.
United States Patent [191
[11] Patent Number:
Dragone
[45]
A
W
[ 541 PERIODI
ELEMEN'? PKTTERAYRNHH
A NEARLY IDEAL
_
[75] Inventor‘
[73] Assignee:
.
.
Comldo Dragon’ Lmlc sllvcr' NJ‘
AT&T Bell Laboratories, Murray
?ill, N_J_
Date of Patent:
5,039,993
Aug. 13, 1991
OTHER PUBLICATIONS
“Theory & Analysis of Phased Array Antennas”, N. Y.
Wiley,‘ Publisher, 1972, Introduction to array Theory,
N’ Amilay ct 81-, PP- l0—14
“Efficient Multichannel Integrated Optics Star Coupler
on Silicon”, IEEE Photonics Technology Letters, vol.
[21] APPl- N°-= “0,825
[22] Filcd:
N0" 24’ 1989
51
1, No. 8, Aug. 1989, c. Dragone etal., pp. 241-243.
Primary Examiner-Michael C. Wime'r
Int. Cl.’ ........................................... .. H01
13/00
iszi US. 01. ............................... .. 343/776; (3243/7315
[58]
[56]
Field of Search .............. .. 343/776, 786, 772, 909
References Cited
U S_ PATENT DOCUMENTS
y
3mg" Jr" ' ' ' ' '
811976 biiirs'zh' ‘ ' ' ‘ ' '
[57]
ABSTRACT
A waveguide array comprising a plurality of wave
guides which are each outwardly tapered at the aper
'
Re. 23,051 11/1948 Carter ....................... .. 343/786
3’977’0o6
Assistant Exami”e'_H°anga“h Le
At'or'ley' 4488'", 0’ Fi""—-E1i Weiss
'' '' "
ture of the array in accordance with a predetermined
criteria chosen to incl-ease waveguide efficiency‘ The
tapering serves to gradually transform a fundamental
' ‘ ' ‘ " 345853
Bloch mode, propagating through the waveguide array,
4’259’674 3/19“ Dragon
" 343/909
4:369:4l3 l/l983 Devan m1. .'...'.'...............
333/34
into a plane wave in a predetermined direction, and then
1° launch the Plane Wave in“) free space in the predeter
4,737,o04 4/1988 Amitay =1 al. ................. .. 350/96.15
mined direction‘ In another embodiment, the wave
4,s7s,059 l0/l989 Yukl ........ ..' ........................ .. 343/756
guides are positioned relative to one another in order to
FOREIGN PATENT DOCUMENTS
satisfy the predetermined criteria.
60-196003 10/1985 Japan ................................. .. 343/776
7 Claims, 6 Drawing Sheets
""10
0-
_.lc\'
US. Patent
FIG.1
PRIOR ART
Aug. 13, 1991
Sheet 1 of 6
5,039,993
US. Patent
Aug. 13,1991
Sheet 2 of 6
5,039,993
FIG.2
PRIOR ART
ENERGY
IDEAL
RESPONSE
202
TYPICAL ACTUAL
Il
RESPONSE
201
US. Patent
FIG.3
PRIOR ART
Aug. 13, 1991
Sheet 3 of 6
5,039,993
US. Patent
FIG.4
Aug. 13, 1991
Sheet 4 of 6
5,039,993
US. Patent
Aug. 13, 1991
FIG 7
WAVEGUIDES
101
—
Sheet 6 of 6
5,039,993
5,039,993
1
2
i?ed by the relationship ¢=kasin(00), m=il, i2 . . .
PERIODIC ARRAY WITH A NEARLY IDEAL
ELEMENT PATTERN
BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention relates to waveguides, and more par
, and k=27T/>\ in the medium occupying the positive 2
plane. The direction of 00, and consequently of all the
other plane waves emanating from the waveguide ar
5
ray, can be adjusted by adjusting the phase difference ¢
between the inputs to adjacent elements. It can be
shown that the fraction of the power radiated at direc
tion 00 when the inputs are excited in a linear phase
ticularly, a technique for maximizing the ef?ciency of
progression is N(0), de?ned previously herein'for the
an array of waveguides.
2. Description of the Prior Art
10 case of excitation of only one of the waveguides with
Waveguide arrays are used in a wide variety of appli
cations such as phased array antennas and optical star
couplers. FIG. 1 shows one such waveguide array com
prising three waveguides 101-103 directed into the x—z
plane as shown. The waveguides are separated by a
distance "a" between the central axis of adjacent wave
guides, as shown. A ?gure of merit for such a wave
guide array is the radiated power density P(0) as a func
tion of 0, the angle from the z-axis. This is measured by
the fundamental mode.
The relationship between the response of the array to
excitation of a single waveguide with the fundamental
mode, and the response of the array to the fundamental
Bloch mode can be further understood by way of exam
ple. Suppose in a Bloch mode excitation 4) is adjusted
' according to ¢=kasin 00 such that 00 is 5°.
The power radiated at 5° divided by the total input
power=N(5°). However, if only one waveguide is ex
exciting one of the waveguides in the array, i.e. wave
cited, and a response similar to response 201 of FIG. 2
guide 102, with the fundamental input mode of the
waveguide, and then measuring the radiated pattern.
Ideally, it is desired to produce a uniform power distri
bution as shown in ideal'response 202 of FIG. 2, where
is produced in the Brillouin zone, then at 0=|5°, P09)“.
'“a1/P(e)!'deal=N(5°)
The fractional radiated power outside the central
Brillouin zone of FIG. 2, or equivalently, the percent
(y) is speci?ed by the well-known equation
age of the power radiated in directions other than 00 in
FIG. 3, should be minimized in order to maximize per
[a] sin ('y)=)\/2.
(1)
formance. In a phased array radar antenna, for example,
false detection could result from the power radiated in
where A is the wavelength of the radiated power in the
directions
other than then 00. It can be shown that the"
medium occupying the positive 2 plane of FIG. 1. The 30
wavefront in the direction 61 of FIG. 3 comprises most
angular distance from —y to 'y is known as the central
of the unwanted power. Thus, it is a goal of many prior
Brillouin zone. In practice, it is impossible to produce
art waveguide arrays, and of this invention, to eliminate
ideal results. An exemplary response from an actual
as much as possible of the power radiated in the 01
array would look more like typical actual response 201
of FIG. 2. The ef?ciency of the array, N(0), when one 35 direction, and thus provide a high ef?ciency waveguide
array.
waveguide is excited, is the ratio of the actual response
Prior art waveguide arrays have attempted to attain
divided by the ideal response, for all 6 such that
the goal stated above in several ways. One such prior
—'yé0§y. Of course, this neglects waveguide attenua
art array is described in N. Amitay et al., Theory and
tion and reflection losses. With this background, the
operation of phased array antennas is discussed below. 40 Analysis of Phased Array Antennas, New York, Wiley
Publisher, 1972, at pp. 10-14. The array achieves the
The operation of a prior art phased array antenna can
goal by setting the spacing between the waveguide
be described as follows. The input to each waveguide of
centers equal to 7t/2 or less. This forces 7 to be at least
FIG. 1 is excited with the fundamental mode of the
90°, and thus the central order Brillouin zone occupies
input waveguides. The signal supplied to each wave
guide is vinitially uncoupled from the signals supplied to 45 the entire real space in the positive 1 plane of FIG. 1.
This method, however, makes it difficult to aim the
the other waveguides and at a separate phase, such that
beam in a narrow desired direction, even with a large
a constant phase difference d) is produced between
25
number of waveguides. The problem that remains in the
prior art is to provide a waveguide array which, when
guide 101 could be excited with a signal at zero phase,
waveguide 102 with the same signal, at 5° phase, wave 50 excited with a Bloch mode, can con?ne a large portion
of its radiated power to the direction 00 without using a
guide 103 with the same signal at 10° phase, and so forth
adjacent waveguides. For example, in FIG. 1, wave
for the remaining waveguides in the array (not shown).
large number of waveguides. Equivalently, the problem
This would imply a phase difference of 5° between any
is to provide a waveguide array such that when one
two adjacent waveguides. The input wave produced by
waveguide is excited with the fundamental mode, a
large portion of the radiated power will be uniformly
this excitation is known as the fundamental Bloch mode,
or linear phase progression excitation. When the input
excitation is the fundamental Bloch mode, the output
from the waveguide array, part of which is illustrated in
distributed over the central Brillouin zone.
FIG. 3, will be a series of plane waves, e. g., at directions
SUMMARY OF THE INVENTION
The foregoing problem in the prior art has been
00,01 and 02, each in a different direction, where the
direction of the in"I plane wave is speci?ed by:
solved in accordance with the present invention which
relates to a highly ef?cient waveguide array formed by
shaping each of the waveguides in an appropriate man- ‘
' (2)
21r
a
ner, or equivalently, aligning the waveguides in accor
dance with a predetermined pattern. The predeter
65 mined shape or alignment serves to gradually increase
the coupling between each waveguide and the adjacent
and the wavefront radiated in the direction of 00 is the
only wavefront in the central Brillouin zone and is spec
waveguides as the wave propagates through the wave
guide array towards the radiating end of the array. The
3
5,039,993
ef?ciency is maintained regardless of waveguide spac
4
The gradual taper in each waveguide, shown in FIG.
ing.
4, can be viewed as an in?nite series of in?nitely small
discontinuities, each of which causes some energy to be
BRIEF DESCRIPTION OF THE DRAWING
FIG. 1 shows an exemplary waveguide array of the
transferred from the fundamental mode to the ?rst un
wanted mode. However, because of the difference in
propagation constants between the two modes, the en
ergy transferred from the fundamental mode to the ?rst
unwanted mode by each discontinuity will reach the
aperture end of the waveguide array at a different
prior art;
FIG. 2 shows the desired response and a typical ac
tual response to the excitation of a single waveguide in
the array of FIG. 1;
FIG. 3 shows a typical response to the excitation of
all the waveguides of FIG. 1 in a Bloch mode;
FIG. 4 shows an exemplary waveguide array in ac
phase. The waveguide taper should be designed such
that the phase of the energy shifted into the ?rst un
wanted mode by the different discontinuities is essen~
tially uniformly distributed between zero and 211'. If the
foregoing condition is satis?ed, all the energy in the ?rst
unwanted mode will destructively interfere. The design
procedure for the taper is more fully described below.
FIG. 6 shows a plot of the function
cordance with the present invention;
FIG. 5 shows the response to the waveguide array of
FIG. 4 as compared to that of an ideal array;
FIG. 6 shows, as a function of x, the refractive space
pro?les of the waveguide array in two separate planes
orthogonal to the longitudinal axis; and
FIG. 7 shows an alternative embodiment of the in
ventive waveguide array.
20
DETAILED DESCRIPTION
FIG. 4 shows a waveguide array in accordance with
as a function of x at the points z=c and 2:0’ of FIG. 4,
the present invention comprising three waveguides
where n is the index of refraction at the particular point
401-403. The signi?cance of the points z=s,t,c, and 0’ 25 in question along an axis parallel to the x axis at points
will be explained later herein, as will the dashed portion
0 and c’ of FIG. 4, and z is the distance from the radiat
of the waveguides to the right of the apertures of the
ing end of the array. For purposes of explanation, each
waveguides at the x axis. In practical arrays, it is impos
sible to achieve perfect performance throughout the
central Brillouin zone. Therefore, a 70 is chosen, and
represents some ?eld of view within the central Bril
louin zone over which it is desired to maximize perfor
mance. As will be shown hereinafter, the choice of 70
will effect the level to which performance can be maxi 35
of the graphs of FIG. 6 is de?ned herein as a refractive
space pro?le of the waveguide array. The designations
n1 and n2 in FIG. 6 represent the index of refraction
between waveguides and within waveguides respec
tively. Everything in the above expression is constant
except for n, which will oscillate up and down as the
mized. A procedure for choosing the “best” '70 is also
waveguides are entered and exited, respectively. Thus,
each plot is a periodic square wave with amplitude
discussed hereafter. FIG. 5 shows the response curve of
proportional to the square of the index of refraction at
FIG. 2, with an exemplary choice of yd. Assuming 70
has been chosen, the design of the array is more fully
described below.
coupling produces a plane wave in a speci?ed direction 45
which is based on the phase difference of the input
signals. However, the gradual transition from uncou
the particular point in question along the x axis. Note
the wider duty cycle of the plot at 2:0’, where the
waveguides are wider. Specifying the shape of these
plots at various closely spaced points along the z-axis,
uniquely determines the shape of the waveguides to be
used. Thus, the problem reduces to one of specifying
the plots of FIG. 6 at small intervals along the length of
the waveguide. The closer the spacing of the intervals,
the more accurate the design. In practical applications,
?fty or more such plots, equally spaced, will suf?ce.
pled signals to a plane wave also causes unwanted
higher order Bloch modes to be generated in the wave
Referring to FIG. 6, note that each plot can be ex
panded into a Fourier series
Returning to FIG. 3, as the fundamental Bloch mode
propagates in the positive 2 direction through the wave
guide array, the energy in each waveguide is gradually
coupled with the energy in the other waveguides. This
guide array, and each unwanted mode produces a plane
wave in an undesired direction. The directions of these
unwanted modes are speci?ed by Equation (2) above.
These unwanted plane waves, called space harmonics,
(3)
21
reduce the power in the desired direction. The ef?
ciency of the waveguide array is substantially maxi 55 Of interest is the coef?cient of the lowest order Fourier
mized by recognizing that most of the energy radiated
term V1 from the above sum. The magnitude of V1' is
in the unwanted directions is radiated in the direction of
denoted herein as V(z).
’
01. As described previously, energy radiated in the
V(z) is of interest for the following reasons: The
direction of 01 is a direct result of energy converted to
phase difference v between the ?rst unwanted mode
the ?rst higher order Bloch mode as the fundamental 60 produced by the aperture.of the waveguide array and
Bloch mode propagates through the waveguide array.
the ?rst unwanted mode produced by a section dz lo
Thus, the design philosophy is to minimize the energy
cated at some arbitrary point along the waveguide array
transferred from the fundamental Bloch mode to the
is
?rst higher order Bloch mode, denoted the ?rst un
wanted mode, as the energy propagates through the
(4)
waveguide array. This is accomplished by taking ad
vantage of the difference in propagation constants of
the fundamental mode and the ?rst unwanted mode.
where the integral is taken over the distance from the
arbitrary point to the array aperture, and B0 and B1 are
5
5,039,993
the propagation constants of the fundamental and ?rst
unwanted mode respectively. The total amplitude of the
?rst unwanted mode at the array aperture is
will be discussed later herein, however, for purposes of
the present discussion, F, can be assumed zero, corre
sponding to an untruncated waveguide. It can be veri
(5)
where v L is given by Equation (4) evaluated for the case
where dz is located at the input end of the waveguide
where n1=index of refraction in the waveguides,
n2=index of refraction in the medium between the
waveguides, and l is the distance between the outer
array, i.e., the point z=s in FIG. 4, and t is given as
walls of two adjacent waveguides as shown in FIG. 4.
41r"(sin7 - may
15
Thus, from equations (9) and (l l),
l2
(7)
21r2[sin 7Sin'— 7sinog
and 0 is an arbitrary angle in the central Brillouin zone,
discussed more fully hereinafter. Thus, from equations
\| l
I:
J:
_ p20,)
( )
("1 + "2)("1
— "2) kzazsin[ lgzpr
J
411'
0
5-7, it can be seen that the total power radiated in the 0
direction, is highly dependent on V(z). Further, the
ef?ciency N(0) previously discussed can be represented
as
Thus, after specifying 0B and ‘yo, and, assuming that
25 F 1:0, Equation 12 can be utilized to specify l(z) at
various points along the z axis and thereby de?ne the
30
This is the reason V(z) is of interest to the designer, as
stated above.
In order to maximize the ef?ciency of the array, the
shape of the waveguides.
Throughout the previous discussion, three assump
tions have been made. First, it has been assumed that "yo
was chosen prior to the design and the ef?ciency was
width of the waveguides, and thus the duty cycle in the
maximized over the chosen ?eld of view. Next, 03 was
assumed to be an arbitrary angle in the central Brillouin
zone. Finally, F, was assumed to be zero, corresponding
V(z) substantially satis?es the relationship
ence the performance of the array. Further, the perfor
corresponding plot, V(z) should be chosen such that at 35 to an untruncated waveguide. In actuality, all of these
three parameters interact in a complex manner to in?u
any point z~ along the length of the waveguide array,
mance may even be de?ned in a manner different from
that above. Therefore, an example is provided below of
the design of a star coupler. It is to be understood that
the example given below is for illustrative purposes of
demonstrating the design procedure may be utilized in a
(9)
7
sin ‘y — sinOg
= [T][T“i“—$l
wide variety of other applications.
where
.
One ?gure of merit, M, for an optical star coupler is
(10)
45 de?ned as
sin-ya
(13)
M = Mm) Sm
To maximize M, the procedure is as follows: Assume
F;=0, choose an arbitrary 0B, and calcualte N(0) using
L is the length of the waveguide after truncating, i.e.,
excluding the dashed portion in FIG. 4, F, and F,are the
fractions of the waveguide remaining and truncated,
respectively. More particularly, the length of the wave
guide before truncation would include the dashed por
tion of each waveguide, shown in FIG. 4. This can be
calculated easily since, at the point when the wave
guides are tangent, (z=t in FIG. 4), V(z) will equal 0 as
the ‘plot
.equations 5-8, for all angles 0 within the Brillouin zone.
Having obtained these values of N(0), vary 70 between
zero and ‘y to maximize M. This gives the maximum M
55
for a given F, and a given 013. Next, keeping F, equal to
zero, the same process is iterated using various 0B’s un'til
every, 03 within the Brillouin zone has been tried. This
gives the maximum M for a given F, over all 93s. Fi
nally, iterate the entire process with various Ffs until
the maximum M is achieved over all 03s and F,s. This
can be carried out using a computer program.
It should be noted that the example given herein is for
illustrative purposes only, and that other variations are
possible without violating the scope or spirit of the
65 invention. For example, note from equation 12 that the
is a constant. Thus, by ?nding the leftmost point z=t
along the z axis such that V=0, one can determine the
required property of V(z) can be satis?ed by varying
“a” as the waveguide is traversed, rather than varying l
length before truncation. The length after truncation
as is suggested herein. Such an embodiment is shown in
7
5,039,993
FIG. 7, and can be designed using the same methodol
ogy and the equations given above. Further, the value
of the refractive index, it, could vary at different points
in the waveguide cross-section such that equation (12) is
satis?ed. Applications to radar, optics, microwave, etc.
where 63 is an arbitrary angle within a predeter
mined range of angles de?ned by a minimum and
are easily implemented by one of ordinary skill in the
art.
maximum angle, 7 is the maximum angle,
The invention can also be implemented using a two
dimensional array of waveguides, rather than the one
dimensional array described herein. For the two-dimen
sional case, equation (3) becomes
L is a predetermine length of each waveguide, |z|
where ax is the spacing between waveguide centers in
the x direction, and ayis the spacing between waveguide
20
centers in the y direction. The above equation can then
be used to calculate Vlvo, the ?rst order Fourier coef?
cient in the x direction. Note from equation (14) that
this coef?cient is calculated by using a two-dimensional 25
Fourier transform. Once this is calculated, the method
set forth previously can be utilized to maximize the
efficiency in the x direction. Next, ax in the left side of
equation (14) can be replaced by ay, the spacing be
tween waveguide centers in the second dimension, and
the same methods applied to the second dimension.
is a perpendicular distance between the refractive
space pro?le and the second end ofthe waveguide,
F, is equal to L/(L+b), b is a perpendicular‘ dis
tance which an outer surface of each waveguide
would have to be extended in order to become
tangent to an outer surface of an adjacent wave
guide, and F,=l-F,.
2. A waveguide array according to claim 1 wherein
the waveguides are all aligned substantially parallel to
each other in a predetermined direction, and wherein
the input ports of all the waveguides substantially de?ne.
a ?rst plane substantially normal to the predetermined
direction, and the output ports of all the waveguides
substantially de?ne a second plane substantially normal
to the predetermined direction, and each waveguide
comprises a diameter which varies along said predeter
The waveguides need not be aligned in perpendicular
rows and columns of the x,y plane. Rather, they may be
aligned in several rows which are offset from one an 35 mined direction such that a predetermined criteria is
substantially satis?ed.
other or in any planar pattern. However, in that case,
3. A waveguide array according to claim 1 wherein
the exponent of the two-dimensional Fourier series of ,
equation (14) would be calculated in a slightly different
the waveguides are aligned substantially radially with
manner in order to account for the angle between the x
and y axes. Techniques for calculating a two-dimen
sional Fourier series when the basis is not two perpen
each other, and wherein the input ports of the wave
guides substantially de?ne a ?rst arc and the output
ports of the waveguides substantially de?ne a second
are, substantially concentric to and larger than the ?rst
arc, such that a predetermined criteria is substantially
dicular vectors are well-known in the art and can be
used to practice this invention.
'
satis?ed.
_
I claim:
45
4. A waveguide array according to claim 1 wherein
each waveguide includes a predetermined index of re
1. A waveguide array including an associated ef?
fraction which varies along the predetermined direction
such that a predetermined criteria is substantially satis~
?ed.
5. A waveguide array according to claim 1, 2, 3 or 4
wherein the length of each waveguide is chosen in ac
cordance with a predetermined criteria such that the
ciency and comprising:
a plurality of waveguides, each waveguide including
an input port at a ?rst end thereof for receiving
electromagnetic energy, and an output port at a
second end tlierqof for launching the electromag
netic energy, the waveguide array including a pre
ef?ciency of the waveguide array-is substantially maxi
determined series of refractive-space pro?les ar
guide array, each refractive-space pro?le including
mized.
6. A waveguide array according to claim 5 wherein
the plurality of waveguides are arranged in a AXB
a separate Fourier series expansion which com
prises a lowest order Fourier term that is deter
arbitrary integers.
ranged at spaced-apart locations across the wave
two-dimensional array where A and B are separate
7. A waveguide array according to claim 1, 2, 3 or 4
wherein the plurality of waveguides are arranged in an
mined to substantially maximize the associated
ef?ciency of the waveguide array, whereby said
AXB two-dimensional array where A and B are sepa
lowest order Fourier term, denoted V(,), is de?ned ‘
by
rate arbitrary integers.
*
65
it
t
‘I
*