## Real-time terahertz material characterization by numerical three-dimensional optimization |

Optics Express, Vol. 19, Issue 11, pp. 10647-10655 (2011)

http://dx.doi.org/10.1364/OE.19.010647

Acrobat PDF (1318 KB)

### Abstract

Terahertz time domain spectroscopy allows for characterization of dielectrics even in cases where the samples thickness is unknown. However, a parameter extraction over a broad frequency range with simultaneous thickness determination is time consuming using conventional algorithms due to the large number of optimization steps. In this paper we present a novel method to extract the data. By employing a three dimensional optimization algorithm the calculation effort is significantly reduced while preserving the same accuracy level as conventional approaches. The presented method is even fast enough to be used in imaging applications.

© 2011 OSA

## 1. Introduction

1. S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. **70**(5), 559–561 (1997). [CrossRef]

4. M. Scheller, J. M. Yarborough, J. V. Moloney, M. Fallahi, M. Koch, and S. W. Koch, “Room temperature continuous wave milliwatt terahertz source,” Opt. Express **18**(26), 27112–27117 (2010). [CrossRef]

5. X. Wang, L. Hou, and Y. Zhang, “Continuous-wave terahertz interferometry with multiwavelength phase unwrapping,” Appl. Opt. **49**(27), 5095–5102 (2010). [CrossRef] [PubMed]

6. M. Scheller, K. Baaske, and M. Koch, “Multifrequency continuous wave terahertz spectroscopy for absolute thickness determination,” Appl. Phys. Lett. **96**(15), 151112 (2010). [CrossRef]

7. D. H. Auston and K. P. Cheung, “Coherent time-domain far-infrared spectroscopy,” J. Opt. Soc. Am. B **2**(4), 606–612 (1985). [CrossRef]

10. P. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photonics Rev. **5**(1), 124–166 (2011). [CrossRef]

10. P. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photonics Rev. **5**(1), 124–166 (2011). [CrossRef]

11. L. Duvillaret, F. Garet, and J. L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. **38**(2), 409–415 (1999). [CrossRef]

13. M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100µm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. **282**(7), 1304–1306 (2009). [CrossRef]

11. L. Duvillaret, F. Garet, and J. L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. **38**(2), 409–415 (1999). [CrossRef]

13. M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100µm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. **282**(7), 1304–1306 (2009). [CrossRef]

14. I. Duling and D. Zimdars, “Terahertz imaging: revealing hidden defects,” Nat. Photonics **3**(11), 630–632 (2009). [CrossRef]

## 2. Algorithm

11. L. Duvillaret, F. Garet, and J. L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. **38**(2), 409–415 (1999). [CrossRef]

13. M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100µm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. **282**(7), 1304–1306 (2009). [CrossRef]

**282**(7), 1304–1306 (2009). [CrossRef]

*n*and imaginary part

*κ*of the refractive index together with the thickness

*L*, constitute the parameter set to calculate the theoretically complex transfer function that describes the sample response:

*M*is the maximal FP echo pulse that appears in the time window of the measurement. For the optimization we use three scalar parameters

*ξ*,

*ψ*and

*ζ*:

16. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the nelder–mead simplex method in low dimensions,” SIAM J. Optim. **9**(1), 112–147 (1998). [CrossRef]

*ΔH*as:

*n*,

*κ*and

*L*, significantly influence the error function. Thus, the optimization is expected to derive the best values for all three fitting coefficients

*ξ*,

*ψ*, and

*ζ*.

*ξ*,

_{opt}*ψ*, and

_{opt}*ζ*, the thickness

_{opt}*L*is known and the resulting material parameters can be calculated:

_{opt}*n*and

_{opt}*κ*originate from the filtered initial values. Thus, absorption features of the material under investigation may be corrupted due to the filtering. To obtain precise material parameters, we apply in the final step a single successive optimization similar to [13

_{opt}**282**(7), 1304–1306 (2009). [CrossRef]

*L*. Here,

_{opt}*n*(

*f*) and

_{j}*k*(

*f*) are optimized at every frequency step

_{j}*f*using the error function:

_{j}## 3. Results

10. P. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photonics Rev. **5**(1), 124–166 (2011). [CrossRef]

**282**(7), 1304–1306 (2009). [CrossRef]

**282**(7), 1304–1306 (2009). [CrossRef]

*ξ*,

*ψ*,

*ζ*) constant (Figs. 3(a) –3(c)). The figure shows that a distinctive minimum occurs for values close to the real physical ones. Particularly, the choice of

*ξ*and

*ζ*, which denote the thickness and the refractive index of the sample, respectively, influence the error function noticeably. Due to the definition of

*ξ*and

*ζ*(cf. Eq. (5)) circularly shaped sets of local minima result. Yet, the remaining error value of the global minimum that corresponds to the real physical thickness is significantly lower than the errors for the other minima. The parameter

*ψ*, as measure for the imaginary part of the refractive index, only weakly influence the error function in this case due to the relatively high transparency of the sample. The global minimum is given for a defined value of

*ψ*though.

*n*and

_{0}*κ*result from the choice of

_{0}*L*, we vary this parameter. We iteratively perform the computation for initial values of the thickness between 30 µm and 600 µm. The resulting thickness of the algorithm is shown in Fig. 6 together with the remaining error function. As one can see in the figure the algorithm converges to the correct value within the range between 30 and 451 µm. For other initial values, it converges to another local minimum. Yet, the remaining value of the error function allows us to distinguish between the global minimum and other minima. Using more sophisticated optimization algorithms that are more robust against local minima distortion would therefore extend the convergence range for the initial values.

_{0}17. M. Scheller and M. Koch, ““Fast and accurate thickness determination of unknown materials using terahertz time domain spectroscopy,” J. Infrared Milli. Terahz. Waves **30**(7), 762–769 (2009). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. |

2. | S. Verghese, K. A. McIntosh, S. Calawa, W. F. Dinatale, E. K. Duerr, and K. A. Molvar, “Generation and detection of coherent terahertz waves using two photomixers,” Appl. Phys. Lett. |

3. | K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. |

4. | M. Scheller, J. M. Yarborough, J. V. Moloney, M. Fallahi, M. Koch, and S. W. Koch, “Room temperature continuous wave milliwatt terahertz source,” Opt. Express |

5. | X. Wang, L. Hou, and Y. Zhang, “Continuous-wave terahertz interferometry with multiwavelength phase unwrapping,” Appl. Opt. |

6. | M. Scheller, K. Baaske, and M. Koch, “Multifrequency continuous wave terahertz spectroscopy for absolute thickness determination,” Appl. Phys. Lett. |

7. | D. H. Auston and K. P. Cheung, “Coherent time-domain far-infrared spectroscopy,” J. Opt. Soc. Am. B |

8. | B. Sartorius, H. Roehle, H. Künzel, J. Böttcher, M. Schlak, D. Stanze, H. Venghaus, and M. Schell, “All-fiber terahertz time-domain spectrometer operating at 1.5 μm telecom wavelengths,” Opt. Express |

9. | M. Walther, B. M. Fischer, A. Ortner, A. Bitzer, A. Thoman, and H. Helm, “Chemical sensing and imaging with pulsed terahertz radiation,” Anal. Bioanal. Chem. |

10. | P. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photonics Rev. |

11. | L. Duvillaret, F. Garet, and J. L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. |

12. | T. D. Dorney, R. G. Baraniuk, and D. M. Mittleman, “Material parameter estimation with terahertz time-domain spectroscopy,” J. Opt. Soc. Am. A |

13. | M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100µm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. |

14. | I. Duling and D. Zimdars, “Terahertz imaging: revealing hidden defects,” Nat. Photonics |

15. | P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. |

16. | J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the nelder–mead simplex method in low dimensions,” SIAM J. Optim. |

17. | M. Scheller and M. Koch, ““Fast and accurate thickness determination of unknown materials using terahertz time domain spectroscopy,” J. Infrared Milli. Terahz. Waves |

**OCIS Codes**

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(300.6495) Spectroscopy : Spectroscopy, teraherz

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: March 8, 2011

Revised Manuscript: May 5, 2011

Manuscript Accepted: May 6, 2011

Published: May 16, 2011

**Citation**

Maik Scheller, "Real-time terahertz material characterization by numerical three-dimensional optimization," Opt. Express **19**, 10647-10655 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10647

Sort: Year | Journal | Reset

### References

- S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559–561 (1997). [CrossRef]
- S. Verghese, K. A. McIntosh, S. Calawa, W. F. Dinatale, E. K. Duerr, and K. A. Molvar, “Generation and detection of coherent terahertz waves using two photomixers,” Appl. Phys. Lett. 73(26), 3824–3826 (1998). [CrossRef]
- K. J. Siebert, H. Quast, R. Leonhardt, T. Löffler, M. Thomson, T. Bauer, H. G. Roskos, and S. Czasch, “Continuous-wave all-optoelectronic terahertz imaging,” Appl. Phys. Lett. 80(16), 3003–3005 (2002). [CrossRef]
- M. Scheller, J. M. Yarborough, J. V. Moloney, M. Fallahi, M. Koch, and S. W. Koch, “Room temperature continuous wave milliwatt terahertz source,” Opt. Express 18(26), 27112–27117 (2010). [CrossRef]
- X. Wang, L. Hou, and Y. Zhang, “Continuous-wave terahertz interferometry with multiwavelength phase unwrapping,” Appl. Opt. 49(27), 5095–5102 (2010). [CrossRef] [PubMed]
- M. Scheller, K. Baaske, and M. Koch, “Multifrequency continuous wave terahertz spectroscopy for absolute thickness determination,” Appl. Phys. Lett. 96(15), 151112 (2010). [CrossRef]
- D. H. Auston and K. P. Cheung, “Coherent time-domain far-infrared spectroscopy,” J. Opt. Soc. Am. B 2(4), 606–612 (1985). [CrossRef]
- B. Sartorius, H. Roehle, H. Künzel, J. Böttcher, M. Schlak, D. Stanze, H. Venghaus, and M. Schell, “All-fiber terahertz time-domain spectrometer operating at 1.5 μm telecom wavelengths,” Opt. Express 16(13), 9565–9570 (2008). [CrossRef] [PubMed]
- M. Walther, B. M. Fischer, A. Ortner, A. Bitzer, A. Thoman, and H. Helm, “Chemical sensing and imaging with pulsed terahertz radiation,” Anal. Bioanal. Chem. 397(3), 1009–1017 (2010). [CrossRef] [PubMed]
- P. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging—modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011). [CrossRef]
- L. Duvillaret, F. Garet, and J. L. Coutaz, “Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy,” Appl. Opt. 38(2), 409–415 (1999). [CrossRef]
- T. D. Dorney, R. G. Baraniuk, and D. M. Mittleman, “Material parameter estimation with terahertz time-domain spectroscopy,” J. Opt. Soc. Am. A 18(7), 1562–1571 (2001). [CrossRef]
- M. Scheller, C. Jansen, and M. Koch, “Analyzing sub-100µm samples with transmission terahertz time domain spectroscopy,” Opt. Commun. 282(7), 1304–1306 (2009). [CrossRef]
- I. Duling and D. Zimdars, “Terahertz imaging: revealing hidden defects,” Nat. Photonics 3(11), 630–632 (2009). [CrossRef]
- P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30(1), 29–31 (2005). [CrossRef] [PubMed]
- J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the nelder–mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998). [CrossRef]
- M. Scheller and M. Koch, ““Fast and accurate thickness determination of unknown materials using terahertz time domain spectroscopy,” J. Infrared Milli. Terahz. Waves 30(7), 762–769 (2009). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.