Abstract
 Top of page
 Abstract
 1. Introduction
 2. Simulation Approach
 3. Measurement Setup and Procedure
 4. Comparisons in a Bent Stoneware Tube
 5. Comparisons in a Model Tunnel Built of Straight and Curved Sections
 6. Conclusions
 References
 Supporting Information
[1] This work is concerned with the comparison of measurements and predictions of highfrequency electromagnetic wave propagation in curved confined spaces. For this purpose, measurements have been performed at 120 GHz (Dband) in curved prefabricated stoneware tubes. A raytracing tool based on geometrical optics and ray density normalization is used for the simulations. The comparisons reveal that an appropriate raytracing approach is able to precisely predict wave propagation in curved geometry. It is shown that the actual shape of the boundaries has a major impact on the propagation characteristics compared to, for example, the materials parameters. Furthermore, the effect of the focusing of energy was observed in both measurements and simulations. In this case, the ray density normalization allows us to overcome one of the major disadvantages of conventional geometrical optics: its failure at caustics.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Simulation Approach
 3. Measurement Setup and Procedure
 4. Comparisons in a Bent Stoneware Tube
 5. Comparisons in a Model Tunnel Built of Straight and Curved Sections
 6. Conclusions
 References
 Supporting Information
[2] Rayoptical electromagnetic wave propagation modeling has become a powerful and widely used tool for various scenarios and applications including radio communications, optics, remote sensing, etc. It is applicable whenever the wavelength is small compared to significant dimensions of the objects with which the field interacts [Deschamps, 1972; Ling et al., 1989]. Based on the principles of geometrical optics (GO) [Balanis, 1989] and its extensions by the (uniform) geometrical theory of diffraction (GTD/UTD) [Keller, 1962; McNamara et al., 1990], the propagation phenomena freespace propagation, reflection, refraction, and diffraction can be treated with sufficient accuracy. In fact, the determination of the distribution of electromagnetic fields in confined spaces like buildings, tunnels, or cavities, which are large compared to the wavelength, is almost solely feasible by ray tracing [Glassner, 1989] because of the computational constraints of fullwave solutions [Yee, 1966; Taflove, 1995; Harrington, 1968; Mittra, 1975]. However, whenever the boundaries of the confined spaces are curved, rayoptical wave propagation modeling becomes quite difficult or even impossible.
[3] Conventional raytracing techniques, based on ray launching [Zhang et al., 1998], imaging [Mariage, 1992; Rembold, 1993; Mariage et al., 1994; Klemenschits, 1993], or a combination of both [Chen and Jeng, 1995, 1996], have in common that they can only treat reflections at plane boundaries in multipath environments. The reason for this is the socalled multipleray problem [Didascalou et al., 2000], where adjacent rays are too closely spaced to be considered as independent [Deschamps, 1972]. The multipleray problem can be overcome as indicated by Honcharenko et al. [1992], Seidl and Rappaport [1994], Cichon et al. [1995], and Suzuki and Mohan [1997], at least considering exclusively planar geometries. Besides the multipleray problem, the application of geometrical optics (GO) for the calculation of reflection at curved boundaries leads to unphysical results near caustics, i.e., regions where focusing of energy occurs.
[4] In order to handle realistic curved geometries the concept of ray density normalization (RDN) was introduced recently [Didascalou, 2000; Didascalou et al., 2000]. The approach has been validated theoretically and by initial measurements in scaled model tunnels. In this paper, the results of extensive measurements at 120 GHz in several tubes built of stoneware are presented and compared to simulations. The performance of the modeling approach in real threedimensional curvature is therefore of primary interest. The choice of prefabricated sewer tubes is primarily due to availability and to ensure reproducibility. Furthermore, the possibility of investigating different arrangements and scenarios together with the ability to perform analyses over the entire cross section of the tubes are additional advantages of such a compact test scenario.
[5] The paper is organized as follows. First, the simulation approach is briefly introduced in section 2. The measurement setups and procedures are described in section 3. This is followed by the comparison of measurements and simulations in various configurations. In section 4 the modeling approach is validated in a curved stoneware tube. Finally, an entire model tunnel, including a straight and a curved stoneware section, with a concrete road lane is examined in section 5.
2. Simulation Approach
 Top of page
 Abstract
 1. Introduction
 2. Simulation Approach
 3. Measurement Setup and Procedure
 4. Comparisons in a Bent Stoneware Tube
 5. Comparisons in a Model Tunnel Built of Straight and Curved Sections
 6. Conclusions
 References
 Supporting Information
[6] A rayoptical modeling approach, based on stochastic ray launching with ray density normalization, is used for all simulations in this paper. The method is intensively treated by Didascalou [2000] and Didascalou et al. [2000]. It allows simulating wave propagation of highfrequency electromagnetic waves in arbitrarily shaped geometries, for example, tunnels, considering freespace propagation and reflection phenomena. The actual curved geometry of the boundary walls, the building materials, and the positions and directional patterns of the transmitting and receiving antennas are taken into account by the simulation approach. Furthermore, the RDN allows us to overcome one of the major disadvantages of geometrical optics: its failure at caustics. In a caustic the predicted GO field, and therefore the received power, approaches infinity [Balanis, 1989]. Using the RDN approach, however, the received power is determined by the number of rays that actually reach the receiver. This number is always finite and smaller than the number of originally launched rays. Consequently, the maximum received power cannot approach infinity either. Since the RDN is based on GO, it is also only applicable as long as the dimensions of the interacting boundaries are large compared to the wavelength. This is always ensured in the following comparisons of measurements with simulation results.
3. Measurement Setup and Procedure
 Top of page
 Abstract
 1. Introduction
 2. Simulation Approach
 3. Measurement Setup and Procedure
 4. Comparisons in a Bent Stoneware Tube
 5. Comparisons in a Model Tunnel Built of Straight and Curved Sections
 6. Conclusions
 References
 Supporting Information
[7] The measurement setup is shown in Figure 1, with the model tunnel of Section 5 as device under test (DUT). A standard gain pyramidal Dband (110–170 GHz) horn antenna is used as transmitter at a frequency of f = 120 GHz. The input power of the antenna P_{T} ≈ 10 dBm is generated by a backward wave oscillator (BWO). The receiver is a Dband rectangular waveguide probe, which is displaced computer controlled to generate twodimensional (2D) scans with a resolution of 2 mm × 2 mm. The received power level is measured with a vector network analyzer (VNWA). The measurement equipment was developed at the Institut für Hochleistungsimpuls und Mikrowellentechnik, Forschungszentrum Karlsruhe, Germany [Arnold, 1997; Schindel, 1999]. To avoid noticeable side effects by the edges of the tubes like diffraction, etc., the transmitter as well as the waveguide probe are positioned at least 1 cm inside the tubes. The measured directional patterns of the horn antenna and the waveguide probe [Didascalou, 2000, Appendix D] are considered in the simulations. They match very well the theoretical patterns for pyramidal horns as, for example, calculated by the equivalence principle techniques [Balanis, 1997, chapter 13].
[8] Three different tubes are utilized in the comparison: (1) a bent stoneware tube with angle of curvature of 45°, length of 30 cm, and diameter of 20 cm (DN 200 45°), (2) a bent stoneware tube with angle of curvature of 90°, length of 47 cm, and diameter of 20 cm (DN 200 90°), and (3) a straight stoneware tube with length of 60 cm and diameter of 20 cm (DN 200). The allowable tolerances for this type of tubes are given by Deutsches Institut für Normung e.V. (DIN) [1999]; for example, the curvature may vary up to ±5°. The material parameters of the tubes have been determined at 200 MHz to 40 GHz via measurements of reflection coefficients with a HP 8510 VNWA (assuming a relative permeability of μ_{r} = 1). The resulting averaged relative permittivity of ε_{r} ≈ 8 is used in the simulations at 120 GHz. Various simulations suggested that the influence of the permittivity, even on the order of 50% variation, is of minor importance compared to the influence of geometrical parameters (compare section 4), which coincides with results obtained in the literature [Mariage, 1992; Klemenschits, 1993; Mariage et al., 1994]. The surface roughness (RMS height) of the stained stoneware tubes is σ_{h} ≤ 0.05 mm, which is ≤ λ/50 and thus negligible [Geng and Wiesbeck, 1998].
[9] Despite the tolerances in the geometry of the different tubes and the extremely small wavelengths (λ = 2.5 mm) a coherent analysis is chosen for comparison purposes in the sections 4 and 5. The measured and simulated power levels are normalized to their respective maximum values P_{M}. The agreement of the absolute values is confirmed by an initial freespace measurement.
4. Comparisons in a Bent Stoneware Tube
 Top of page
 Abstract
 1. Introduction
 2. Simulation Approach
 3. Measurement Setup and Procedure
 4. Comparisons in a Bent Stoneware Tube
 5. Comparisons in a Model Tunnel Built of Straight and Curved Sections
 6. Conclusions
 References
 Supporting Information
[11] To validate the RDNbased modeling in real 3D curvature, the bent stoneware tube depicted in Figure 2 is used as device under test in this section. Figure 3a shows a schematic plot of the longitudinal profile of such a tube. A closer inspection reveals that the tube is actually composed of two short straight sections and an intermediate (generally noncircular) bend. For the modeling, the curvature is approximated by a circular arc, resulting in the geometry plotted in Figure 3b.
[12] In order to validate the performance of the modeling, and, at the same time, to examine the influence of the actual shape of the DUT on the propagation properties, the bent tube of Figure 2 has been measured and compared with three different simulation setups: (1) a straight tube with the same length and diameter as the actual probe, (2) a tube composed of one single circular arc, having the same angle of curvature (45°), the same length, and the same diameter as the actual probe (d = 20 cm), and (3) a bent stoneware tube modeled according to Figure 3b, with a first straight section of length l_{1} = 7.5 cm, a second curved section of length l_{2} = 14.85 cm, a radius of curvature r_{c} = 18.91 cm, and a third straight section of length l_{3} = l_{1} = 7.5 cm.
[13] The transmitter is positioned at the entrance of the tube in an eccentric position 5 cm from the center in direction of the bend. Figure 4 depicts the measurements and the simulations at the other end of the tube. Although the actual geometry of the bent stoneware tube is only roughly approximated by the trisectional geometry of Figure 3b, the measurement in Figure 4a and the simulation in Figure 4b suggest a good agreement over a large area. The effects of the relatively strong bend on the propagation characteristics can be identified by comparing the images with the simulation results obtained for the pure (and therefore less distinct) bend in Figure 4c, and for the equivalent straight tube in Figure 4d. The correlations and the standard deviations between the measurement and the calculations are given in Table 1. In addition to the comparison of the entire images, the values are also determined for the portions of the images containing the most distinct parts of the interference patterns, −2 cm ≤ x ≤ 6.5 cm and −4 cm ≤ y ≤ 4 cm (the axes' orientations are depicted in Figure 3a). It is apparent from the figures that the actual shape (i.e., the course) of the geometry is of major importance. The differences between the straight tube, the pure bend, and the trisectional geometry are significant. The comparison between the measurement in the bent tube and the simulation of the straight section even leads to a negative correlation. In addition to the course, the crosssectional shape has also a major impact on the propagation behavior, manifested by the interference pattern in the straight tube in Figure 4d. (Further results in straight tubes including symmetrical arrangements are given by Didascalou et al. [1999] and Didascalou [2000]). Without the tube, simply the antenna pattern would have been measured, which in this case would lead to an eccentric intersection of the main lobe. Thus a precise modeling of the actual geometry including curvature, as given by the RDN approach, is mandatory to predict the results with sufficient accuracy in curved confined spaces. For all simulations, 5 × 10^{7} rays have been launched, and up to 20 reflections have been considered. The simulation times are given in Table 1.
Table 1. Correlations and Standard Deviations Between the Measurements and the Simulations of Figure 4 Together With Computation Times of the Simulations on a Hewlett Packard (HP) CSeries Workstation With 240 MHz Clock Rate  Straight Section (Figure 4d)  Pure Bend (Figure 4c)  Trisectional Bend (Figure 4b) 

Entire Images 
ρ_{M}  −0.4  0.89  0.91 
ρ  −0.43  0.94  0.95 
σ_{M}  11.9 dB  3.3 dB  3.0 dB 
σ  11.7 dB  2.5 dB  2.2 dB 

Parts of the Images (−2 cm ≤ x ≤ 6.5 cm, −4 cm ≤ y ≤ 4 cm) 
ρ  −0.33  0.68  0.74 
ρ_{M}  −0.39  0.8  0.88 
σ_{M}  7.7 dB  3.4 dB  2.5 dB 
σ  7.4 dB  2.3 dB  1.6 dB 

Simulation Times 
t  1 hour 4 min  1 hour 10 min  1 hour 20 min 
[14] The results are also very sensitive to the location of the transmitter, in addition to the geometry of the tube. In Figure 5, simulations and measurements are compared with a different transmitter position at 5 cm above the center of the tube. Apart from that, the same scenario as for the previous figures is assumed. The measured (Figure 5a) and the simulated (Figure 5b, trisectional geometry) power distribution differ significantly from the ones obtained in Figures 4a and 4b. The agreement between the measurements and the simulation is very encouraging (correlations and standard deviations between measurement and simulation in Figure 5: ρ_{M} = 0.89 dB, ρ_{M} = 0.96 dB, σ_{M} = 4.9 dB, and σ_{M} = 3.2 dB). Again, 5 × 10^{7} rays have been launched, resulting in a simulation time of 1 hour 20 min.
5. Comparisons in a Model Tunnel Built of Straight and Curved Sections
 Top of page
 Abstract
 1. Introduction
 2. Simulation Approach
 3. Measurement Setup and Procedure
 4. Comparisons in a Bent Stoneware Tube
 5. Comparisons in a Model Tunnel Built of Straight and Curved Sections
 6. Conclusions
 References
 Supporting Information
[15] After the analysis of a single, curved piece of tube, two stoneware tubes are combined to build a more complex model tunnel. The model tunnel consists of a straight section of length l_{s} = 60 cm and a 90° bend of length l_{b} = 47 cm, resulting in a total length of l_{t} = 107 cm. Because of the tolerances in the geometry of the two tubes, the transition from the straight to the curved tube is not homogeneous, but leaves a gap of approximately 0.5 cm width. Furthermore a concrete road lane is cemented into the tunnel up to a height of 5.5 cm above the lowest point in the tubes. Figure 1 shows the measurement setup with the entire model tunnel.
[16] The frequency of f = 120 GHz in the scaled geometry is comparable to a frequency of 1–3 GHz in real tunnels. However, in contrast to scaled measurements [Jacard and Maldonado, 1984; Yamaguchi et al., 1989; Klemenschits, 1993; Klemenschits et al., 1993], that is, using the values measured at high frequency in a scaled geometry corresponding to a lower frequency in the unscaled geometry, the measurements and calculations in this paper are both performed at f = 120 GHz. For scaled measurements, one has to ensure a correct scaling of the equivalent conductivity of the building materials, otherwise leading to false imaginary parts of the permittivity [Klemenschits, 1993].
[17] The aim of the measurements in the model tunnel is, in analogy to section 4, to validate the RDNbased modeling approach. Moreover, the focusing of energy introduced by a concave curvature is investigated. For this purpose, horizontal scans in the tunnel are carried out. The geometry for this type of analysis is depicted in Figure 6.
[18] The first horizontal scan is performed before the concrete road lane is put into the model tunnel. The transmitter is situated in a centric position, the scan is taken 10 cm above the lowest point of the (still circular) cross section. After the inclusion of the road lane, a second horizontal scan is measured over the same area (i.e., 4.5 cm above the road), with the transmitter 9.5 cm above the road at the tunnel entrance. Figure 7 shows the measured results (Figures 7a and 7b) together with the corresponding predicted results (Figures 7c and 7d).
[19] For the simulations the bent tube is approximated by a trisectional geometry according to Section 4, that is, a first straight section with l_{1} = 5.5 cm, a second curved section with l_{2} = 36.1 cm and r_{c} = 21.8 cm, and a third straight section with l_{3} = l_{1} = 5.5 cm. The relative permittivities are ε_{r} = 8 for the stoneware tubes and ε_{r} = 5 for the concrete road lane in the simulations. The roughness of the surfaces is σ_{h} < 0.05 mm, which is again negligible. A horizontal reception plane is used in the simulation. A significant part of the launched rays is intersecting the horizontal reception plane near grazing incidence. Therefore the number of rays has to be significantly large to ensure a sufficient convergence compared to a crosssectional analysis [Didascalou, 2000]. For the results shown in Figures 7c and 7d, 5 × 10^{8} rays have been launched, and up to 20 reflections were traced. The computation time was about 4 days.
[20] Bearing in mind that the real geometry of the model tunnel cannot be constructed perfectly in the simulation (especially the gap on the order of two wavelengths and the geometry of the bend), the measurements and the simulations in Figure 7 agree surprisingly well. The standard deviations and the correlations between the measurements and the simulations are calculated for the right halves of the images, containing the most distinct parts of the interference patterns. Otherwise, the strong influence of the shadow regions, indicated by the dark areas in the lower left corners of the images, would dominate the determination of the correlation. The values are given in Table 2. In the upper parts of the images the focusing effect of the bend becomes clearly visible by the light “stripes” (focal lines), indicating a high level of received power. Thus, for the first time, the RDNbased rayoptical modeling approach enables a precise prediction of electromagnetic wave propagation in curved confined spaces including focusing of energy, without the problems of traditional GO solutions in the vicinity of caustics [Balanis, 1989]. Figure 8 depicts the same simulations but over the whole bent area of the model tunnel. It is seen that the transitions from the straight to the curved section can be clearly distinguished, and the influence of the road lane is pronounced.
Table 2. Correlations and Standard Deviations Between the Measurements and the Simulations for the Right Halves of the Images in Figure 7  ρ_{M}  ρ  σ_{M}  σ 

Without road lane  0.33  0.53  7.8 dB  5.5 dB 
With road lane  0.5  0.63  6.7 dB  4.9 dB 