# Terahertz Imaging of Thin Film Layers with Matched Field Processing

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## Abstract

**:**

## 1. Introduction

#### 1.1. Matched Field Processing (MFP)

#### 1.2. Applying MFP Techniques to THz Tomography

## 2. Methodology

#### 2.1. Conventional THz TOF Processing

#### 2.2. Mathematical Model for Terahertz Measurement Data

#### 2.3. Generating THz Replica Spectra

#### 2.4. Sample Covariance Matrix

#### 2.5. Objective Functions

#### 2.5.1. Bartlett Processor

#### 2.5.2. Minimum Variance (MV) Processor

#### 2.6. Ambiguity Surfaces

#### 2.7. Accuracy Limitations

## 3. Results

#### 3.1. Terahertz Measurement System

#### 3.2. Layered Media Samples

^{®}Removable Double Sided Tape 667, Cat. 238) with a thickness of $2.4$ mils (61 $\mathsf{\mu}$m), per the manufacturer.

#### 3.3. Measurement Data Processing

#### 3.4. Generating Replica Spectra

#### 3.5. THz MFP Results for Sample A

#### 3.6. THz MFP Results for Sample D

#### 3.7. Error Analysis for All THz MFP Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Propagation within Layered Media

## References

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**Figure 1.**Illustration of the THz NDE experiment configuration (not to scale). (

**Left**) Measurement configuration for the THz reference, which is used to approximate the THz source signal in MFP processing. (

**Right**) Measurement configuration for the layered sample under test. A calibration layer (air) with unknown thickness, ${d}_{0}$, accounts for the offset distance between the reference mirror and the surface of the sample. A shim (Scotch

^{®}Removable Double Sided Tape) located a distance of ${d}_{1}$ below the sample surface creates an air gap with thickness, ${d}_{2}$. THz MFP is used to estimate the thicknesses of all three layers (${d}_{0},{d}_{1},$ and ${d}_{2}$), simultaneously.

**Figure 2.**Measured data for Sample A. (

**Top**) Mean of measured THz waveforms. Note that reflected THz pulse 1 (illustrated in Figure 1) arrives at approximately 15 ps, and pulses 2 and 3 overlap with one another at approximately 25 ps; (

**Middle**) Spectrum of mean waveform in the top panel; (

**Bottom**) Covariance matrix computed with Equation (7) using the spectrum of each 300 measured waveform.

**Figure 3.**Measured data for Sample D. (

**Top**) Mean of measured THz waveforms. Note that reflected THz pulses 1, 2 and 3 (illustrated in Figure 1) all overlap on one another; (

**Middle**) Spectrum of mean waveform in the top panel; (

**Bottom**) Covariance matrix computed with Equation (7) using the spectrum of each 300 measured waveform.

**Figure 4.**Matched field ambiguity surfaces provide estimates for the thickness of the calibration layer, ${d}_{0}$, and polycarbonate layer, ${d}_{1}$, as illustrated in Figure 1 with layer thickness for Sample A given in Table 1. (

**Top**) Bartlett processor has a global maximum ${d}_{0}=190$ $\mathsf{\mu}$m and ${d}_{1}=750$ $\mathsf{\mu}$m; (

**Bottom**) MV processor has a global maximum ${d}_{0}=180$ $\mathsf{\mu}$m and ${d}_{1}=750$ $\mathsf{\mu}$m. The results for ${d}_{1}$ are consistent with ground truth measurements with a Vernier caliper. Ground truth data are not available for the calibration layer, but the these results are reasonable, and consistent between both Bartlett and MV processors. See Table 2 for a comparison of measurement errors for layers in all samples.

**Figure 5.**Matched field ambiguity surfaces provide estimates for the thickness of the polycarbonate layer, ${d}_{1}$ and the air gap, ${d}_{2}$, as illustrated in Figure 1 with layer thickness for Sample A given in Table 1. (

**Top**) Bartlett processor has a global maximum ${d}_{1}=750$ $\mathsf{\mu}$m and ${d}_{2}=70$ $\mathsf{\mu}$m; (

**Bottom**) MV processor has a global maximum ${d}_{1}=750$ $\mathsf{\mu}$m and ${d}_{2}=70$ $\mathsf{\mu}$m. These results are consistent with ground truth measurements with a Vernier caliper. See Table 2 for a comparison of measurement errors for layers in all samples.

**Figure 6.**Matched field ambiguity surfaces provide estimates for the thickness of the polycarbonate layer, ${d}_{1}$, and the air gap, ${d}_{2}$, as illustrated in Figure 1 with layer thickness for Sample D given in Table 1. (

**Top**) Bartlett processor has a global maximum ${d}_{1}=250$ $\mathsf{\mu}$m and ${d}_{2}=60$ $\mathsf{\mu}$m; (

**Bottom**) MV processor has a global maximum ${d}_{1}=240$ $\mathsf{\mu}$m and ${d}_{2}=60$ $\mathsf{\mu}$m. These results are consistent with ground truth measurements with a Vernier caliper. See Table 2 for a comparison of measurement errors for layers in all samples.

**Figure 7.**For each of the Vernier caliper measurements listed in Table 2, the corresponding thickness estimates from THz MFP with the Bartlett and MV processors are plotted as blue circles and red squares, respectively. The dashed line represents an ideal case of equality of the Vernier caliper measurement and the THz MFP thickness estimate. Thus, all of the measurement data from the THz MFP approach is in close agreement with the Vernier caliper data.

**Table 1.**Four samples of thin polycarbonate films (layer ${d}_{1}$ in Figure 1) were evaluated in this study. Each of the samples was measured with a digital Vernier caliper. The mean of 10 measurements for each sample film is listed in the table.

Sample | Manuf. Spec. (mils) | Vernier Cal. ($\mathsf{\mu}$m) |
---|---|---|

A | 30 mil | 740 |

B | 20 mil | 520 |

C | 15 mil | 390 |

D | 10 mil | 260 |

**Table 2.**Thickness estimates obtained from THz MFP with the Bartlett and MV objective functions for the experiment configuration shown in Figure 1. All of the differences between the THz MFP results and the Vernier caliper measurements are within the measurement resolution of the Vernier caliper (20 $\mathsf{\mu}$m).

Sample | Layer | Vernier Cal. | THz MFP Bartlett | THz MFP: MV |
---|---|---|---|---|

ID | ID | ($\mathsf{\mu}$m) | ($\mathsf{\mu}$m) | ($\mathsf{\mu}$m) |

A | ${d}_{1}$ | 740 | 750 | 750 |

A | ${d}_{2}$ | 60 | 70 | 70 |

B | ${d}_{1}$ | 520 | 510 | 510 |

B | ${d}_{2}$ | 60 | 70 | 70 |

C | ${d}_{1}$ | 390 | 380 | 370 |

C | ${d}_{2}$ | 60 | 60 | 70 |

D | ${d}_{1}$ | 260 | 250 | 240 |

D | ${d}_{2}$ | 60 | 60 | 60 |

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Schecklman, S.; Zurk, L.M.
Terahertz Imaging of Thin Film Layers with Matched Field Processing. *Sensors* **2018**, *18*, 3547.
https://doi.org/10.3390/s18103547

**AMA Style**

Schecklman S, Zurk LM.
Terahertz Imaging of Thin Film Layers with Matched Field Processing. *Sensors*. 2018; 18(10):3547.
https://doi.org/10.3390/s18103547

**Chicago/Turabian Style**

Schecklman, Scott, and Lisa M. Zurk.
2018. "Terahertz Imaging of Thin Film Layers with Matched Field Processing" *Sensors* 18, no. 10: 3547.
https://doi.org/10.3390/s18103547