Abstract:
A traveling wave device for the combining or splitting of symmetric and asymmetric traveling wave energy includes a feed waveguide for traveling wave energy, the feed waveguide having an input port and a launching port, a reflector for coupling wave energy between the feed waveguide and a final waveguide for the collection and transport of wave energy to or from the reflector. The power combiner has a launching port for symmetrical waves which includes a cylindrical section coaxial to the feed waveguide, and a launching port for asymmetric waves which includes a sawtooth rotated about a central axis.

Description:
This invention was made with Government support under grant DE-FG03-97ER82343 awarded by the Department of Energy. The government has certain rights in this invention. 

   FIELD OF THE INVENTION 
   The current invention is directed to the class of power combiners comprising a plurality of input waveguides, hereafter referred to as feed waveguides summing input power into a single output waveguide, hereafter called a final waveguide. Because of symmetrical behavior in the present invention between input and output ports, the relevant field of the present invention also includes power splitters having a single input port dividing the power applied to this port into a plurality of output ports, dividing the power according to a desired ratio between these ports. 
   The present invention includes the class of power combiners which sum wave energy from a plurality of waveguides, each carrying traveling TE, TM, and HEmn mode electromagnetic waves. The traveling electromagnetic waves may be propagating either in a symmetric mode or in an asymmetric mode. The present power combiner has several feed waveguides, a reflector for each feed waveguide, and a single final waveguide. 
   BACKGROUND OF THE INVENTION 
   In applications requiring the summing of a large number of output from klystrons launching TE01 mode waves into cylindrical waveguides, it has been necessary to first convert the waves to TE00 functional waves, and summing according to prior art techniques. 
   Examples of prior art power combiners are the class of circular power combiners such as U.S. Pat. No. 5,446,426 by Wu et al, which describes a device accepting microwave power from the resonant cavity of a microwave oscillator, and summing into a circularly symmetric waveguide for delivery to an output port. U.S. Pat. No. 4,175,257 by Smith et al describes another circular power combiner comprising radial input ports which furnish microwave power which is summed along a principal axis. U.S. Pat. No. 4,684,874 by Oltman describes another radially symmetric power combiner/divider, and U.S. Pat. No. 3,873,935 describes an elliptical combiner, whereby input energy is provided to one focus of the ellipse, and removed at the other focus. In all of these combiners, the output port is orthogonal to the input port, and the wave mode is TM, rather than TE. 
   U.S. Pat. No. 4,677,393 by Sharma describes a power combiner/splitter for TE waves comprising an input port, a parabolic reflector, and a plurality of output ports. 
   For complete understanding of the present invention, a review of well-known traveling wave principles relevant to the prior art should be explained. References for traveling wave phenomenon are “Fields and Waves in Communication Electronics” by Ramo, Whinnery, and Van Duzer, Chapter 7 “Gyrotron output launchers and output tapers” by Mobius and Thumm in “Gyrotron Oscillators” by C. J. Edgcombe, and “Open Waveguides and Resonators” by L. A. Weinstein. 
   Circular waveguides support a variety of traveling wave types. Modes are formed by waves which propagate in a given phase with respect to each other. For a given free-space wavelength λ, a circular waveguide is said to be overmoded if the diameter of the waveguide is large compared to the wavelength of a wave traveling in it. An overmoded waveguide will support many simultaneous wave modes traveling concurrently. If the wave propagates axially down the waveguide, the wave is said to be a symmetric mode wave. If the wave travels helically down the waveguide, as shown in  FIG. 16 , the wave is said to be an asymmetric mode wave. In the case where two identical asymmetrical helical waves are combined, the result is an asymmetric wave mode propagating axially. In the case of the present invention, helically propagating waves will be considered. 
   Transverse electric, transverse magnetic, or hybrid modes propagating in cylindrical waveguides have two integer indices. The first index is the azimuthal index m which corresponds to the number of variations in the azimuthal direction, and the second index is the radial index n that corresponds to the number of radial variations of the distribution of either the electric or magnetic field component. While the radial index n always has to be larger than zero, the azimuthal index m can be equal to zero. Due to their azimuthal symmetry, modes with m=0 are called symmetric modes whereas all other modes are called asymmetric. Asymmetric modes can be composed of a co- and counter-rotating mode with has the consequence that—as in the case of symmetric modes—the net power flow (real part of the poyntingvector) only occurs in the axial direction. However, if either to co- or counter-rotating mode is present there is a net energy flow in axial and azimuthal direction, hence we obtain a helical propagation. For the present invention helically propagating or symmetric modes are considered. 
   When using a ray-optical approach to the modes, a decomposition of the modes as plane waves with the limit of zero wavelength rays are obtained. In general, these are tangent to a caustic with a radius:
 
 Rc=Rw ( m/Xmn ) 
 
where:
         Rc is the radius of the caustic   Rw is the radius of the waveguide   Xm is the eigenvalue of the mode       

   This has the consequence that the geometrical rays have an azimuthal, radial, and axial coordinate. However, in the case of symmetric modes, the radius of the caustic becomes zero, and hence the rays representing symmetric modes only have a radial and an axial component. In the design of a reflector, the phase front of the rays tangent to a caustic is required. In an asymmetric mode, this phase front is the involute of the caustic. For a symmetric mode, the phase front reduces to a point representing the caustic with a radius=0. 
   In a cylindrical waveguides, the radial component of the ray does not contribute to the net flow. This however changes as soon as the waveguides has a port which causes a net power flow in the radial direction. 
   The phase front for an asymmetric mode wave is described by an involute in free space, a shape which is inwardly curled towards the center of the waveguide. The particular shape for the phase front for each wave mode unique, and is generally numerically calculated. The important aspect of the phase front is that it defines a particular surface, and this phase front will be used later for construction of certain structures of the invention. 
   Traveling waves can also be described in terms of the propagation velocity in a particular direction. Symmetric waves traveling down the axis of the waveguide have a purely axial component, and no perpendicular component. Asymmetric waves traveling helically down the axis of a waveguide have both an axial component, and a perpendicular component. There is a wave number k=2π/λ, where λ is the wavelength of the traveling wave. In each axial (parallel) direction and transverse (perpendicular) direction of travel, the following wave numbers may be computed:
 
 k   perp   =X   mn   /Rw  
 
 k   par   =sqrt{k   2   =k   perp   2 }
 
In these calculations,
         X mn  is the eigenvalue of the mode   m is the azimuthal index   Rw is the waveguide radius.       

   For asymmetric mode waves, the internally reflecting waves define a circle within the waveguide radius Rw known as a caustic. The radius of the caustic for an asymmetric mode wave is
 
 Rc=Rw ( m/X   mn ) 
 
Where
         Rc=radius of caustic   Rw=radius of waveguide   m=azimuthal index   n=radial index   X mn  is the eigenvalue of the mode       

   In cylindrical waveguides, the distance Lc represents the length of waveguide for which propagating TEmn, TMmn, or HEmn waves propagating in a cylindrical wavelength complete a 2n phase change. The formula for Lc is
 
 Lc= 2π Rw{k   par   sqrt {1=( m/X   mn)   2   }}/{k   perp   cos   −1 ( m/X   mn )}
 
where
         Rw, m, n, X mn , k perp , k par  are as previously defined       

   OBJECTS OF THE INVENTION 
   A first object of the invention is the summation of a plurality of symmetric waves such as TE01, TE02, TE03, etc. from a plurality of feed waveguides into a single final waveguide. 
   A second object of the invention is the summation of a plurality of asymmetric waves with azimuthal index m&gt;0 such as TE11, TE12, TE21, etc. from a plurality of feed waveguides into a single final waveguide. 
   A third object of the invention is the summation of a plurality of either traveling symmetric or traveling assymetric waves, each traveling wave coupled into a feed waveguide, thereafter coupled to a feed waveguide launching port, thereafter to a reflector, and thereafter to a summing final waveguide. 
   A fourth object of the invention is the splitting of a plurality of either traveling symmetric or traveling asymmetric waves applied to a final waveguide, these traveling waves thereafter coupled to a reflector, and thereafter coupled to a plurality of feed waveguides, 
   SUMMARY OF THE INVENTION 
   A power combiner has a plurality of feed waveguides, each feed waveguide having an input port and a launching port. The input port accepts either symmetric or asymmetric traveling waves, and the launching port emits these traveling waves to a focusing reflector. Each launching port has its own focusing reflector. A plurality of feed waveguides and focusing reflectors is arranged about a central axis. A final waveguide is disposed on this central axis for the transport of combined wave energy reflecting of the reflectors. Each feed waveguide is energized with a source of traveling wave energy, and this traveling wave energy is directed to the reflectors by the launching port of the feed waveguide, combining in the final waveguide. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  shows a single feed waveguide and a reflector for symmetric mode waves. 
       FIG. 1   a  shows the detail of a feed waveguide when unfolded into a plane. 
       FIG. 2  shows a cross sectional views of  FIG. 1   
       FIG. 3  shows a power combiner which sums input power from three symmetric wave sources. 
       FIG. 4  shows the cross sectional views of  FIG. 3   
       FIG. 5  shows a power combiner which combines input power from four symmetric wave sources. 
       FIG. 6  shows the cross sectional views of FIG.  5 . 
       FIG. 7  shows the details of the reflector construction in a collapsed section view. 
       FIG. 8  shows a collapsed section view of the reflector, feed waveguides, and final waveguides for the power combiner of FIG.  5 . 
       FIG. 9  shows a single feed waveguide, a reflector, and a final waveguide for asymmetric waves. 
       FIG. 10  shows a feed waveguide for asymmetric wave sources, the feed waveguide shown unwound onto a planar surface for clarity. 
       FIG. 11  shows a final waveguide for asymmetric wave summing, the final waveguide shown unwound onto a planar surface for clarity. 
       FIG. 12  shows final waveguide of  FIG. 11  unwound onto a planar surface, and with shaded areas showing the progressions of traveling wave energy 
       FIGS. 13   a  and  13   b  show different views of a power combiner for asymmetric mode input power which is summing asymmetric mode input power from 3 sources. 
       FIGS. 14   a  and  14   b  show a power combiner for asymmetric mode input power which is summing asymmetric mode input power from 4 sources. 
       FIG. 15  shows wave propagation in a waveguide as the geometrical optical summing of a plurality of individual geometric optic waves into a helically traveling wave. 
       FIG. 16  shows the helically traveling wave in a waveguide. 
       FIG. 17  shows the collapsed section view of 4 feed waveguides, the final waveguide, and the reflectors. 
       FIG. 18  shows the details of construction of a single reflector. 
       FIG. 19  shows power summing in the final waveguide. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
     FIG. 1  shows a feed waveguide  10  arranged about a feed waveguide axis  18 , and  FIG. 2  shows the cross sections of the related structures of FIG.  1 . Typically, these feed waveguides are fed by high power klystrons in TE 01  mode from a cylindrical waveguide. The feed guide  10  has a radius  13 , an input port  15 , and a launching port  12  centered on the feed waveguide axis  18 . In one embodiment optimized for symmetric waves, the feed waveguide  10  has a cylindrical part L 1   16  which is of a sufficient length to remove higher mode waves that may be present in the feed waveguide, a feed port  15  for receiving input power, and a launch port  12  for directing wave energy towards a reflector  14 . The first section of the feed waveguide is shown in section A—A of FIG.  2 .  FIG. 1  shows a launch port section  12  which comprises a cylindrical section having the same diameter and waveguide axis  18  as the input section, and further has a length L launch  of the launch port which is optimally
   L   launch   =Lc/ 2  
where
         L launch  is the length of the feature  20  in  FIG. 1         
   Lc=2πRf{k par sqrt{1−(m/X mn)   2 }}/{k perp cos −1 (m/X mn )}. As described earlier, Lc represents the length of a waveguide section for which propagating TEmn, TMmn, or HEmn waves propagating in a cylindrical wavelength complete a 2π phase change.
         Rf is the radius of the feed waveguide   k par  is the parallel, or axial wave number   m is the azimuthal index of the mode   X mn  is the eigenvalue of the mode   K perp  is the perpendicular wave number       

   For a symmeteric move wave, m=0, and so the equation for Lc simplifies to
 
 Lc= 4 Rf{k   par   }/{k   perp }
 
and therefore
 
L launch =2 Rf{k   par   }/{k   perp }
 
     FIG. 1   a  shows the feed waveguide  10  unfolded onto a planar surface with the features dimensioned for clarity. 
     FIG. 2  shows the cross section B—B of the second section having an included angle α1  24  which is preferably 180 degrees. The angular extent of the reflector  14  may be greater or smaller than 180 degrees, depending on the location of the center of the reflector with respect to the feed waveguide axis  18 , and the spatial requirements of the other reflectors. In general, the available included angle for each reflector will be 360/k degrees, where k is the number of feedguides present, as will be explained later with FIG.  8 . In  FIG. 2 , focusing reflector  14  may comprise an elliptical surface having an included angle α2  26  determined by the included angle  64   a  and  64   a ′ of  FIG. 8 , which will be 360/k degrees, where k is the number of feed waveguides present. The length L 3   22 , should be of sufficient length to enable reflection of most of the incident power from a launching port  12  into a final waveguide. The launching port  12  may be defined as the cylindrical section formed by sweeping a line of length L launch , with a separation from the feed waveguide axis  18  equal to feed waveguide radius  13  about an included angle α1  24 . Focusing reflector  14  is disposed about feed waveguide axis  18 , and has a length L 3  sufficient to reflect waves leaving the feed waveguide  10  into the final waveguide. 
     FIG. 3  shows a power combiner comprising three feed waveguides  30   a,    30   b,  and 30 c.  Incoming sources of symmetric wave energy enter each of the three feed waveguides  30   a,    30   b,  and  30   c,  which are arranged symmetrically about a power combiner central axis  36 , also shown in section E—E of FIG.  4 . The symmetric wave energy exists at the feed waveguide launching port, shown in section F—F of FIG.  4 . Focusing reflectors  32   a,    32   b,  and  32   c  act on energy exiting each of feed waveguides  30   a,    30   b,  and  30   c  respectively. Each feed waveguides is arranged with its feed waveguide central axis parallel to the power combiner central axis  36 . The focusing reflectors direct wave energy to final waveguide  34 .  FIG. 4  shows the section details of the structures of FIG.  3 . Section E—E shows the feed waveguides  30   a,    30   b,  and  30   c  of FIG.  3 . Each of the feed waveguides  30   a,    30   b,  and  30   c  has an identical radius  38 , shown only on waveguide  30   a  as  38   a  for clarity. Section F—F shows the launching ports of feed waveguides  30   a,    30   b,  and  30   c.  Section G—G shows the arrangement of focusing reflectors  32   a,    32   b,  and  32   c,  which will be described in detail later. Section H—H shows the cylindrical sectional view of final waveguide  34 , which has a radius  40 , and is disposed about the central axis  36 . In accordance with best mode shown in  FIG. 4  section F—F, the launching ports are convex with respect to the power container central axis  36 , while the reflectors  32   a,    32   b,    32   c  of section G- 13  G are concave with respect to the power combiner central axis  36 . In an alternate construction, each of the feed waveguides could be rotated 180 degrees about its own respective waveguide axis to produce launch ports which are concave when viewed in section F—F of  FIG. 4 , and each of the reflectors could be rotated 180 degrees about each feed waveguide central axis to produce reflectors which are convex with respect to the power combiner central axis  36 . As is clear to one skilled in the art, this arrangement would produce a feed waveguide launching port which directs energy towards the central axis  36 , and would be reflected by each reflector to the final waveguide  34 . However, it is believed that the arrangement of  FIG. 3  would produce the best power combiner. Also, while the feed waveguide radius  38  is shown as equal for each of the feed waveguides, it is possible for the power combiner to have unequal feed waveguide radii for each feed waveguide. While the feed waveguides of  FIG. 3  are shown distributed equally about the central axis  36  as is believed to be the best mode, it is also possible to arrange the feed waveguides with an unequal angular distribution. This angular distribution could be described in terms of the included angle formed between the planes which include each feed waveguide axis and the power combiner axis  36 . 
   In the final waveguide  34 , different wave modes may be present than were present in the feed waveguide  30 , so that wave mode in the final waveguide will be described in TEpq, where p &amp; q are the final waveguide mode numbers. For the final waveguide, the radius Rfinal and wave mode indices p and q should be chosen such that the brillouin angle for the mode in the final waveguide matches the brillouin angle for the mode in the feed waveguide. Since the radius Rfinal is generally larger than the radius of the individual feed waveguides, the mode indices will be higher as well. If the two feed waveguides carry TE 01  mode, and it is desired to carry TE 02  in the final guide, then R final  may be determined by
 
 R   final   =R   feed ( X   02   /X   01 ). 
 
   In general,
 
 R   final   =R   feed ( X   mn   /X   pq ) 
 
where
         R final =radius of final waveguide   R feed =radius of feed waveguide   X mn =eigenvalue of mode in feed waveguide   X pq =eigenvalue of mode in final waveguide       

   In addition to the above selection or Rfinal, the additional constraint Lfeedhelix=Lfinaldepth must be met. Since this criterion will generally not be met for a given feed waveguide mode and final waveguide mode, this is accomplished by utilizing the observation that the spectrum of eigenvalue of the various modes is dense. This constraint is met by making an appropriate selection between the available wave modes found in the feed waveguide and final waveguide, and the feed and final waveguide radii. 
     FIG. 5  shows a power combiner with 4 feed waveguides  50   a,    50   b,    50   c,  and  50   c.  Symmetric mode wave energy enters each of the feed waveguides  50 , and is directed to a launching port, as before. The wave energy leaving each launching port  50   a,    50   b,    50   c,  and  50   d  is sent to each reflector  52   a,    52   b,    52   c,  and  53   d,  and thereafter is reflected to final waveguide  54 .  FIG. 6  shows the cross sectional views of the power combiner/splitter of FIG.  5 . Section J—J shows the arrangement of feed guides  50   a - 50   d,  including the launching ports of section K—K. Section L—L shows the reflectors  52   a - 52   d,  and section M—M shows the output guide  54 . 
     FIG. 7  shows the construction details for a single reflector, shown as reflector  52   a  of FIG.  5 . The reference points of  FIG. 7  are the final waveguide axis  56  and the feed waveguide axis  51   a.  Wave energy leaves the center of feed guide  51   a  and is directed to the center of final waveguide  54 . These two points are used to construct the locus of points which define the reflector  52 . By the geometric optics technique of ray tracing, the reflector  52  is formed by the locus of points forming an equidistant total path from a first focus  51   a,  to the reflector  52   a,  and to the center of the final waveguide  54 . In  FIG. 7 , each exit path  60   a,    60   a′,    60   a″  is reflected from reflector  52   a,  and is directed to second focus  56  via reflected path  62   a,    62   a′,  and  62   a″,  respectively. The total path length  60   a + 62   a = 60   a′ + 62   a′ = 60   a″ + 62   a″,  etc. Feed guide radius  38   a  and final guide radius  40  are also shown. The extent of reflector  52   a  is typically determined by the included angle about reflector reference plane  64   a,  formed by sweeping a plane which includes the main axis  56  about waveguide axis  51   a.  The solid angular extent of the reflector  50   a  is shown as the included angle from reflector extent  64   a ′ to reflector extent  64   a″,  which is typically symmetric about the reflector axis  64   a.  The angle from  64   a ′ to  64   a ″ is determined by the number of reflectors present. In the case p=3 of 3 reflectors and 3 feed waveguides, the included angle of the reflector is 360/3=120 degrees. For the case p=4 of 4 reflectors and 4 feed waveguides, the included angle is 360/4=90 degrees. Any number of feedguides and reflectors may be accommodated in this manner. The reflector  52   a  comprises the locus of points providing equal path length from first focus to second focus, and is truncated by the included angle formed by  64   a′.  to  64   a″,  which enables the reflectors for the other feed guides to utilize the remaining space. 
   Once the locus of points, which defines the reflector  52   a  is determined as described above, it may be used to form the shape of the reflector along the waveguide axis  56 . The formation of the reflector solid  52  from the locus of reflector points may be thought of as an extrusion of the locus of points along the power combiner axis  56  to form the reflectors  52   a,   52   b,   52   c,   52   d  of  FIG. 5 , or any of the other reflectors shown in previous figures. The axial extent of the reflector may be chosen based on minimum power loss when coupling energy from the launching ports to the final waveguide. This axial extent is approximately the value Lc defined earlier. 
     FIG. 8  shows the arrangement of feed guides, reflectors, output guides for the case where k=4. Each feed guide  50   a,    50   b,    50   c,    50   d  has a central axis, and reflectors  52   a,    52   b,    52   c,  and  52   d  respectively dispose wave energy to the central axis of final waveguide  54 . Each reflector is symmetrically located about the connecting line between the two focal points, one at the central axis  56  and the other located at each feed guide center  51   a,    51   b,    51   c,  and  51   d.  These are also shown by the lines  64   a,    64   b    64   c,  and  64   d.  Typically, each feed waveguide and each reflector waveguide is coaxially arranged, although other arrangements, such as an angular offset between feed waveguides and reflectors could be accommodated. The result of the arrangement of feed waveguides, reflectors, and final waveguides in  FIG. 8  is that input power from each feed waveguide  50   a-d  is reflected by reflector  52   a-d,  and is focused at the center of final waveguide  54 . 
     FIG. 9  shows the power summer/splitter for asymmetric mode waves. In the general case, a plurality of feed waveguides  70  would be used, but only one is shown in this figure for clarity. Asymmetric mode waves travel in a helical path, as will be described later. Feed waveguide  70  includes a feed waveguide axis  73 , and a reference line  72  is shown to assist in understanding the actual shape of the feed guide. If feed guide  70  were unfolded about reference line  72 , the shape would be as shown in FIG.  10 . The circumference of feed guide  70  is equal to the number of wavelengths of the azimuthal mode, which is m wavelengths, or 2*pi*m radians in phase, and includes an exit surface of length  78  for the launching of waves towards the reflector  74  of FIG.  9 . Feed guide central axis  73  is shown offset from main axis  71 . Final waveguide  88  may be constructed on one of two different ways. For the special case where
         (φ c )/2π=(1/n)arc cos (m/X mn ) is an integer, where   m=azimuthal index   n=radial index   X mn =the eigenvalue of the mode
 
the final waveguide may be a simple cylinder without the multicuts  88   a,    88   b,    88   c,  etc. For all other cases, the final waveguide includes a multi-cut input wave surfaces  88   a,    88   b,    88   c,  and  88   d,  as shown in FIG.  9 .
       
   The feed waveguide  70  of  FIG. 9  includes a helical launch port which may be described by sweeping a line of length L feedlaunch =θ*L feedhelix /2n at the radius of the launch port from and parallel to said feed guide axis, where 0≦θ≦2π and θ is the angle in radians about the feed waveguide axis  73  and L feedhelix  is the depth of the helical cut  78 . L feedhelix  may be computer by
 
L feedhelix =Lc 
 
where
         Lc=2πR feed {k par sqrt{1−(m/X mn ) 2 }}/{k perp cos −1 (m/X mn )}   k par  is the parallel, or axial wave number   R feed  is the radius of the feed waveguide   m is the azimuthal index of the mode   X mn  is the eigenvalue of the mode   K perp  is the perpendicular wave number       

   Sweeping the line L feedlaunch  produces the helical launch ramp shown in  FIGS. 9 and 10 . 
   As shown in  FIG. 9 , the multicuts  88   a,    88   b,    88   c,    88   d  of the reflector port of the final waveguide may be constructed by sweeping a line of varying length L finalmulticut  at the final waveguide radius from said central guide axis about an angle θ:
 
L finalmulticut =(Lc/k)*(θ/(k*2*pi)) for 0≦θ≦2*pi/k 
 
where
         Lc=9πR final {k par sqrt{1−(p/X pq ) 2 }}/{k perp cos −1 (p/X pq )}   (Lc/k) is the multicut depth  77     k par  is the parallel, or axial wave number   R final  is the radius of the final waveguide   p is the azimuthal index of the mode   q is the radial index of the mode   X pq  is the eigenvalue of the mode   K perp  is the perpendicular wave number   k is the number of multicuts       

   The multicut of the final waveguide is formed by joining end-for-end k said surfaces of rotation to form a cylindrical solid, as shown in  FIG. 9  for the case k=4. 
     FIG. 9  also defines a drop and a ramp, which will be used later to show orientation of the helix in projection with respect to the helical cut. The drop may also be defined to be the location where θ=0 in the earlier definition of L feedlaunch . 
   As was described earlier for the symmetric mode case, final waveguide  88  may have different wave modes present than were present in the feed waveguides  70 , so the wave mode in the final waveguide will be described as TEpq, where p &amp; q are the final waveguide mode numbers. For the final waveguide, the radius Rfinal and wave mode indices p and q should be chosen such that the brillouin angle for the mode in the final waveguide matches the brillouin angle for the mode in the feed waveguide. Since the radius Rfinal is generally larger than the radius of the individual feed waveguide, the mode indices will be higher as well. If the two feed waveguides carry TE 01  mode, and it is desired to carry TE 02  in the final guide, then R final  may be determined by
 
 R   final   =R   feed ( X   02   /X   01 ). 
 
   In general,
 
 R   final   =R   feed ( X   mn   /X   pq ) 
 
where
         R final =radius of final waveguide   R feed =radius of feed waveguide   X mn =eigenvalue of mode in feed waveguide   X pq =eigenvalue of mode in final waveguide       

   In addition to the above selection or Rfinal, the additional constraint Lfeedhelix=Lfinaldepth must be met. Sine this criterion will generally not be met for a given feed waveguide mode and final waveguide mode, this is accomplished by utilizing the observation that the spectrum of eigenvalues of the various modes is dense. By making an appropriate selection between the available wave modes found in the feed waveguide and final waveguide, and the feed and final waveguide radii, it is possible to meet this constraint. 
     FIG. 11  shows the final waveguide  88  unfolded to a planar surface about reference line  89 . In practice, helically propagating waves exit feed waveguide  70 , are reflected by helical reflector  74 , and are collected by multicut input final waveguide  88 , entering at multicut surface  88   a  and other surfaces  88   b,    88   c,  and  88   d,  as shown by the ray traces  80 ,  82   84 , and  86 . These rays enter at angle α4  81 . The value of angle α4  81  is not the same as the brillouin angle but can be computed from
 tan α4={k par sqrt{1−{p 2 /X pq   2 }}}/{k perp  cos −1 {p/X pq }} 
where p≢0, and the other variables are as earlier defined. The final waveguide has final multicuts  88   a,   88   b,   88   c,   88   d,  of depth
   L   finaldepth   =L   c   /k,    
with parameters as defined earlier.
 
     FIG. 12  shows the path of input waves collected by each multicut collection surface, and includes an input surface for the multicut, each multicut surface corresponding to a surface collecting wave energy from each reflector, and directing it to each multi-cut surface, as will be described later. The angular hatch patterns represent approximations of wave energy as it travels through the structure. For example, examining the multicut port  84 , the series of identical hatch patterns correspond to the wave energy propagating through this path, which continues at the connection point at the top 4 bands to the right. Lc is shown graphically as the width of k bands (shown as k=4), and the Lfinaldepth  77  is Lc/k, as shown in FIG.  11 . φ c    83  is shown for reference, and will be described in detail later in FIG.  15 . The circumference of final waveguide  88  is shown in  FIGS. 11 and 12  as L launch . 
     FIG. 13   a  shows for k=3 an asymmetric mode, 3 port power summing/dividing structure. Each feed guide  100   a,    100   b,  and  100   c  has helically traveling waves which launch at the helical cut end  114  of each feed guide. The helical cut angle and feed guide diameter is designed as described in FIG.  10 . The three reflectors  102   a,    102   b,  and  102   c  capture and reflect wave energy leaving each feed guide  100   a,    100   b,  and  100   c  respectively, and feed this energy into each multicut surface of the multicut final guide  116 . Each multicut  118  is arranged to capture traveling wave energy from each reflector  102 .  FIG. 13   b  shows a different perspective view of  FIG. 13   a  for clarity in viewing the multicut final waveguide, and it can be seen that wave energy leaving each reflector  102   a,    102   b,    102   c  is captured by each multicut face  118   a,    118   b,  and  118   c,  respectively. The summed wave energy from each feed guide  100   a-c  thereafter travels down final guide  116 . 
     FIG. 14   a  shows the same power summer/divider for the case where k=4. As before, each feed guide  120   a-d  has a feed end and a helically cut output and described by the unwound detail of FIG.  10 . The reflectors  122   a-d  capture and reflect traveling wave energy to each of the 4 multicuts  124   a-,  respectively.  FIG. 14   a  and  14   b  show different views of the identical set of structures to enable clarity in viewing the helical cuts in the feed guide output waveguides  112 , as well as the multicuts  124  of the final guide  126 . The details of construction for the reflectors will be described later. 
     FIG. 15  shows the geometric optic ray-tracing case for a single ray  150  entering the waveguide  140  having a wall radius  146 , reflecting from the walls of waveguide  140 , and eventually exiting the waveguide at point  148 .  FIG. 15  shows this internal reflection in the projection view, where in addition to the internal reflection, the ray is also traveling down the longitudinal axis of the waveguide. A plurality of such geometric optic rays traveling through the waveguide, with all such waves sharing the same length angle and helical angle, would sum to produce traveling waves with helical propagation, with the mean radius of the traveling wave helix being located at a caustic radius Rc  144 . The included angle between wall reflections is shown as Φ c    143 , where
 Φ c /2=2*arc cos (Rw/Rc)=2* arc cos (p/X pq ).  
   The overall effect of summing many such rays  150  is the helical wave propagation shown in  FIG. 16 , where the cylindrical waveguide  140  is shown having a waveguide radius Rc  146 , and a caustic radius Rc  144 , and the wave energy enters at entry locations  160   a  and  160   b,  travels helically along the paths shown, and exits at egress locations  160   a′  and  160   b′.  The waves maintain their caustic radius Rc  144 , a characteristic of the launch angle at entry point  160   a  and velocity of propagation in the medium carrying the wave energy, which is typically air. 
     FIG. 17  shows the construction details for the reflectors of asymmetric combiners of  FIGS. 9 ,  13  and  14 . The symmetric mode reflector of  FIG. 7  was formed using a locus of points which reflect wave energy from a first focus  51   a  to a second focus  56 . In the construction of reflector of  210   a  of  FIG. 17 , feed guide  212   a  has a caustic Rc(feed)  218   a  as was described in  FIGS. 15 and 16 . Waves traveling in the feed waveguide  212   a  have a constant phase front  240 , shown as an involute which starts at point  242  and curls outward to a point  252  on the waveguide wall. Similarly, final waveguide  200  has a caustic  202  with Rc(final)  204 , and waves traveling in the final waveguide have a phase front  250 , shown as an involute starting at point  248 ″ and ending at point  242 ″′. The feed waveguide phase front  240  and final waveguide phase front  250  are specific to the mode of wave traveling in the respective waveguide, and are shown in  FIG. 17  only to clarify construction details of the reflectors  210   a.  In ray tracing construction of the reflectors, the feed guide phase front  240  and final guide phase front  250  are perpendicular to the feed guide ray paths  242 ,  244 ,  246 , and  248 . When the reflector is formed to create equal optical path lengths from the phase front of the wave in the feed guide to the phase front of the wave in the final guide, maximal power summing is achieved. The reflector is formed by a locus of points which satisfy the following criteria for each locus point:
         1) a first line segment starts at a given reflector locus point, passes tangent to the feed waveguide caustic Rc(feed), and terminates at the phase front of the feed waveguide, and a second line segment which starts at the same given reflector locus point, passes tangent to the final waveguide caustic Rc(final), and terminate on the phase front of the final waveguide.   2) the path length of the first line segment added to the second line segment is a constant. This constraint makes the electrical distance from the a point on the feed waveguide phase front to the same phase point on the final waveguide phase front equal for all such phase front points, thereby ensuring constructive addition of the wave in the final waveguide.   3) At each locus point, an intersection point is defined by the intersection of the locus point of the reflector and a line which is tangent to the reflector curve at the locus point, and a perpendicular line which is perpendicular to the tangent line at the locus point, the perpendicular line bisecting the angle formed by the first line segment and the second line segment. This constraint ensures the reflector surface at the given locus point will act to reflect energy from the feed waveguide phase front to the appropriate point on the final waveguide phase front. Using this metric, the construction of the reflector is formed by the locus of points shown on FIG.  17 . Reflector  210   a  is illustrated for simplicity by 4 points which are used as examples to show how these constraints are used to construct the reflector. Phase front  240  and caustic  214   a  Rc(feed)  218   f  of the feed waveguide and phase front  250  and caustic  202  Rc(final)  204  of the final guide are known from the characteristics of the desired input and output wave mode patterns. A first line segment starts at reflector locus point  242 ′, passes tangent to the feed caustic  214   a,  and terminates on the feed phase front point  242 . A second line segment starts at reflector locus point  242 ′, passes tangent to Rc(final)  242 ″, and terminates at final waveguide phase front  242 ″′. Similarly, for given reflector locus points  244 ′,  246 ′,  248 ′, there are respective first segments formed by lines which start at the reflector locus points  244 ′,  246 ′, and  248 ′ respectively, pass tangent to the feed caustic Rc(feed)  214   a,  and terminate on the feed guide phase front  240  on points  244 ,  246 , and  248 . Respective second lines are formed by lines which start at respective locus points  244 ′,  246 ′,  248 ′, pass tangent to the final waveguide caustic Rc(final)  202  on points  244 ′,  246 ′,  248 ′, and terminate on the final waveguide phase front  250  on points  244 ″,  246 ″,  248 ″ respectively. At each given point, the reflector surface  210   a  has a tangent line which includes the given point, and a line perspective to this tangent line which includes the given point on the reflector. The angle formed by the first and second line which includes the given reflector point is bisected by the perpendicular line, as is clear to one skilled in the art of reflectors and ray tracing. Thus, the entire reflector surface  210  is formed by the locus of points which meet the constraints described earlier: for each given reflector locus point, the sum of the first and second line segment lengths is equal, and at the given locus point of the reflector, a line perpendicular to the reflector surface at the given locus point bisects the angle formed by the first and second line at each given point. The locus of points which meet these criteria from the reflector surface.       
   Generalizing to the earlier symmetric mode case, we can further say that the reflectors follow the same constraint, where the feed and final guides for the symmetric case have a feed caustic Rc(feed) and a final caustic Rc(final) equal to 0. This simplification produces the reflectors earlier shown in  FIGS. 7 and 8 .  FIG. 17  shows the projection view looking through the input side of the feed waveguides, through the reflector  210   a,  and finally through the final waveguide. In this view, the additional detail of the location and orientation of the helical ramp of the feed guide and the multicut ramps of the final waveguide are shown. Point  215  is shown as the tip of the helical feed waveguide, showing the “ramp” side and the “drop” side, and points  221  and  223  indicate the relative locations of the tips of two multicuts, also showing the “ramp” and “drop” side, corresponding to the features of the multicut. The points  215 ,  221 , and  223  are shown only to aid in the understanding of the relationship between the angular orientations of the ramps on each of the structures, and may be in different places than shown in FIG.  17 . In practice, the angular positions of these points is determined by maximizing power transfer from the feed waveguides, through the reflectors, and to the final guide. 
     FIG. 18  shows the collapsed section view for all reflectors and feed guides, for the case where p=4. 
     FIG. 19  shows power summing in the final waveguide, for the case where p=4. Wave energy enters each multicut  124   a,    124   b,    124   c,    124   d  from each reflector  120   a,    120   b,    120   c,  and  120   d  as in  FIG. 14 , and these sum respectively into the traveling wave groups shown entering as  168   a,    168   b,    168   c,  and  168   d,  and exiting as  170   a,    170   b,    170   c,  and  170   d.