]. In fact, since the electro- 
magnetic system under consideration is by hypoth- 
esis a linear system, the relation between the set 
of out-going wave-amplitudes and the set of incident 
amplitudes must be a linear relation. We write this 



&o — $o(A)~+ 



■/s* 



(m,K)a m (K>ZK (18a) 



b m (K) = S 10 (m,K)a +jz:S n (m,K;n > L)a n (L)dL, 

(18b) 

thereby defining the scattering matrix for the trans- 
ducer considered. Figure 1 may be helpful in fixing 
the significance of the quantities involved in (18). 
It is convenient and it seems appropriate to use the 



term "matrix" here even though one must think of 
rows and columns labeled both by discrete indices 
and by indices having continuous ranges. Evi- 
dently the functions S i, £ 10 , and S n respectively 
embody the receiving properties, the radiating 
properties, and the (space-side) scattering properties 
of the transducer involved ; S 00 is an ordinary wave- 
guide reflection coefficient defined at the terminal 
surface S and expressing the "antenna mismatch." 
Equation (18) can be represented in terms of S 00 

and linear functional operators S 0i , S m , and S n such 
that 



&o— $00&0 + Sold, 

b = S 10 a J rS n d, 



(19a) 
(19b) 



where b and a are understood as function vectors 
corresponding to the functions b m (K) and a m (K). 
This compact notation is used later primarily as a 
convenience in some of the more formal and general 
parts of the discussion. 

E. Reciprocity. Radiating and receiving charac- 
teristics are related by the reciprocity condition, 
which here takes the form 



YjQi 



Soi(m,K) = i ?ro (K)S 10 (m, -K) 



(20) 



Since this particular form of the condition appears 
to be new, a derivation is given below (appendix). 
The occurrence of the factors rj and jy m (K) in (20) 
can be regarded as a consequence of the particular 
normalizations adopted in setting up the basis fields. 
(The same type of relationship holds for the elements 
of the scattering matrix of an ordinary waveguide 
junction with discrete modes [6].) The occurrence 
of the argument — K in one side of (20) means that 
the equation relates radiating and receiving charac- 
teristics in the line of a given propagation vector k 
(if a radiated wave has propagation vector k, 
the received wave in the same line has the propaga- 
tion vector — k). 

The plane-wave into plane-wave scattering func- 
tion #ii is also subject to reciprocity; the relation is 

Vm (K)S n (m,-K; n } L)= Vn (L)S n (n,-L;m } K). (21) 

Comments similar to those following (20) apply here 
also. This relation is not used in the present paper 
but it may well be of interest in other diffraction 
problems. 

F. Determination of scattering matrix. Although 
the basic viewpoint of this paper is that S 00 , S ou etc., 
are to be considered known, it seems well to take some 
note of the problem of determining these quantities 
from empirical data in the arrangements of particular 
interest. 

Concerning S o and S n there is not much to be 
said. #oo is not only relatively easily measurable 
but also experimentally controllable (by means of 
tuning elements), whereas in general the same is not 
at all true of S n - However, the desired operating 
condition that S u be effectively negligible is ap- 



proximately attainable and to some extent subject 
to experimental verification (by observation of the 
effects of multiple reflections). 

Concerning # i and S i0 we first note: (1) Either 
one of these functions may be determined readily 
from the other with the aid of the reciprocity rela- 
tion; (2) in the present context these functions need 
be evaluated only for K 2 <ik 2 , the effects of evan- 
escent waves being avoided by keeping rf>>\ (this 
is not inconsistent with Fresnel-region operation, 
cf sec. 1). 

A direct approach to the determination of # i 
is implied by the definition (18): # i represents the 
received signal b as a function of the direction and 
the polarization of incident plane waves of suitably 
normalized amplitudes. 

According to (18), #i (m,K) represents the 
transverse components of the vector spectrum of 
outgoing waves under the conditions a m (K) = and 
a =l. If we define the transverse vectorial spectrum 



S 10 (K) = S 10 (l,K)ic 1 + # 10 (2,K)ic 2 



then, from (15), 



S 10 (K)= i ^J , E l (B)«-«^B; 



(22) 



(23) 



that is, Sio may be represented as the Fourier trans- 
form of the transverse components of the electric 
field obtaining on the reference plane in the absence 
of incident waves, normalized to unit a . This is, 
of course, essentiall}- a vector form of the well-known 
relation between "aperture" distribution and spec- 
trum. 

The vectorial spectrum Si (K) is also closely 
related to the far electric field by a well-known 
type of relation. Under certain restrictions E(r)< 
has for large r the asymptotic form [7] 



E(r) lf 



: - 2-wik cos 6 B (Rk/r) e ikr /r; (24) 



the angle 6 introduced here is the polar angle of r 
relative to the 2-axis. By rewriting this equation 
and dividing by a we obtain a formula for S 10 (K) 
in terms of the asymptotic form of E*: 

S 10 (K) = i(27ra )- 1 7 - 1 r6-^E(kr/Z:), asymp , (25) 

for K 2 <k 2 . 

Finally we note that the familiar "power radiation 
pattern" or "polar diagram" of antenna theory, 
defined as radiated power per unit solid angle as a 
function of direction, is given by 



P = 2( € /m)* (tt& cos0) 2 bb, 



(26) 



where b = B J rb z e 2 and b 2 =— B-K/y (bis the com- 
plete vectorial angular spectrum, including the 
^-component). Clearly this equation is not suf- 
ficient by itself to determine S 10 (K); polarization 
and phase information is required in addition. 



6 



3. Applications 

A. Reflection systems. One obtains a form of the 
first basic arrangement described in section 1 by 
placing an infinite plane reflecting- surface "in front 
of" a radiating-receiving system of the type con- 
sidered in section 2. This represents a problem 
of intermediate complexity, from which the basic 
equations for tin* Michelson arrangement may be 
obtained by specialization. If the reflecting surface 
is at z=d and has reflection coefficient p(w,K), then, 
transforming the reflection coefficient to the plane 
z = 0, we have 



flh (K)=p(jn,K)^'MK) 



(27) 



(the dependence of y on its arguments is now indi- 
cated explicitly). Upon substituting (27) into (18) 
one obtains 

b () =S oao+ j j:S {n (m ,K) p(m ,K)^y^ l bJK) ( /K f 

(28a) 

UK) = S 10 (m,K)a 



+ 



f ^S n (m^]ri,L)p(nX)e 2J ^ L)d b f XL)</L. (281 



>) 



The last line represents two simultaneous, inhomo- 
geneous, linear integral equations for the determina- 
tion of b m (K) (a being prescribed). The Liouville- 
Xcumann series solution of these equations may be 
obtained by a process of successive approximations. 
For the first approximation one takes 



6£>(K) = S f 10 (m,K)rv 



(29) 



the second approximation is obtained by substituting 
the first into the right-hand side of (28b), 

b^(K) = S 10 (m,K)a 

+a jj:S n (m,K;n,L)p(n ) L)e*->™< i S 10 (n,L)dL-, 

(30) 

and so on: & ( ^(K) accounts for the first n reflections 
at the reflecting surface. Once b m (K) is obtained, 
approximately or otherwise, it is to be substituted 
into (28a), thus determining the received wave- 
amplitude b in the waveguide at So- 
Useful approximate equations describing the be- 
havior of the Michelson instrument may now be 
obtained. The appropriate conditions are (1) that 
the effects of multiple reflections be negligible and 
(2) that the reflecting surface at z=d be a mirror, 
for which we may put p= — 1. The first condition 
means that (29) is already a good approximation 
for 6„,(K); this substituted into (28a) yields 



a 



=S o~ [S S {n (m,K)e^^ d S l() (m,K)dK. (31) 



The second term on the right will be called the "re- 
flection integral" and denoted by $(d). Either S oi 
or iSio may be eliminated by means of the reciprocity 
condition (20); for the purposes of the present dis- 
cussion it seems preferable to eliminate S in . We 
then have 

* (d) =vo l Ji: rj m (K) S 10 (m,K) S 1Q (m -K) e*r««K. 

(32) 

This is the main equation for the Michelson. It 
will be illustrated below by means of examples ob- 
tained by choosing specific mathematical expres- 
sions for Si (m,K). 

Problems involving reflecting or scattering objects 
other than an effectively infinite reflecting surface 
are important not only in microwave interferometry 
but also in other fields. Consequently the following 
formulation of the general case where the plane 
reflecting surface considered above is replaced by an 
arbitrary reflecting object is of interest. Let the 
general reflecting object be characterized by means 
of a scattering function R(m,K; n, L) defined with 
2=0 as terminal surface. Instead of (27) we now 
have the linear transformation 



a*(10=JSfl(ro,K;w,L) b n {h)dh 



(33) 



as boundary condition. In operator notation (33) 
is written 

A A A 

a= Ho 
and the equations corresponding to (28) are 

&o=#o(A>+$oi#&, (34a) 



b = S 10 a +S u /ib. 



(34b) 



The Liouville-Neumann series solution of (34b) 
may be written 



whei 



b = Si a -\- LSio^Of 






(35) 
(36) 



is the operator corresponding to what is called the 
"resolvent kernel" in the theory of integral equations. 
Finally, for b we obtain 

b = S oa,+ S J?S lo a d +S JlLS 10 a d . (37) 

In this form the last term (specifically the operator 

L) accounts for multiple reflections. 

An instance of (37) is the case of a large but 
finite mirror in the Michelson interferometer, which 
has been considered using an approximate expres- 
sion for 7? and neglecting multiple reflections [8]. 



B. Transmission systems; Fabry-Perot interferom- 
eter. We now consider a general transmission sys- 
tem consisting of a radiating system and a receiving 
system with an arbitrary (electromagnetically linear) 
intervening structure or medium. For the active 
terminal the pertinent descriptive equation is 
(19b), which for convenience is repeated here: 



b=S 1( flo+Sn<i. 



(196) 



This equation is understood to be set up with 
reference to a coordinate system Oxyz and space- 
side reference plane 2=0, as detailed in section 2. 
Using primes to distinguish quantities associated 
with the passive terminal, we may write 



K- 



■ Sold' , 



y=s, 



& 



(39a) 



(39b) 



as the equations corresponding to (19). For these 
equations the space-side reference plane is z=d in 
the above-mentioned coordinate system and the 
general arrangement is shown in figure 3. Next, 
let the structure and/or medium between the 
terminals be described by a set of linear operators 

Tij, defined with respect to z = and z=d as reference 
planes, such that 



a=T n b + T l2 b', 
a' = fj+f 22 b'. 



(40) 



(The Fabry-Perot interferometer considered below 
will furnish an example of these equations. It will 
be a very special example, however, since specular 
reflection and transmission will be assumed, so that 

the operators T i3 - will be diagonal and (40) is then 
reducible to a family of ordinary equations.) 

A method of solving the problem described in 
(19), (39), and (40) may be indicated as follows. 

The result of eliminating b and o from (19b), (39b), 
and (40) may be put in the form 



p [f n S l0 a 
U'J LT 2 Aoao_ 



+ 



-* 11^11 -* 12^11 



(41) 



As this form suggests, these equations may be solved 
for & and a by a process of successive approxima- 
tions similar to that used above. The received 
wave amplitude b' is directly determinable from 
(39a) as soon as 6! is known. (b is also directly 
determinable.) 

To pass to the consideration of a highly simplified 
model of the Fabry-Perot interferometer, we first 

assume that S n and S' n are effectively so small 
that reflections at the terminals of the system may 
be neglected; that is, we assume that the second 
term on the right in (41) may be neglected. We 



then have the explicit expression 



a — ^2i^io^o> 



(42) 



and consequently for the received wave-amplitude 
we have 



6 — &01-L 21^10^0- 



(43) 



Next, we assume that the structure described by 
the T^ is a Fabry-Perot "e talon", consisting of a 
pair of elements corresponding to the two plates of 
an optical Fabry-Perot interferometer. Such ele- 
ments might be, for example, perforated metal sheets 
or stacked quartz plates [8]. We assume that each 
element is symmetric with respect to the ^-direction 
and characterizable by means of a (specular) reflec- 
tion coefficient p(ra,K) and a (specular) transmission 
coefficient r(m,K) defined at the symmetry plane 
of the element as reference plane. It is not as- 
sumed that the elements are lossless; p and r are 
subject merely to realizability conditions for passive 
elements. We let the two elements be located so 
that their reference planes coincide with z=0 and 
with z=d, respectively. It may be noted that so 
locating the elements implies no real loss of gener- 
ality, since the planes z = and z=d are arbitrarily 
located with respect to the physical arrangements 
with which they are respectively associated. From 
the symmetry of the etalon as a whole with respect 

to z = d/2 and the fact that the Tij must be diagonal 
we have 



T 11 =T 22 =5 Bm 5(K-L)^ 11 (m,K) 

T 2l =T l2 =6 m J(K-L)t 2l (m,K), 
where <5(K — L) means 8(k x —l x ) 8(k y —l v ) and 
\ + (r 2 -p 2 )e 2iyd 



tn = P 



k 



1- P V 



1-pV 



(44a) 
(44b) 

(45a) 
(45b) 



as may be found by ordinary methods. In these 
expressions y, p, and r in general depend upon K. 
When evaluated for K = and simplified somewhat 
as they may be for lossless elements, the expressions 
reduce to ones frequently used in discussions of the 
optical Fabry-Perot interferometer. 

Using (44b) and writing out (43) we obtain 

K=a, fs SUrn,K)t 21 (m,K)S 10 (m,K)dK; 

finally, inserting (45b) and defining V(d) = b' /a () , 



S' ol (m,K)T 2 (m,K)e*™ d S w (mJL)dK 
l- P 2 (m,K)e 2i ^ K)d 



(46) 



8 



This is the "transmission integral" for a Fabry- 
Perot interferometer. An example of the analytical 
evaluation of (46) will be given below, assuming 
constant p and r and choosing very simple expressions 
for $01 and Si . (It may be observed that in regard- 
ing (46) as a function of d, it is implicit thai the 
terminal apparatuses remain fixed relative to the 
reference planes with which they are respectively 
associated.) 

The examples that follow have been chosen to 
illustrate various features of the theory and of the 
instruments considered. Inasmuch as the examples 
are rather hypothetical in nature and mainly illus- 
trative, no thorough or rigorous discussions are 
attempted. 

C. Examples, Michelson case. 

C.l. Dipole. The following example seems well- 
suited to illustrate the theory, inasmuch as it 
involves both TE and TM field-components in a 
fairly complicated way, the integrals involved can 
be evaluated, and the form of the answer can be 
anticipated. In this example the radiated field is 
assumed to be identical to that of an elementary 
electric dipole of moment p located at (). 

To find the angular spectrum of the electric field 
we may proceed as follows. The appropriate Hertz 
potential is [9] 

Il=(4ire)- l pe ikr lr; 

the representation of the spherical wave exp(i&r)/r 
in terms of plane waves is [10], for 2>0, 

e ikr /r= - (2iri)- 1 J y- l e ik - T (/K; 

where k = k + is understood. Since E^VXVXII 
we have 

E=c7rkX(kXp)T- 1 ^' k ' r ^K, (47) 

where C=(ST 2 ei)~ 1 . Hence the complete vectorial 
spectrum is b=6 y kX (kXp)Y -1 . This result holds for 
all|K; immediate confirmation for K 2 <^k 2 may be 
obtained from (25) using the asymptotic form 

-F(47re)- 1 rX(rXp)^^7r 3 

forE [9]. 

According to the definition of Sio(m, K), we have 
in this example 

S 10 (m, K) = (tf/oo)* w - [kX(kXp)h" 1 (48) 

for ra=l, 2. These spectral components are to be 
substituted into the reflection integral (32) for the 
Michelson instrument. The coefficient of exp (2iyd) 
in the integrand of (32), after some vector-algebraic 
labor, is found in the present instance to be expres- 
sible in the form 

-co,(r/a ) 2 p.[kX(kXp , )]7- 1 , 



where p' is the negative mirror image of p, with 
components (—p x ,—Py,Pz)' Thus for (32) we have 

Hd)=-o>erio 1 (C/ao) 2 p • fkX (kXp'h~ V'^/K. (49) 

Comparing this expression with (47) it is seen that 



*($=- 



Cu 



Vodo 



p.E,0,2d), 



(50) 



where E' is the electric field of a dipole of moment 
p' located at O, and E' is evaluated at the image of 
O in the reflecting surface of the instrument. This 
result is indeed of a form that might be anticipated. 
If for simplicity we take p to be transverse, (50) 
becomes more explicitly [9], 

$ (d) = C\[{2ikd) ~ 3 - {2ikd)~ 2 + (2ikd) ~ l ]^ M 9 

where (\ is independent of d. It is clear that the 
result in this example is not physically meaningful 
as kd-^0, for the magnitude of <£ can not properly 
exceed unity. Tins defect is attributable to the 
neglect of scattering or re-radiation by the dipole; a 
plausible extension of the theory of this example 
taking scattering into account gives results qualita- 
tively well-behaved for all values of kd. 

C.2 Two-dimensional Gaussian. — To provide a 
reasonably simple analytical illustration of the be- 
havior of a Michelson instrument operated in the 
Fresnel region, we consider a two-dimensional case 
with the pure TM "Gaussian" spectrum 






(51) 



The subscripts 1, 2 here refer to e x , e y , respectively, 
and A is an arbitrary amplitude. The correspond- 
ing distribution of E, on the reference plane 2—0 
is also Gaussian, 



E T 



--Const, e ~ x2/{2a2 \ 



Ey=0, 

as follows from (14 2 ) (see appendix, B). The param- 
eter a is, in a well-known manner, a measure of the 
sharpness of the angular spectrum and a measure 
of the width of the distribution of E,. 

In this example it will be interesting to determine 
the absolute magnitude of $ explicitly. We need 
the relation between \a \ and \A\ and we obtain this 
from a consideration of energy balance under free- 
space radiation conditions. If the fraction h of the 
net input power at S is radiated, we have 



7rT7o|ao| 2 (l — \S 0I 



\ 2 )h= 



"£ s,?w 



K(k x )\ 2 dk x 



(52) 



532245—60- 



where the left- and the right-hand sides of this equa- 
tion come from (4 2 ) and (16 2 ), respectively. 

The appropriate two-dimensional form of (32) is 

*W=-^jS^WS 10 (m,ys 10 (m r i,)^^4 

(53) 

where we still have r] 1 = c J oey~ 1 and rj 2 = 7(w/x)~ 1 . In 
the present case this becomes 



*(d)= 



Vtfil 



|JV« e> 



exp {—d 2 k 2 x -\-2iyd)dk x 



Letting <j> denote the phase of A/a and using (52) to 
eliminate |^4/a | from the last equation, one obtains 



*(<*)=- 



k a- 



|$oo| )h 



I 7 * exp (— a : 



'k 2 x + 2iyd)dk x 



L 



7 ' exp (— a?kl)dk x 



(54) 



If &a»l, this expression yields $(0)^ — e 2i<f> 
(l — \So \ 2 )h — a quantity that may approximate unity 
in magnitude. 

Since we are interested in Fresnel-region behavior, 
an asymptotic expansion of the numerator of (54) 
in terms of inverse powers of a is appropriate. This 
asymptotic expansion may be found with the aid of 
Watson's lemma, as given by Jeffreys and Jeffreys 
[11]. One finds 



*(<*)= C5 



\e 2iM r 



H 



\-2ikd 

4(ka) 2 

. 9-lSikd- 



\2(kd) 2 



Z2{ka) A 



} 



(55) 



where C x does not depend upon d. For this expres- 
sion to yield a good approximation it is necessary 
that &a>>l and that &a 2 »rf (these inequalities 
are equivalent to those given in section 1 in the 
description of Fresnel-region operation). To the 
second order in ka, 

&Tg <$>(d)^2kd[l-{2ka)~ 2 }. 

Thus the "diffraction correction" to the phase can 
be expressed in terms of a small increase in effective 
wavelength, which in this approximation and in 
this example is independent of d. 

It is of some interest to evaluate this result for 
values of k and a that might be considered typical 



of optical cases, even though the formula does not 
apply, or at least does not apply directly, to any 
optical instrument. If one takes A =5000 A and a = 5 
cm, the fractional increase in effective wavelength 
given by the formula is approximately 6X10 -13 . 

C.3 TE 10 aperture-distribution. A somewhat more 
realistic — and much more intractable — example of 
Fresnel region behavior is afforded by the assump- 
tion of a TE 10 -mode distribution in a square aper- 
ture in a conducting screen. This example is sug- 
gested by an experimental arrangement in which 
a square horn-lens radiator is fed by a taper from 
rectangular waveguide supporting the TE 10 mode [8]. 
If the aperture is bounded by \x\=a/2 and \y\=a/2 
in the plane 3=0, we may take 

E / =cos(7ra;/a)e l/ 

in the aperture and E,=0 elsewhere in the plane. 
From (22) and (23) it follows that 

S lQ (l,K)=kAK)/\K\, 

S 10 (2,K)=k x f(K)/\K\, 



where 



/(K)=<7 



7 cos (k x a/2) sin (k v a/2\ 



(»/«)*-« 



(Here and subsequently unimportant constant factors 
are denoted C, C' } etc., without explicit definition 
in each case.) Hence (32) becomes 

*(d) = C" J(^ kl+^ **) [ ^^ e^ d dK. (56) 

This integral has been evaluated numerically. The 
numerical analysis and the programming required 
for this difficult task were performed by Paul F. 
Wacker and William W. Longley, Jr. Some of the 
results are shown in table 1, wherein the quantity 
labeled Ad is calculated in accordance with the 
definition 

Ad=(2k)- l [sLYg$(d) — arg$(0)]— d. 



Table 1 



X 


a 


d 


Ad 


cm 


cm 


m 


Microns 


0. 6278 


60 


2 


-56.96 


.6278 


60 


10 


-193.80 


.6278 


30 


2 


-155.93 


.6278 


30 


10 


-503. 28 


. 1 


60 


2 


-2.32 


.1 


60 


10 


-7. 56 



10 



The fad thai (his quantity is negative corresponds 
to the genera] fact that the effective wavelength 

lends to be greater than the free-space wavelength, 
so that the observed phase increases with d more 
slowly than 2kd. 

D. Fabry-Perot with line source. In this example 
the radiated field of the radiating system is taken to 
be identical to t hat of a simple line current coinciding 
with the y-£Lxis, and it is assumed that the receiving 
system is the same as the radiating system. These 
assumptions yield a two-dimensional, pure TE prob- 
lem in which the free-space radiation pattern has no 
Fresnel region. Although only the most readily 
obtainable results are given here, the example already 
affords an interesting illustration of the behavior of a 
Fabry-Perot instrument in the presence of a continu- 
ous angular spectrum. 

The assumed field being pure TE, we certainly 
have S 10 (\,k x ) = 0; and since E is asymptotically pro- 
portional to r~ 1/2 e ttr e y , it follows from (25 2 ) that 
we may write Si Q (2,k x ) = y~ 1 , at least for k 2 x <Ck 2 . 
We shall employ this expression for the whole range 
of k x , choosing to justify this analytic continuation 
by the results to be obtained in a moment. Using 
the reciprocity relation (20 2 ) we obtain for the re- 
ceiving system So 1 (2,k x )=y(ufi)- 1 V o 1 S' 10 {2,--k x ) = 
(co/xt/o) l . Hence the transmission integral (46) he- 
comes 



*(d) = C 



r r tV] 

J-oo 1 — f 



l y 



2 e 2iyd 



dk x . 



(58) 



The integral diverges for d=0) this behavior is 
attributable to the implicit assumption of an en- 
forced current in the radiating element (as in the 
dipole example above). If we assume r and p con- 
stants independent of k X) rf>0, and |p|<l, then (58) 
may be written 



9(d) = Cr 






■/: 



y -l e H2n+l)yd dkx 



By introducing a new variable of integration a, such 
that k sin a= — y, one may transform the integral 
to one of the standard representations of the Hankel 
function [12] of the first kind and order zero. Thus 



m)-- 



:C' , r 2 2p 2 ^H(2ft+l)W]. 
n=0 



(59) 



At this point we note that for p=0, ^(d) becomes 
proportional to H^ikd), as might be expected for 
free-space transmission between systems of the des- 
crip t ion considered . 

For \p\ approaching unity, features peculiar to the 
Fabry-Perot may be expected to appear — and the 
series becomes very slowly convergent. A thorough 
discussion of (59) might be premature and will not 
he attempted here (one might wish to consider a 
more realistic model of the Fabry-Perot, for example). 
We shall assume td»l and approximate the Han- 
kel function by means of the first term of its asymp- 



totic expansion. Instead of (59) we then have 

^(d)^C /f r 2 p- 1 (kd)-^F( P ,kd), (60a) 



where 



F( P , kd)=J2 (2n+ l)"*(p^) 2w+1 . (60b) 

71 = 



Tn (60b) a phase angle in p obviously is equivalent 
to an additive constant in d\ having noted this, we 
assume the p 2 is real and positive. 

For p 2 <l, the series (606) converges for all values 
of d and \F(p,kd)\ has maxima for kd=mir, where m 
is an integer (for p 2 =l, the series still converges for 
kd^mir but diverges infinitely at the points where 
kd=mir). These maxima arc 4 certainly the principal 
maxima and correspond to the passage of axial rays; 
it is not known whether there are subsidiary maxima. 

Since |r| 2 <l — |p| 2 (the equality holding for a 
lossless etalon), r and hence ty must approach zero 
as p 2 — >1 except possibly at the values for d for 
which the series diverges. An estimate of F(p,mir) 
as p 2 — >1 indicates that ^ must approach zero at 
these points also. This decrease of transmission 
at the maxima as p 2 — >1 differs from the result 
given by the optical formula (45b) and may be 
surprising. It can he explained in terms of increasing 
selectivity for axial rays, such that a decreasing 
portion of the incident spectrum, and hence de- 
creasing power, is transmitted. 

Finally it may be observed that F(p,kd) is a 
periodic function of d, so that the factor d~\ in (60a) 
gives the general trend of ^ with d. In the optical 
ease, >J> itself would be periodic. 

4. Appendix 

A. Reciprocity relations. Let E', H', and E", H" 
denote any two electromagnetic fields (of the same 
frequency) that can exist in the interior of the 
waveguide-space transducer considered. In virtue 
of the hypotheses imposed in section 2, tin 1 Lorentz 
relation 

V-(E'XH"-E"XH')=0 

will hold in V[6\. Therefore, using the divergence 
theorem, one has 



x 



(E'XH"-E"XH')-n</S=0, 



«0+«o+ S l+ N oo 



where n is the inward unit normal on the boundary 
of V and the parts So, S' , etc., making up the bound- 
ary of \ ' are as defined in section 2. Now, the contri- 
bution of the integral over S f vanishes, since the 
integrand vanishes there. By using the asymp- 
totic form of the fields for large r, it can be shown 
that the S^ -integral vanishes for fields generated 
by any distribution of sources confined to a sphere 
of finite radius centered at O. This restriction 



11 



apparently would rule out excitation of the system 
by plane waves incident on Si, which we wish to 
consider, but since we can imagine a plane wave 
approximated arbitrarily closely by a source of 
finite dimensions at a sufficiently large distance, 
there is no real limitation. Hence we may employ 
the relation 



x 



So+Si 



(E'XH"-E"XH') -ndS=0 



(61) 



without explicit restrictions. If in this expression 
one replaces E', H' and E", H" by their representa- 
tions on S and£i, equations (1) and (12), one finds 
after some analysis 

-<4'(K)&;(-K)]dK=0. (62) 

In obtaining this result the use of the integral 
representation (17) of the 5-function is helpful. 

We now assume that E', H' and E", H" are the 
fields corresponding to excitation by the following 
particular sets of incident waves 

a =l, #0=0; 

<4(K)=0; <f m {K)=6 mn 8(K--L). 

From the scattering equations (18) we obtain 
&;(K) = S 10 (m,K), K=S 01 (n,L). 

Upon substituting all these quantities into (62) and 
observing that rj m (L) = rj m (—Ij), one obtains the 
reciprocity relation (20) of the text. 

In a very similar manner one can derive (21) of 
the text. 

B. Two-dimensional formulation. For the discus- 
sion of two-dimensional cases many, if not most, of 
the formulas of the text require modification, and 
almost all the modifications may readily be obtained 
by reduction from formulas given in the text or by 
parallel development. Certain key formulas and 
minor subtleties are discussed here. 

It is assumed that all quantities are independent 
of the transverse coordinates y and k y , so that these 
coordinates will be absent from all formulas. 

Under the above assumption the unit vectors * u 
k 2 designating the "parallel" and "perpendicular" 
electric-field directions degenerate to Ki=e x sgn k x , 
K 2 =e y sgn k x . The inconvenience of the sign re- 
versal is avoided by adopting e x , e y as the basis 
vectors for the two polarizations. Sums over the 
polarization index m become sums over x- and y- 
components. This change induces a few further 
sign changes, the key one being in the reciprocity 
relation (given below). 



The two-dimensional forms of (14) and (15) may 
be written 

E(r) t =jB(k x )e ik **+^dk x , (14 2 ) 

B(k x ) = (2w)- 1 e-^ f E(r) l €r*'Sfe (15 2 ) 

where r=xe x -{-ze z , y 2 =k 2 —k 2 x , and integrations 
with respect to k x and x are understood to be taken 
over the range (— °°, <»). (The numbering of these 
two equations indicates the scheme to be followed 
in this appendix.) 

Power expressions such as (16) and (4) must be 
reinterpreted as power per unit length in the ^-direc- 
tion. Equation (16) becomes 

P r =Tc\^ m \b m {k x )\ 2 dk x - (16 2 ) 

k\<¥ 

It is convenient to match the loss of a factor of 2t 
between (16) and (16 2 ) by renormalizirg the basis 
fields at S so that (4) becomes 



Po=T7 lo (\a \ 2 -\b \ 2 )- 



(4 2 ) 



Witli this renormalization the numerical factors in 
the reciprocity relation remain unchanged. 
The reciprocity relation becomes 



VoSoi(m,k x )=7i m S lo (m, — k x 



(20 2 ) 



This is probably best established by a derivation 
parallel to that used for (20). The disappearance 
of a minus sign between the members of (20 2 ) as 
compared with (20) is due to the adoption of e x 
and e v as basis vectors for the two polarizations. 

In the two-dimensional case the asymptotic form 
of E(r); for large r is 

E(r) tt asym P =(|:) 2 k cos B f r'h ikr B(k sin 6') , (24 2 ) 
where 0' = tan -1 Or/2). From this follows 

S io(fc)=(^) a ^J-'rh-^Eikr/k)^^ (25 2 ) 

for kl<Ck 2 , where k=k x e x +ye 2 . 

The power radiation pattern becomes 

p=w(e/fx)ik eos 2 0'b-b (26*) 

where b, the complete vectorial spectrum in the two- 
dimensional case, is b — b x e x -\-b z e z and b z =—k x b x y~ 1 . 
The two-dimensional forms of equations such as 
(18), (32), and (46) may be written without difficulty. 
[The equation corresponding to (32) is written as (53) 
in the text.] 



12 



5. References and Notes 

[1] For a general description of this work, see W. Culshaw, 
J. M. Richardson, and D. M. Kerns, Precision milli- 
meter-wave interferometry at the U.S. National Bu- 
reau of Standards, Proc. of the Symposium on Inter- 
ferometry at the National Physical Laboratory, 
Teddington, Middlesex, England (June 1959). 

[2] A preliminary solution to the problem posed here using 
scalar waves and an asymptotic expansion of the 
reflection integral was obtained by DayhofF (unpub- 
lished National Bureau of Standards Office of Basic 
Instrumentation report. This office supported the 
earlier phases of this work.) 

[3] Microwave versions of the partially reflecting plates of 
the optical Fabry-Perot are discussed in the paper 
cited in note 1; see also W. Culshaw, Trans IRE, 
MTT-7, 221 (1959). 

[4] See, e.g., J. A. Stratton, Electromagnetic theory, p. 361ff 
(McGraw-Hill Book Co., New York, N.Y., 1941). 

[5] See e.g., B. Friedman, Principles and techniques of 
applied mathematics (John Wiley & Sons, New York, 
N.Y., 1956). 



[6] D. M. Kerns, Basis of the application of network equa- 
tions to waveguide problems, J. Research NBS, 42, 
515 (1949). 
[7] This can be derived formally from (14) by application 
of methods originally devised for single integrals. For 
these methods see e.g., H. and B. S. Jeffreys, Methods 
of mathematical physics (Cambridge Univ. Press, 
1950). For discussion of multiple integrals see D. S. 
Jones and M. Kline, Asymptotic expansion of multiple 
integrals and the method of stationary phase, N.Y.U., 
Inst. Math. Sci., Research Report No. EM-100 (1956). 
[8] Discussed briefly in the reference cited in note 1. 
[9] A convenient reference is J. A. Stratton, op. cit. f Chap. 8. 
[10] This is readily derivable from, e.g., eq (26), p. 578, in 

Stratton, op. cit. 
[11] H. and B. S. Jeffreys, op. cit., p. 501ff. 
[12] R. Courant and D. Hilbert, Methods of mathematical 
physics, 1st English Ed., p. 468 (Interscience Pub- 
lishers, Inc., New York, 1953). 



Boulder, Colo. 



(Paper 64B1-15) 



13 



About Timothytrespas

I am a victim of human experimentation MK-ultra mind control Morgellons nanotechnology syndrome & remote neural connectivity. I am an artist, inventor, musician, thinker, lover, human being who cares for all humanity & all life. I believe people should endeavor to live in peaceful cooperation rather than brutal waring survival of the most brutal. We live in a forced false-paradigm and I desire to wake people up from the 'trance hypnotic mind control programming' to the 'TRUTH of light and love'! Blessing and peace. Justice to all who suffer under tyranny. Compassion for all beings. May GOD have mercy on us all.
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