## Material thickness optimization for transmission-mode terahertz time-domain spectroscopy

Optics Express, Vol. 16, Issue 10, pp. 7382-7396 (2008)

http://dx.doi.org/10.1364/OE.16.007382

Acrobat PDF (678 KB)

### Abstract

The thickness of a sample material for a transmission-mode terahertz time-domain spectroscopy (THz-TDS) measurement is the subject of interest in this paper. A sample that is too thick or too thin can raise the problem of measurement uncertainty. Although greater thickness allows the terahertz radiation—or T-rays—to interact more with bulk material, the SNR rolls off with thickness due to signal attenuation. A sample that is too thin renders itself nearly invisible to T-rays, in such a way that the system can hardly sense the difference between the sample and a free space path. The optimal trade-off is analyzed and revealed in this paper, where our approach is to find the optimal thickness that results in the minimal uncertainty of measured optical constants. The derived model for optimal thickness is supported by the results from experiments performed with polyvinyl chloride (PVC), high-density polyethylene (HDPE), and lactose samples.

© 2008 Optical Society of America

## 1. Introduction

7. R. H. Jacobsen, D. M. Mittleman, and M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. **21**, 2011–2013 (1996). [CrossRef] [PubMed]

8. C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, “Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity,” J. Pharm. Sci. **94**, 837–846 (2005). [CrossRef] [PubMed]

9. S. Gorenflo, U. Tauer, I. Hinkov, A. Lambrecht, R. Buchner, and H. Helm, “Dielectric properties of oil—water complexes using terahertz transmission spectroscopy,” Chem. Phys. Lett. **421**, 494–498 (2006). [CrossRef]

10. M. van Exter, C. Fattinger, and D. Grischkowsky, “Terahertz time-domain spectroscopy of water vapor,” Opt. Lett. **14**, 1128–1130 (1989). [CrossRef]

16. P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. **30**, 29–31 (2005). [CrossRef] [PubMed]

## 2. Uncertainty in optical constants

17. W. Feller, “The fundamental limit theorems in probability,” Bull. Amer. Math. Soc. **51**, 800–832 (1945). [CrossRef]

18. H. F. Trotter, “An elementary proof of the central limit theorem,” Arch. Math. **10**, 226–234 (1959). [CrossRef]

*E*

_{ref}(

*ω*) and

*E*

_{sam}(

*ω*) are the reference and sample signals in the frequency domain,

*l*is the sample thickness,

*n*(

*ω*) and

*κ*(

*ω*) are the refractive index and the extinction coefficient of the sample,

*n*

_{0}is the refractive index of air, and

*τ*and

*τ*′ are the transmission coefficients at the sample interfaces. The refractive index and the extinction coefficient can be deduced from Eq. (1) as

*s*

^{2}

*(*

_{n}*ω*), and in extinction coefficient,

*s*

^{2}

*(*

_{κ}*ω*), can be derived from Eq. (2) using the law of propagation of uncertainty. In brief, from the signal amplitudes in the time domain, the variance is transferred to the variance of the magnitude and phase spectra in the frequency domain via Fourier transform, as shown in [19

19. W. Withayachumnankul, B. M. Fischer, H. Lin, and D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B (2008). (In press). [CrossRef]

*k*) and

*k*) are the variances associated with the reference and sample signals, respectively;

*k*is the sampling index number and Δ is the sampling interval, and thus

*k*Δ is the time; ℜ

^{2}and ℑ

^{2}denote the square of real and imaginary parts, respectively. The summation is carried out over the time duration of the recorded T-ray signal. In the equations, all parameters utilize mean values. The proposed model in Eq. (3) is successfully validated using a Monte Carlo method [19

19. W. Withayachumnankul, B. M. Fischer, H. Lin, and D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B (2008). (In press). [CrossRef]

19. W. Withayachumnankul, B. M. Fischer, H. Lin, and D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B (2008). (In press). [CrossRef]

## 3. Optimisation of the sample thickness

*s*, is simulated and demonstrated via a contour plot in Fig. 1. As we can see, at every frequency, there is an optimum sample thickness that gives the lowest

_{n}*s*. Figure 2 reveals the magnitude of

_{n}*s*and

_{n}*s*as a function of the thickness, estimated at three different example frequencies. The optimum thicknesses for the simulated sample at these frequencies approximately span from 300

_{κ}*µ*m to 1 mm. Moving towards a thicker sample by an order of magnitude sees an increment of the standard deviation by three orders. Moreover, moving towards a thinner sample by an order of magnitude results in increased standard deviation by one order of magnitude. Selecting the sample thickness to correspond to desired minimum in Fig. 2 is therefore advantageous.

*s*

^{2}

_{E}in

*A*(

*ω*) is not a function of

*l*, we are now able to minimise

*s*

^{2}

*(*

_{n}*ω*) with respect to the thickness

*l*. By taking the derivative of Eq. (6) with respect to

*l*, we arrive at

*A*

_{sam}(

*ω*) on the sample thickness is an approximation, and does not affect the optimisation. Substituting Eq. (9) into Eq. (8) and equating to zero gives,

*e*. The optimum thickness turns out to be a distance that is equal to twice the penetration depth. Optimization of the sample thickness by starting from Eq. (3b) also delivers the same outcome. Notice that the optimum thickness,

*l*

_{opt}, relies on neither the index of refraction,

*n*(

*ω*), nor the signal magnitude, |

*E*(

*ω*)|. This is because the transmittance at the sample interfaces is not a function of thickness. In addition, the sensitivity of the detector

23. S. P. Mickan, R. Shvartsman, J. Munch, X. C. Zhang, and D. Abbott, “Low noise laser-based T-ray spectroscopy of liquids using double-modulated differential time-domain spectroscopy,” J Opt. B-Quantum S. O. **6**, S786–S795 (2004). [CrossRef]

*κ*=0.013 or

*α*=5.45 cm

^{-1}, is reported to be 4 mm, and the optimum thickness for water,

*κ*=0.478 or

*α*=200 cm

^{-1}, is 100

*µ*m [23

23. S. P. Mickan, R. Shvartsman, J. Munch, X. C. Zhang, and D. Abbott, “Low noise laser-based T-ray spectroscopy of liquids using double-modulated differential time-domain spectroscopy,” J Opt. B-Quantum S. O. **6**, S786–S795 (2004). [CrossRef]

*µ*m, respectively.

## 4. Experiments and results

### 4.1. Polyvinyl chloride: PVC

24. R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. **28**, 363–371 (2007). [CrossRef]

24. R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. **28**, 363–371 (2007). [CrossRef]

11. W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE **95**, 1528–1558 (2007). [CrossRef]

^{-5}, and that for the thinner sample is ≈2×10

^{-3}, or the improvement of the standard deviation is by almost two orders of magnitude.

### 4.2. High-density polyethylene: HDPE

24. R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. **28**, 363–371 (2007). [CrossRef]

**28**, 363–371 (2007). [CrossRef]

### 4.3. Lactose

*α*-lactose monohydrate is selected for this experiment, because it has a distinctive absorption spectrum due to intermolecular resonance modes at lower T-ray frequencies. The sample pellets of lactose with different thicknesses are prepared from lactose powder ground together with ultra-high molecular weight (u.h.m.w.) polyethylene powder in the mass ratio of 1:3. The mixture powder is pressed at 10 tonnes by a hydraulic press to produce six pellets with the diameter of 13 mm and the thicknesses from 0.4 (0.440), 0.8 (0.876), 1.6 (1.658), 2.4 (2.385), 3.2 (3.196), to 4.0 (4.014) mm, all with optical-graded surfaces. The pellets are measured with a focused T-ray beam. Eight scans are recorded for each pellet, and another eight for the reference. For this experiment, the ambient measurement atmosphere is purged with nitrogen to reduce the effects of water vapor absorption. It is worth noting that the measured results are not for pure

*α*-lactose monohydrate, but rather a

*α*-lactose monohydrate/u.h.m.w. PE mixture. However, for conciseness, this mixture is referred to as the

*lactose mix*hereafter.

^{-1}. The strong absorption peaks, clearly observable at 0.53, 1.2, and 1.37 THz, reproduce the results published in [25

25. B. M. Fischer, M. Hoffmann, H. Helm, G. Modjesch, and P. U. Jepsen, “Chemical recognition in terahertz time-domain spectroscopy and imaging,” Semicond. Sci. Technol. **20**, S246–S253 (2005). [CrossRef]

## 5. Usage of the model

26. A. G. Markelz, “Terahertz dielectric sensitivity to biomolecular structure and function,” IEEE J. Sel. Top. Quantum Electron. **14**, 180–190 (2008). [CrossRef]

27. L. Duvillaret, F. Garet, and J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. **2**, 739–746 (1996). [CrossRef]

28. 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**, 409–415 (1999). [CrossRef]

## 6. Conclusion

## Acknowledgements

## References and links

1. | D. Abbott and X.-C. Zhang, “Scanning the issue: T-ray imaging, sensing, and retection,” Proc. IEEE |

2. | R. Huber, A. Brodschelm, F. Tauser, and A. Leitenstorfer, “Generation and field-resolved detection of femtosecond electromagnetic pulses tunable up to 41 THz,” Appl. Phys. Lett. |

3. | K. Liu, J. Xu, and X.-C. Zhang, “GaSe crystals for broadband terahertz wave detection,” Appl. Phys. Lett. |

4. | Z. Jiang and X.-C. Zhang, “Measurement of spatio-temporal terahertz field distribution by using chirped pulse technology,” IEEE J. Quantum Electron. |

5. | A. Bartels, A. Thoma, C. Janke, T. Dekorsy, A. Dreyhaupt, S. Winnerl, and M. Helm, “High-resolution THz spectrometer with kHz scan rates,” Opt. Express |

6. | T. Yasui, E. Saneyoshi, and T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. |

7. | R. H. Jacobsen, D. M. Mittleman, and M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. |

8. | C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, “Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity,” J. Pharm. Sci. |

9. | S. Gorenflo, U. Tauer, I. Hinkov, A. Lambrecht, R. Buchner, and H. Helm, “Dielectric properties of oil—water complexes using terahertz transmission spectroscopy,” Chem. Phys. Lett. |

10. | M. van Exter, C. Fattinger, and D. Grischkowsky, “Terahertz time-domain spectroscopy of water vapor,” Opt. Lett. |

11. | W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE |

12. | M. van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microw. Theory Tech. |

13. | A. Poppe, L. Xu, F. Krausz, and C. Spielmann, “Noise characterization of sub-10-fs Ti:sapphire oscillators,” IEEE J. Sel. Top. Quantum Electron. |

14. | N. Cohen, J. W. Handley, R. D. Boyle, S. L. Braunstein, and E. Berry, “Experimental signature of registration noise in pulsed terahertz systems,” Fluct. Noise Lett. |

15. | D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. |

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

17. | W. Feller, “The fundamental limit theorems in probability,” Bull. Amer. Math. Soc. |

18. | H. F. Trotter, “An elementary proof of the central limit theorem,” Arch. Math. |

19. | W. Withayachumnankul, B. M. Fischer, H. Lin, and D. Abbott, “Uncertainty in terahertz time-domain spectroscopy measurement,” J. Opt. Soc. Am. B (2008). (In press). [CrossRef] |

20. | L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. |

21. | B. M. Fischer, “Broadband THz time-domain spectroscopy of biomolecules — A comprehensive study of the dielectric properties of biomaterials in the far-infrared,” Ph.D. thesis, Department of Molecular and Optical Physics, Freiburg Materials Research Center, Universität Freiburg (2005). |

22. | Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, “Terahertz dielectric properties of polymers,” J Korean Phys. Soc. |

23. | S. P. Mickan, R. Shvartsman, J. Munch, X. C. Zhang, and D. Abbott, “Low noise laser-based T-ray spectroscopy of liquids using double-modulated differential time-domain spectroscopy,” J Opt. B-Quantum S. O. |

24. | R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. Kürner, “Properties of building and plastic materials in the THz range,” Int. J Infrared Milli. |

25. | B. M. Fischer, M. Hoffmann, H. Helm, G. Modjesch, and P. U. Jepsen, “Chemical recognition in terahertz time-domain spectroscopy and imaging,” Semicond. Sci. Technol. |

26. | A. G. Markelz, “Terahertz dielectric sensitivity to biomolecular structure and function,” IEEE J. Sel. Top. Quantum Electron. |

27. | L. Duvillaret, F. Garet, and J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. |

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

**OCIS Codes**

(000.2170) General : Equipment and techniques

(120.4530) Instrumentation, measurement, and metrology : Optical constants

(300.1030) Spectroscopy : Absorption

(080.1753) Geometric optics : Computation methods

(300.6495) Spectroscopy : Spectroscopy, teraherz

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: March 20, 2008

Revised Manuscript: April 30, 2008

Manuscript Accepted: May 1, 2008

Published: May 6, 2008

**Virtual Issues**

Vol. 3, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Withawat Withayachumnankul, Bernd M. Fischer, and Derek Abbott, "Material thickness optimization for
transmission-mode
terahertz time-domain spectroscopy," Opt. Express **16**, 7382-7396 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-10-7382

Sort: Year | Journal | Reset

### References

- D. Abbott and X.-C. Zhang, "Scanning the issue: T-ray imaging, sensing, and retection," Proc. IEEE 95, 1509-1513 (2007). [CrossRef]
- R. Huber, A. Brodschelm, F. Tauser, and A. Leitenstorfer, "Generation and field-resolved detection of femtosecond electromagnetic pulses tunable up to 41 THz," Appl. Phys. Lett. 76, 3191-3193 (2000). [CrossRef]
- K. Liu, J. Xu, and X.-C. Zhang, "GaSe crystals for broadband terahertz wave detection," Appl. Phys. Lett. 85, 863-865 (2004). [CrossRef]
- Z. Jiang and X.-C. Zhang, "Measurement of spatio-temporal terahertz field distribution by using chirped pulse technology," IEEE J. Quantum Electron. 36, 1214-1222 (2000). [CrossRef]
- A. Bartels, A. Thoma, C. Janke, T. Dekorsy, A. Dreyhaupt, S. Winnerl, and M. Helm, "Highresolution THz spectrometer with kHz scan rates," Opt. Express 14, 430-437 (2006). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-14-1-430. [CrossRef] [PubMed]
- T. Yasui, E. Saneyoshi, and T. Araki, "Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition," Appl. Phys. Lett. 87, 061101 (2005). [CrossRef]
- R. H. Jacobsen, D. M. Mittleman, and M. C. Nuss, "Chemical recognition of gases and gas mixtures with terahertz waves," Opt. Lett. 21, 2011-2013 (1996). [CrossRef] [PubMed]
- C. J. Strachan, P. F. Taday, D. A. Newnham, K. C. Gordon, J. A. Zeitler, M. Pepper, and T. Rades, "Using terahertz pulsed spectroscopy to quantify pharmaceutical polymorphism and crystallinity," J. Pharm. Sci. 94, 837-846 (2005). [CrossRef] [PubMed]
- S. Gorenflo, U. Tauer, I. Hinkov, A. Lambrecht, R. Buchner, and H. Helm, "Dielectric properties of oil-water complexes using terahertz transmission spectroscopy," Chem. Phys. Lett. 421, 494-498 (2006). [CrossRef]
- M. van Exter, C. Fattinger, and D. Grischkowsky, "Terahertz time-domain spectroscopy of water vapor," Opt. Lett. 14, 1128-1130 (1989). [CrossRef]
- W. Withayachumnankul, G. M. Png, X. X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, and D. Abbott, "T-ray sensing and imaging," Proc. IEEE 95, 1528-1558 (2007). [CrossRef]
- M. van Exter and D. Grischkowsky, "Characterization of an optoelectronic terahertz beam system," IEEE Trans. Microw. Theory Tech. 38, 1684-1691 (1990). [CrossRef]
- A. Poppe, L. Xu, F. Krausz, and C. Spielmann, "Noise characterization of sub-10-fs Ti:sapphire oscillators," IEEE J. Sel. Top. Quantum Electron. 4, 179-184 (1998). [CrossRef]
- N. Cohen, J. W. Handley, R. D. Boyle, S. L. Braunstein, and E. Berry, "Experimental signature of registration noise in pulsed terahertz systems," Fluct. Noise Lett. 6, L77-L84 (2006). [CrossRef]
- D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, "T-ray imaging," IEEE J. Sel. Top. Quantum Electron. 2, 679-692 (1996). [CrossRef]
- P. U. Jepsen and B. M. Fischer, "Dynamic range in terahertz time-domain transmission and reflection spectroscopy," Opt. Lett. 30, 29-31 (2005). [CrossRef] [PubMed]
- W. Feller, "The fundamental limit theorems in probability," Bull. Amer. Math. Soc. 51, 800-832 (1945). [CrossRef]
- H. F. Trotter, "An elementary proof of the central limit theorem," Arch. Math. 10, 226-234 (1959). [CrossRef]
- W. Withayachumnankul, B. M. Fischer, H. Lin, and D. Abbott, "Uncertainty in terahertz time-domain spectroscopy measurement," J. Opt. Soc. Am. B (2008). (In press). [CrossRef]
- L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, "THz reflection spectroscopy of liquid water," Chem. Phys. Lett. 240, 330-333 (1995). [CrossRef]
- B. M. Fischer, "Broadband THz time-domain spectroscopy of biomolecules - A comprehensive study of the dielectric properties of biomaterials in the far-infrared," Ph.D. thesis, Department of Molecular and Optical Physics, Freiburg Materials Research Center, Universit¨at Freiburg (2005).
- Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, "Terahertz dielectric properties of polymers," J Korean Phys.Soc. 49, 513-517 (2006).
- S. P. Mickan, R. Shvartsman, J. Munch, X. C. Zhang, and D. Abbott, "Low noise laser-based T-ray spectroscopy of liquids using double-modulated differential time-domain spectroscopy," J Opt. B-Quantum S. O. 6,S786- S795 (2004). [CrossRef]
- R. Piesiewicz, C. Jansen, S. Wietzke, D. Mittleman, M. Koch, and T. K¨urner, "Properties of building and plastic materials in the THz range," Int.J Infrared Milli. 28, 363-371 (2007). [CrossRef]
- B. M. Fischer, M. Hoffmann, and H. Helm, G. Modjesch, P. U. Jepsen, "Chemical recognition in terahertz timedomain spectroscopy and imaging," Semicond. Sci. Technol. 20, S246-S253 (2005). [CrossRef]
- A. G. Markelz, "Terahertz dielectric sensitivity to biomolecular structure and function," IEEE J. Sel. Top. Quantum Electron. 14, 180-190 (2008). [CrossRef]
- L. Duvillaret, F. Garet, and J.-L. Coutaz, "A reliable method for extraction of material parameters in terahertz time-domain spectroscopy," IEEE J. Sel. Top. Quantum Electron. 2, 739-746 (1996). [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, 409-415 (1999). [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.