# 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
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-

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—
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

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
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

>

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 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

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
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 .

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
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

```