# A Microwave Reflectometry Technique for Profiling the Dielectric-Conductivity Properties of the Hagia Sophia Globe

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## Abstract

**:**

## 1. Introduction

_{11}reflection coefficient by use of a metal-backed stratified configuration was proposed, to be followed by a considerable number of studies). Such techniques are also (or even more) popular in optical frequencies (see e.g., Reference [7] and references therein), with applications in the measurement of the thickness or complex permittivity of layered materials. A problem is posed by the fact that, for wall structures, neither of these parameters is a priori known with sufficient accuracy to allow the accurate determination of the other one via some fitting procedure. However, in the present case, the actual goal is the detection of hidden elements rather than the accurate measurement of the corresponding parameters. For the same reason, another limitation of free space methods, namely poor performance for small loss tangents (which is the case for the building materials of interest), may be considered insignificant here.

## 2. Materials and Methods

^{®}simulation software. In the first computation, a canonical planar stratified configuration (infinite planar layers) and linearly polarized plane wave normal incidence is assumed. The second computation assumes a rectangular patch of the layered medium combined with a transmitting/receiving horn antenna of the dimensions actually used, as depicted in Figure 3. A test configuration was adopted, with thickness values of d

_{1}= 0.2 cm (lime), d

_{2}= 0.5 cm (glass), and d

_{3}= 5 cm (red brick) for each layer. To save computation time, a 15 × 15 cm surface patch and a d = 10 cm antenna-surface distance (gap) was assumed for the 3D simulations. The rationale for performing a plane wave calculation, besides more accurate 3D simulations, was based on the advantage of simplicity and speed, providing an independent benchmark to check as well as a possible application with a parameter fitting procedure to obtain quantitative estimates of the thickness or the electric parameters of the wall layers, which is expected to be impractical with the intensely time-consuming 3D simulations.

_{i}(i = 1,2,3) is the wave impedance and k

_{i}is the wavenumber for each layer (see e.g., [12,14]), with ε

_{0}, μ

_{0}the free space permittivity and permeability, respectively, and

_{0}= 120π ≅ 377 Ω

## 3. Results

^{−8}, 10

^{−11}, and 10

^{−7}for the three layers) to check for cases of relatively large reflection. Varying the frequency between 8–12 GHz, results for the reflection coefficient amplitude were first obtained for a reference two-layer configuration (no hidden mosaic), setting d

_{2}= 0 (i.e., removing the intermediate glass layer), as plotted in Figure 4, by 3D and plane wave simulation. Observing the two waveforms, it was confirmed that at nearly the same frequencies (9.2, 10.4, and 11.7 GHz), the reflection coefficient S

_{11}falls below −10 dB.

_{2}= 0.5 cm) are presented in Figure 5. We note that both methods have roughly the same response in terms of input reflection coefficient. In the plane wave approach the S

_{11}response has slipped slightly to the left, while the frequency differences where the reflection coefficient is lower than −10 dB (at 8.8, 9.8, 10.9, 12.1 GHz for the 3D approach and at 8.6, 9.7, 10.6, 11.8 GHz for the plane wave approach) are almost identical (every 1.1 GHz).

- the simulated data (and, of course, the measured ones) exhibit the fluctuating behavior expected for the stratified structure under consideration
- the density of the fluctuations seems to depend mainly on the thickness of the third layer (red brick), which was significantly increased accordingly
- the depth of the fluctuations seems to depend mainly on the difference between the dielectric parameters of the second (glass) and third (red brick) layer; a slight decrease of the glass permittivity and an increase of the brick permittivity beyond the values reported in the literature (which in any event refer to materials and frequency ranges not identical to the ones actually measured in the Hagia Sophia) result in better matching of the experimental results
- matching of the overall form of the fluctuations is improved with a slight increase in the depth of the first (lime) and second (glass) layer (as compared with the initial benchmark values adopted).

_{1,r}= 7.4, ε

_{2,r}= 4.5, ε

_{3,r}= 6.5, σ

_{1}= 2.4 × 10

^{−6}(the mean value between the two extreme literature values), σ

_{2}= 7 × 10

^{−3}, σ

_{3}= 0.034 (the literature value), and d

_{1}= 0.3 cm (lime), d

_{2}= 0.55 cm (glass), d

_{3}= 40 cm (red brick).

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Outside view of the Hagia Sophia; the dome, supported by the four large piers, covers the inner square; (

**b**) Pictures of the measurement site under the dome; (

**c**) Dome detail near the measurement points; (

**d**) The HP 8719D VNA used for measurements.

**Figure 4.**Reflection Coefficient in dB versus frequency by 3D (red continuous line) and plane wave (blue dashed line) simulation results without the glass (mosaic) layer.

**Figure 5.**Reflection Coefficient in dB versus frequency by 3D (red continuous line) and plane wave (blue dashed line) simulation results with the mosaic.

**Figure 6.**Experimental measurements of reflection coefficient in dB vs. frequency without the glass (mosaic) layer.

**Figure 7.**Experimental measurements of reflection coefficient in dB vs frequency with the glass (mosaic) layer.

Material | Relative Permittivity ε_{r} | Conductivity σ (S/m) | Source |
---|---|---|---|

Lime (CaO) | 7.4 | ${\mathsf{\sigma}}_{wet}$ = 4.76 × 10^{−6} | [10] |

${\mathsf{\sigma}}_{\mathrm{dry}}$ = 4.35 × 10^{−8} | |||

Glass | 5–10 | $\mathsf{\sigma}$ = 7 × 10^{−3} | [11,12] |

Red Brick | ≅6 ^{1} | $\mathsf{\sigma}$ ≅ 0.034 ^{2} | [13] |

^{1}Estimated value from Reference [13] corresponding to a frequency of 5.25 GHz; beyond this, no estimate is given, but from measurements up to 7 GHz presented therein, a trend increasing with frequency is implied, so we may expect a larger ε

_{r}value in the X band.

^{2}Based on the estimated value of loss tangent at 5.25 GHz from Reference [13].

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**MDPI and ACS Style**

Vazouras, C.; Kasapoglu, G.B.; Karagianni, E.A.; Uzunoglu, N.K.
A Microwave Reflectometry Technique for Profiling the Dielectric-Conductivity Properties of the Hagia Sophia Globe. *Computation* **2018**, *6*, 12.
https://doi.org/10.3390/computation6010012

**AMA Style**

Vazouras C, Kasapoglu GB, Karagianni EA, Uzunoglu NK.
A Microwave Reflectometry Technique for Profiling the Dielectric-Conductivity Properties of the Hagia Sophia Globe. *Computation*. 2018; 6(1):12.
https://doi.org/10.3390/computation6010012

**Chicago/Turabian Style**

Vazouras, Christos, George B. Kasapoglu, Evangelia A. Karagianni, and Nikolaos K. Uzunoglu.
2018. "A Microwave Reflectometry Technique for Profiling the Dielectric-Conductivity Properties of the Hagia Sophia Globe" *Computation* 6, no. 1: 12.
https://doi.org/10.3390/computation6010012