Broadband Transitions for Micro-machined Waveguides

Broadband Transitions for Micro-machined Waveguides Edward J. Wollack1, IEEE Member, Felice Maria Vanin2, IEEE Member NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA1 Department of Electrical Engineering at the University of Maryland, College Park, MD 20740, USA2 Abstract — When chemically micromachining silicon wafers by wet etching, the naturally occurring shapes are trapezoidal or hexagonal instead of rectangular. Practical applications require interfacing these common crossections with standard rectangular waveguides; therefore, efficient transitions with broadband characteristics are necessary for enhancing reliability, simplicity, and cost-effectiveness. This paper presents a straightforward design approach and measurements for convenient broadband transitions. The uses of such interfaces improve compatibility with existing precision waveguide measurement capabilities and facilitate the realization of broadband transitions. Index Terms — Waveguides, Waveguide Discontinuities

Waveguide

Transitions,

I. INTRODUCTION The terahertz spectrum lies roughly between 100GHz to 3THz and spans the millimeter to submillimeter wavebands. Terrestrial and astronomical remote sensing are practical applications currently using THz technology. Compared to complicated and costly conventional methods for producing waveguides with small dimensions, silicon micromachining has demonstrated the potential to be a more efficient and costeffective fabrication process to produce waveguides with comparable results. Because micromachined components perform at higher frequencies, allow higher levels of circuit integration, and enhance design flexibility in complex systems, micromachining technology is now broadly used as a means of realizing low loss split-block integrated waveguide [1,2,3,4,5] and antenna feed [6,7,8] structures at millimeter and submillimeter wavelengths. By etching a (110) silicon wafer, rectangular waveguide split-block components can be formed [1, 2]. Subsequent metallization of the walls has resulted in insertion losses of ~0.04dB and ~0.08dB per wavelength for WR10.0 and WR04.3 guides. These magnitudes are surprisingly comparable to losses observed in waveguides produced by conventional methods. One example of silicon micromachining is the chemical wet etching of (100)-oriented silicon that results in the formation of V-grooves that can be used to form a diamond shaped guide; however, the result is a relatively narrow single mode bandwidth. To overcome the problem of limited bandwidth, the introduction of vertical fins has been explored [3, 5], but has resulted in increased insertion losses. Because of this problematic tradeoff, an alternative approach that exhibits broad bandwidth and low losses is desirable. A solution to the tradeoff problem is a waveguide geometry

1

Fig.1: Hardware models (a) & 3D-CAD views (b) of an asymmetrical hexagon to rectangular waveguide transition and an isosceles trapezoidal to rectangular waveguide transition.

discussed in [4], where an asymmetric hexagonal waveguide crossection is described. This type of crossection leads to the design discussed in this work. This paper presents simple and efficient transition designs between rectangular waveguides and asymmetric hexagonal or isosceles trapezoidal micromachined waveguides (see Fig. 1). A straightforward design procedure is given, the result of which is a single step transition that exhibits good return loss (RL) levels over a broadband frequency range. If lower RL levels are desired, multiple step transitions are also possible by conceptually following the same procedures described in this work with minor changes. However, only the simplest transition (single step or junction) will be described in this paper. The practical importance of this work is evident in settings where an interface between components formed by conventional and micromachined techniques may occur. II. THEORETICAL APPROACH The intent of this paper is to present an efficient design procedure for the waveguide crossections shown in Fig.1 to rectangular waveguides. The use of conventional mechanical fabrication techniques, laser milling, or deep reactive ion etching (DRIE) can produce θ ≈ 0o wall angle for rectangular and θ ≈ 45o for diagonal guides, resulting in a relatively small correction to the waveguide cutoff frequency [9]. In micromachining silicon by wet chemical etching the naturally occurring wall angles θ ≈ 35.3o or 54.7o are preferred for some practical applications due to ease of fabrication and compatibility with other processing steps. Consider the truncated guide crossections depicted in Fig. 1. The following approach (directed to guides with ~2:1 aspect ratio in cross section to preserve sufficient separation between the dominant and higher order modes in the structure) is employed to derive

θ

a

a′

δ

[ao ]

[ao ]

[ao ]

[ao ]

35.3

0.500

1.177

0.823

0.000

45.0

0.500

1.250

0.750

0.000

54.7

0.500

1.354

0.646

0.000

35.3

0.500

1.299

0.591

0.059

45.0

0.500

1.380

0.380

0.084

54.7

0.500

1.444

0.032

0.114

b = bo

[deg]

(I) Asymmetrical Hexagon

(II) Isosceles Trapezoidal

Tab. 1: Design coefficients normalized by ao for the Asymmetrical Hexagonal (I) & Isosceles Trapezoidal (II) transitions.

Design

parameters are given for the three cases of potential interest.

see the ingenious physically motivated derivation by Levy [12], Eq. (23)). In practice, with the constant guide cutoff condition specified, we analyze the transition’s response with a fullwave EM simulator and minimize the junction reactance by iteration. Fig. 2 shows the normalized λc vs. the

an analytical design formula. For the structures under consideration, the RL is minimized over a finite bandwidth when the dominant mode eigenvalues between the rectangular and truncated guide sections are matched and modal overlap between the guides is maximized. The magnitude of the fundamental truncated waveguide eigenvalue is bound through the use of a stationary approximation. This technique is particularly useful in addressing problems where the cross section or boundary conditions do not lend themselves to the usual method of separation of variables, yet are closely related by symmetry to a variation on a similar problem. In this spirit, an estimation of the eigenvalue for the fundamental mode of propagation for the truncated rectangular guide is:

Normalized Cutoff Wavelength λc / 2a

∫ ∇ψ dS ≤ ∫ ψ dS

normalized a ′ and the good agreement between Eq. (2) and HFSS simulations for the 45o case. In addition, for the 35.3o and 54.7o cases HFSS results are depicted in the graph.

2

β

2 c

(1)

2

where the propagation constant β c is related to the cutoff guide wavelength λc = 2π β c , and where ψ is the trial function, satisfying the Helmholtz equation on the guide crossection defined by region S as in [10,11]. For a truncation angle of θ =45o and small a' a (