The previous studies of backward wave oscillators (BWOs) have been restricted in the operation at the fundamental axisymmetric TM mode. However, non-axisymmetric operations of BWO have been observed in the experiments and should be studied more definitely. In this work, electromagnetic field properties of axisymmetric and non-axisymmetric modes are studied for X-band slow wave structure (SWS), which is used in typical BWO experiments. The wall radius of the structure varies along the axial direction z as R₀+hcos((2π/z₀)z), where average radius R₀=14.45[mm], corrugation amplitude h=4.45[mm], pitch length z₀=16.7[mm]. The electromagnetic fields and their interactions with the beam have commonly been studied based on a Rayleigh-Bessel expansion of space harmonics, so called the Rayleigh-Bessel hypothesis. The theoretical limit of the Rayleigh-Bessel hypothesis has been pointed out and is a modulation depth 2h/z=0.142 for the sinusoidal wall [1]. For the X-band SWS, the modulation depth is 0.533 and is about 4 times the modulation limit. On the other hand, the numerical results based on the Rayleigh-Bessel hypothesis have been in good agreement with experimental results [2, 3].
In this work, we investigate electromagnetic field properties by using two numerical formulations, one is based on the Rayleigh-Bessel hypothesis and the other based on the higher-order implicit difference method, which is free from the Rayleigh-Bessel hypothesis. The field properties are examined for non-axisymmetric as well as axisymmetric cases by comparing two numerical analyses. Although singularities of field appear inside the corrugation due to the Rayleigh-Bessel hypothesis for some modes, the dispersion curves and the fields outside the corrugation obtained from two numerical formalisms are in very good agreement.

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