H 0 i-Eigenwave Characteristics of a Periodic Iris-Loaded Circular Waveguide

H0i-eigenwave characteristics of a periodic iris-loaded circular waveguide (PICW) are examined, as concerns the eigenmode behavior vs arbitrary variations of the geometric parameters and the Bragg bandwidths vs the parameter of filling l d /   extremums.


Introduction
The periodic iris-loaded circular waveguide, Figure 1, has long since found its several important applications, e.g. in the particle acceleration field [1], and thus stimulated its electromagnetics studies.Despite this even its eigenwave characteristics available are not to be regarded as generally satisfactory [1,2]; foremost theoretically and a good deal so [2], whereas exactly knowing the ropes wouldn't do any harm in all respects.
Certain conceptual points as to the eigenwave propagation in PICW are given in [2] to get those waves theory building started.As the next step and immediate continuation, this paper is concerned with characterization of one of the PICW particular wave types -its H 0i -eigenwaves.
It is not that only the PICW asymmetric and symmetric E 0i -waves, in view of their acknowledged complexity [1,3], cannot be properly perceived except by rigorous computations.Any simplified modeling, e.g. as that of l  0, d  0 in [3], and others like it, are rather unsatisfactory, concerning even the simplest guided wave type of H 0i -waves.And in fact, there is no other way at all for dealing adequately with the PICW eigenwave problem except via rigorous computations; which is certainly one of the major difficulties in their investigation.This way, the H 0i -waves are generally looked at on the dispersion side of their electromagnetics; and all of the necessary terms, notions and ways employed are introduced and discussed in detail in [2].

Arbitrary Geometric Parameters
As some work model of PICW to be employed throughout this investigation [2], and in this section in particular, radius b is held constant b  3, the long period l = 3 and the short one l = 0.75 are examined in detail, as one of the wide and one of the narrow cells are considered, and radius a is optimally varied.
The multi-mode Brillouin diagrams is the most suitable instrument for the purpose.
The PICW dispersion curves are drawn below with solid lines, those of the regular waveguide b = 3 with dotted lines, and those of the regular waveguide r = a with dashed ones.

Period l = 3
At the narrow iris for d = 2.8, the effect of radius a variations is represented in Figure 2 for the junior 12 modes and a  {2.8, 2.4, 2, 1.A certain regular-waveguide r = a modeling may be in some validity in this case, whereupon the eigenfrequency equals the regular model's one for the upper boundaries   The piece-wise mode composition due to a lot of the inner Bragg wave-points and the wave propagation up to rather small radius a values, characterize the waves.Two particular cases as to the mode forming are shown in detail in the fragments f-4 and f-5, Figure 4.

Period
The regular-waveguide r = b modeling scheme is not relevant in this case, even to the extent it has been in {l = 3, d = 2.8} event; much less is the r = a scheme.
At the thick iris d = 0.2, the effect of radius a variations is demonstrated in Figure 6 for the junior 12 modes, a  {2.8, 2.4, 2, 1.2, 0.4}; with two detailed fragments on the particularities of the mode forming, f-6 and f-7, Figure 4.
As radius a goes down, the i.p.d. is still mainly in effect up to a = 1.2; which is evidenced by a fairly straight geometry of the dispersion curves.
The regular-waveguide r = a modeling scheme as that in the previous thick-iris event, Figure 3, principally holds true in this case also , and even more accurately.
The fragments f-1 to f-7, Figure 4, exhibit some particular features of the eigenmode formation and transformation in the waveguide.As the values of d and a parameters vary, the standard i.p.d.scheme of the periodicity mode origin in pairs at } 5 .0 , 0 {   , and their further forming at 0 <  < 0.5, somewhat changes to include at least three interacting eigenmodes.As it is in f-1, H 05 being the regular mode (  = 0); in f-4, H 06 the regular mode, in f-6, H 05 the regular mode (0 <  < 0.5); in f-7, H 07 the regular mode (  = 0.5).In f-2, f-3, (  = 0.5), mentioned above, all of the modes involved are the periodicity ones which, at least after the dispersion way of analysis, quite conform to the standard i.p.d.scheme [2].

The Bragg Bandwidths Extremums
Another view on the H 0i -eigenwave behavior is via their Bragg bandwidths ) (  i  extremum characteristics vs the parameter of filling 0 <  = d/l < 1 [4].In essence, this is the d-parameter variation in the waveguide in effect, looked at under a quite promising aspect as to the PICW characterization.For one thing, such graphic representation of those bandwidths behavior as that, e.g., in Figure 7, enables to look simultaneously at both stop and pass bandwidths characteristics.And second, the other PICW eigenwave types do display a good deal of analogical behavior, with certain peculiarities of their own [4].
In this section, the period values considered are l = 5, 3, 1.8, 1, 0.75.According to the classifications in [2], l = 5, 3, 1.8 are the long periods, l = 1, 0.75 are the short ones; and thus, some borderline set of the period values is examined below.
The general rule for the periodicity modes originated by a given regular one in PICW (after the i.p.d.) is that ) (  i  ,  = 0, 0.5, have i maxima and i-1 minima over the interval 0 <  < 1; while , l = 0.75, a = 2.8, are given in Figures 11 (b) and (c).The presence of the inner Bragg wave-points for all of the eigenmodes on the short [2] periods (wherein l = 1, l = 0.75 are such ones), with their asymmetrical partial-wave interactions, does distort the regularity of the max/min pattern of above; which can be seen in ) (

Conclusions
Under the fundamental primary-causal influence of the  period value, in particular, in setting the number of eigenmodes, with all the consequences of the i.p.d.network thus produced [2], and further variations of d and a parameters, the PICW H 0i -eigenwave characteristics can be seen are quite complex; even without any of their power-flow treatment, illustrated in [2].These waves are not to be satisfactorily interpreted by some regular-waveguide modeling schemes, though the latter may be in some validity to this case.
A monotonous response of the H 0i -eigenfrequencies to both d and a variations, 0 , is a major characteristic feature of those waves.variations, to be in their way some 3rd full-right member of the relationship of equivalence in the matter, see, e.g., [2].


i.p.d. of the regular r = a waveguide via the narrow cell)a decreases from b downwards (the regular r = b waveguide modeling, with the narrow-iris l-d effect in the waveguide).As a result, each H 0i , is stable (approximately constant) vs a at its upper iegenfrequency u i 0  , i.e. monotonously and rather slightly grows as a decreases.Since the PICW eigenwaves originate principally due to interactions in the Bragg wave-points (e.g., after the partial-wave model[2]), the Bragg bandwidths ) ( i  extremum law of i/i-1 maxima/minima at   {0.5, 0},presented here in brief, can be treated as the general pe-riodicity law of the Bragg bandwidths variation vs .The limits and specificity of its holding true as radius a varies, are different for different wave types[4].It needs a special power flows investigation in order to further physically interpret this law in proper detail and understanding.And finally, the upper-and-lower-boundary representations of the pass and stop bandwidths,

radius a variations is represented in Figure 3, the
H 03 and H 011 , H 012 by H 01 ( r H 01 ), H 05 , H 06 and H 09 , H 010 by H 04 ( r H 02 ).The modes H 011 , H 012 are the most complex ones due to the effect of junior 12 modes, a{2.8,2.4,2,1.2,0.8,0.4}.Here, the regular waveguide r = a i.p.d.effect is valid up to a = 2 for all of the modes, except in a few of the Bragg bands.At a = 2.8,  = 0, the modes H 01 , H 05 are the regular ones (by , the mode H 05 being only a slightly composed one (the fragment f-1, Figure4); H 02 , H 03 ; H 04 , H 06 ; H 011 , H 012 and H 07 , H 08 ; H 09 , H 010 are the periodicity modes by 02 respectively.The fragments f-1,2,3, Figure4, demonstrate, in particular, a significant localization of the periodicity partial-wave effect closely around the Bragg wave-points; as well as some other exact details of the mode forming.
2, 0.4}.The initial periodicity dispersion (i.p.d.) is quite in effect at a = 2.8, and H 01 , H 04 are the regular PICW modes originated in accordance with the regular waveguide r = r H 01 and r H r H 01 , and the two Bragg bands go one by one.
r H