Calculations on Mode Eigenvalues in a Corrugated Waveguide with Varying Diameter and Corrugation Depth

The present paper addresses numerical calculations on the eigenvalues of hybrid modes in corrugated circular waveguides with varying diameter and corrugation depth. Such calculations are essential for the numerical optimization of advanced mode converters and diameter tapers for future low-loss high-power microwave applications, like broadband high-power radar sensors for space debris observation in low earth orbit (LEO). Corresponding mode converters and diameter tapers may be synthesized based on coupled mode theory. Of particular importance here is the ability to consider varying mode eigenvalues along the perturbed waveguide. The procedure presented here is able to consider arbitrary variations of the corrugation depth as well as the waveguide diameter and therefore is highly flexible. The required computational effort is low. Limitations of the method are discussed.

resolutions can be achieved [7][8][9]. In near future, even W-band transmission powers in the range of 100 kW may be achieved [10]. To realize a W-band radar sensor with such a high transmission power, a suitable high-power amplifier as well as an appropriate transmission line is required. The transmission line connects the output of the highpower amplifier with the antenna feed. Due to the high power, oversized transmission lines are required. A suitable transmission mode is the HE 11 hybrid mode in a corrugated circular waveguide [11][12][13]. Low ohmic loss, small mode conversion, and low-sidelobe radiation patterns can be achieved [11][12][13]. Important components of an overmoded transmission line are miter bends [14,15], polarizers [16,17], rotary joints [18], and waveguide diameter tapers [19,20]. For design verification of such components, low power measurements are required. Here, the fundamental TE 10 mode within a rectangular waveguide, provided by a microwave extension module [21] of a Network Analyzer (NWA), has to be converted into the HE 11 hybrid mode within a highly oversized corrugated circular waveguide. In a simple way, this may be realized by a corrugated horn antenna [22,23], illuminating the aperture of a highly overmoded circular waveguide [24,25]. However, since the component behavior under test may be sensitive in respect to the mode content [14,15], a mode purity as best as possible is required, within the considered frequency range. A concept based on a corrugated horn is limited in respect of the mode purity [24,25]. Therefore, a waveguide mode converter is preferred here, which may attain a significant better mode purity [24]. Such a mode converter may consist of (1) commercial rectangular TE 10 ↦ circular TE 11 transition and (2) customized TE 11 ↦ HE 11 mode converter, based on tapered variation of the corrugation depth [26,27], integrated within a waveguide diameter up-taper.
For the numerical synthesis of such an advanced mode converter with broadband frequency behavior, coupled mode theory [28][29][30] may be applied. However, in calculations on a perturbed corrugated waveguide, varying eigenvalues are of particular importance to consider. The present paper addresses fast numerical calculations on eigenvalues of a corrugated waveguide with varying diameter and corrugation depth. To the best knowledge of the authors, similar calculations along an arbitrary perturbation were not published yet.
The paper is organized as follows: Section 2 introduced the coupled mode formalism, able to calculate mode conversion within multimode waveguides. In Section 3, fundamentals of corrugated waveguides and their eigenvalues are discussed. Section 4 addresses numerical calculations on the eigenvalues of a corrugated waveguide with varying diameter and corrugation depth. Section 5 closes with the conclusion and an outlook for future research activities.

Coupled Mode Theory
The principle concept of the coupled mode theory is to expand an electromagnetic field within a perturbed waveguide as superposition of eigenmodes of a uniform reference waveguide, with the same cross section [28][29][30]. Neglecting ohmic wall losses and backward travelling waves due to reflections (highly oversized waveguide) as well as restricting to adiabatically varying waveguide parameters (characteristic length of variation large compared to the wavelength), the corresponding mode amplitudes satisfy the first-order differential equation [28][29][30]: with I n (z) as complex mode amplitude at the position z, β n as corresponding phase constant, and Κ nm as coupling coefficient [30][31][32] between the nth and mth eigenmode of the uniform reference waveguide. To eliminate fast varying phase terms, the substitution I n (z) = A n (z) · exp[−jβ n · z] is introduced with A n (z) as slowly varying complex mode amplitude. Using the Runge-Kutta integration method [33], such a substitution reduces the required step width and therefore the required computational effort. It follows: with In general, the exponent in Eq.
(2) has to be substituted by [11,34]: with L as the perturbation length. This allows consideration of varying phase constants, for example, caused by a varying waveguide diameter. In matrix representation, Eq. (2) reads: and as the coefficients of the coupling matrix H. Equations (5)- (7) show that knowledge about varying phase constants is of particular importance to control mode conversion within a perturbed waveguide. This results in the particular importance of varying mode eigenvalues since the phase constant depends on the waveguide diameter as well as on the mode eigenvalue [11]: 3 Corrugated Waveguide Figure 1 shows a simplified sketch of a corrugated waveguide with the radius a, the corrugation period c, the slot width b, and the corrugation depth d.
Typically, hybrid modes of a corrugated waveguide are referred according to the balanced hybrid mode condition [22]: The function J m (x) is the Bessel function of the first kind and the order m. The function J 0 m x ð Þ is its first derivation in respect of x. For the nth root and +/−, HE mn / EH mn modes occur [22,35]. Equation (9) is valid for a highly oversized waveguide (ka = 2π · a/λ ↦ ∞) having a longitudinal wall surface impedance Z s = − E z /H φ ≠ 0. For ka ≠ ∞, the surface-impedance model [11,22,31,32], which assumes a uniform surface impedance, is widely used. Corresponding calculations lead to [35]: The function Y m (x) is the Bessel function of the second kind and the order m, and Y 0 m x ð Þ is its first derivation in respect of x.
In principle, Eq. (11) is derived as [35][36][37][38][39] (1) calculation of the field components for r ≤ a as superposition of transversal electric (TE) and transversal magnetic (TM) field components, Fig. 1 Corrugated waveguide with the waveguide radius a, the corrugation period c, the slot width b, and the corrugation depth d (2) calculation of the field components for a < r < (a + d) as a pure TM mode with E z = const., and (3) matching the field components at r = a.
A pure TM mode within the corrugation requires b < λ/2. In this way, a radial TE mode cannot be supported, and no Bragg reflection [40,41] occurs [42,43]. The longitudinal surface impedance Z s at r = a can be represented as [35,44]: For ka ↦ ∞ and Z s ≠ 0, Eq. (11) merges into Eq. (9). Note that the surface-impedance model neglects space harmonics [45,46] and therefore is restricted to c ≈ b (thin corrugation ridges) and b ≪ λ/2 (large number of corrugation slots per wavelength) [22,23,[35][36][37][38][39]. In more general calculations, space harmonics within r ≤ a as well as a < r ≤ (a + d) have to be considered. Corresponding calculations lead to rather elaborate equations [22] which are out of scope of the present paper. Simplified equations result for the restriction to K space harmonics within r ≤ a. Corresponding calculations lead to [22]: In principle, the derivation of Eq. (14) is identical to the mode matched procedure introduced for the surface-impedance model. But now, for r ≤ a, space harmonics are considered with 45,46]. For K = 0 (fundamental space harmonic), c ≈ b, and b ≪ λ/2, Eq. (14) merges into Eq. (11). Figure 2 compares calculated eigenvalues for m = 1 (HE 1n / EH 1n modes) and X ∈ [0, 7], based on Eq. (14) with K = 1 (first spatial harmonic) and the surface-impedance model, given in Eq. (11). The blue data points are calculated using the surface impedance model. The yellow data points are calculated with ka = 10, b = 0.1 · λ/2, and c = 1.1 · b, fulfilling the terms and conditions of the surface-impedance model (c ≈ b, b ≪ λ/2), whereas the brown data points are calculated with ka = 10, b = 0.4 · λ/2, and c = 2 · b, violating the terms and conditions of the surface-impedance model. Figure 2 shows good agreement of the surface-impedance model and Eq. (14), as long as those terms and conditions are fulfilled. Otherwise, significant deviations occur.
The curves in Fig. 2 were calculated using the Newton-Raphson method, addressed in Section 4.1. Calculations with K = 1 and K = 2 revealed that the consideration of an increasing number of space harmonics leads just to small variations compared with the calculations based on the first space harmonic.
In the following, the consideration is restricted to the surface-impedance model. The explicit corrugation geometry is omitted here which leads to a more descriptive representation of the principle procedure. However, this requires that the terms and conditions of the surfaceimpedance model (c ≈ b, b ≪ λ/2) are fulfilled (see Fig. 2). In the case, that they cannot be fulfilled; only Eq. (11) must be replaced by Eq. (14).

Numerical Calculations
To calculate the variation of the eigenvalues along an arbitrary perturbation, Eq. (11) is solved numerically. In Subsection 4.1, the Newton-Raphson method is introduced, which here is used for numerical calculations. Subsection 4.2 addresses corresponding results.

Newton-Raphson Method
The principle concept of the Newton-Raphson method is to linearize the considered function f(x) at a starting point x 0 and use those roots as improved starting point of a second iteration [47,48]. It follows: with The iteration is aborted when the rate of change of the approximated solution falls below a specified value. For numerical calculations, the central difference quotient can be used [49]: To avoid tripping due to a too small h and a not sufficient accuracy through a too large h, h ¼ ffiffiffiffiffiffiffi ffi eps p is used here, with eps as floating point relative accuracy (for double precision: To find all solutions of interest, the Newton-Raphson method is calculated multiple times with random initialization within the range of interest. Figure 3 shows the calculated eigenvalues for m = 1 (HE 1n / EH 1n modes), X ∈ [0, 7], and very large ka = 1000 (b ¼∞, blue curves), as well as for ka = 10 (brown curves) in dependence of the corrugation depth d ∈ [0, λ/2]. For ka ↦ ∞, the eigenvalues are close to constant. For moderate ka, the eigenvalues depend strongly on the corrugation depth. However, for d ≈ λ/4, in each case, broadband frequency behavior is shown, with a relative bandwidth far beyond 10%. For d ↦ 0 or λ/2, Eq. (13) gives Z s = 0, and the HE mn / EH mn modes merge into ordinary TE mn / TM mn modes [51,52]. As shown in Fig. 3, this transition is abrupt at large ka. For a TE 11 ↦ HE 11 mode converter, discussed in Section 1, this is critical in respect of parasitic mode conversion [27,51,52]. To ensure a high mode purity, a customized variation as well on the corrugation depth [26,27,51,52] as the waveguide diameter may be used. However, this requires the ability to calculate the variation of the eigenvalues along an arbitrary perturbation with low computational effort. This is possible with the presented procedure here. As example, Fig. 4 shows numerical calculations for m = 1 (HE 1n / EH 1n modes), X ∈ [0, 7], and ka = 50 (blue color) along a tapered variation of the corrugation depth (brown color), with: Figure 4 shows that due to the tapered variation of the corrugation depth, the strong variation at d ↦ 0 or λ/2 could be smoothened. The eigenvalues change slowly along the perturbation, which leads to low parasitic mode conversion. For a TE 11 ↦ HE 11 mode converter, discussed in Section 1, this is essential [26,27,51,52]. For a numerical synthesis, an even more optimized tapering of the corrugation depth may be applied. However, this is out of scope of the present paper. For comparison, Fig. 5 shows the eigenvalues along a perturbation with ka = 50 and a constant rate of change (linear progression).

Conclusion
The present paper addresses fast numerical calculations on the eigenvalues of hybrid modes in corrugated circular waveguides with arbitrary perturbation of the waveguide diameter and the corrugation depth, in case that the surface-impedance model is valid. Future research activities will address the embedding of the outlined procedure into the numerical synthesis of broadband corrugated waveguide mode converters and diameter tapers. Such passive microwave components will be essential for future broadband high-power microwave applications.
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