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

The periodic iris-loaded circular waveguide,

Certain conceptual points as to the eigenwave propagation in PICW are given in [_{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 [

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 [

As some work model of PICW to be employed throughout this investigation [

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.

At the narrow iris for d = 2.8, the effect of radius a variations is represented in

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 = 3 modes, respectively. All the other eigenmodes are the periodicity ones generated by the former: H_{02}, H_{03} and H_{011}, H_{012} by H_{01} (), H_{05}, H_{06} and H_{09}, H_{010} by H_{04} (). The modes H_{011}, H_{012} are the most complex ones due to the effect of mode involved. Down to a = 2, all the senior modes of those presented are clearly piecewise composed. Ultimately, at a = 0.4, the closedoff H_{05}, H_{06} and H_{011}, H_{012} get in very close vicinities in between.

There are three regular waveguide r = b modes, i = 1,2,3, in the bandwidth. And as radius a decreases, a monotonous growth of all of the eigenfrequences for i = 1,…,12, occurs, except in the regular frequencies: {}|_{k}_{a}_{=0.5} for 3 > a > 1.2, {}|_{k}_{a}_{=0.5} for.

In the waveguide with a fairly thick iris, e.g. d = 0.3, the effect of radius a variations is represented in

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, k = 0, the modes H_{01}, H_{05} are the regular ones (by, respectively), the mode H_{05} being only a slightly composed one (the fragment f-1, _{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 and respectively. The fragments f-1,2,3, _{07}, H_{010} are formed after, the modes H_{08}, H_{09} after, and the corresponding Bragg bands are one inside the other. In f-3, H_{07}, H_{08} are formed after and H_{09}, H_{010} after, and the two Bragg bands go one by one.

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 of the appropriate Bragg bandwidths so that |_{k}_{a}_{ }_{= 0.5}.

At the wide cell d = 0.65, radius a variations are demonstrated in

At a = 2.8, the modes H_{0i}, i = 1,2,3,6,11, are the regular ones in one-to-one correspondence with, i = 1,2,3,4,5, consequently. Of the rest modes, H_{04}, H_{05} (by), H_{07}, H_{08} (by), H_{09}, H_{010} (by) and H_{012} (by) are the periodicity ones. Eventually, at a=0.4, the closed-off H_{04}, H_{05} and H_{07}, H_{08} and H_{011}, H_{012} are very close in between.

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,

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

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,

The fragments f-1 to f-7, _{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 [

Another view on the H_{0i}-eigenwave behavior is via their Bragg bandwidths extremum characteristics vs the parameter of filling 0 < q = d/l < 1 [

In this section, the period values considered are l = 5, 3, 1.8, 1, 0.75. According to the classifications in [

The general rule for the periodicity modes originated by a given regular one in PICW (after the i.p.d.) is that, = 0, 0.5, have i maxima and i-1 minima over the interval 0 < q < 1; while ® 0 as q ® 0

(infinitesimally thin slot) and ® w > 0 as q ® 1 (infinitesimally thin iris).

In

The graphic representation of the PICW pass , and stop bandwidths of

The effects of radius a variation for l = 3, , i = 1,2,3, are shown in Figures 8 (a)-(c), accordingly.

The bandwidths, l = 1.8, a = 2.8, are presented in Figures 9 (b) and (c).

, l = 1, a = 2.9, are presented in Figures 10 (b)-(e), accordingly. The bandwidths for the inner B. w.-p.s are much harder to examine, because their eigenvalue shifts as q varies. Nevertheless certain extremums of the bandwidths are obviously available in this case also.

And finally, , 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 [

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 [_{0i}-eigenwave characteristics can be seen are quite complex; even without any of their power-flow treatment, illustrated in [

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, , is a major characteristic feature of those waves. Wherein, , as (the i.p.d. of the regular r = a waveguide via the narrow cell), , as, monotonously grows as 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, i.e. either at or. In fact 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 [

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, like those in

1. ≡— Bragg wave-number and its ordinate on the Brillouin plane, i.e., the Bragg wave-point (B. w.-p.);

2. — Bragg band, i.e., a (locally) forbidden band; — the i-mode propagation band and all of its possible Bragg bands (the mode being beneath those);

3. periodicity dispersion — the first one of the two factors — periodicity and diffraction — responsible for the waveguide dispersion forming;

4. initial periodicity dispersion (i.p.d.) — the waveguide dispersion at infinitesimal irises;

5. regular mode — the PICW eigenmode in one-to-one correspondence to that of the smooth waveguide;

6. periodicity mode — the PICW eigenmode originating due to the periodicity effect;

7. partial waves — the independent ingredients of a PICW eigenwave.