Simultaneous determination of thickness and refractive index based on time-of-flight measurements of terahertz pulse Babar Hussain,* Mushtaq Ahmed, M. Nawaz, M. Saleem, M. Razzaq, M. Aslam Zia, and M. Iqbal National Institute of Lasers and Optronics, 45650, Nilore, Islamabad, Pakistan *Corresponding author:
[email protected] Received 20 April 2012; revised 19 June 2012; accepted 20 June 2012; posted 21 June 2012 (Doc. ID 167209); published 20 July 2012
We present a simple technique for simultaneous determination of thickness and refractive index of planeparallel samples in the terahertz radiation domain. The technique uses time-of-flight measurements of the terahertz pulse. It has been employed on nine different polymers and semiconductor materials, which are transparent for terahertz frequencies. Our results of thickness measurement are in good agreement with micrometer reading. The accuracy in the determination of refractive index is on the order of two decimal points. © 2012 Optical Society of America OCIS codes: 040.2235, 290.3030, 120.4290, 120.0120.
1. Introduction
Optical methods provide noncontact measurement of thickness and refractive index that has numerous applications in semiconductors, dielectrics, polymers, and many other materials. All these methods have a common limitation, that is, refractive index should be known accurately for the determination of thickness. Many authors have demonstrated different interferometric and other optical techniques [1–6] for simultaneous measurement of thickness and refractive index using two dissimilar experimental setups. These techniques utilize visible or IR radiations and therefore cannot be used for many polymers and semiconductors opaque at these wavelengths. However, the terahertz (THz) radiations offer the advantage that many polymers and semiconductors are transparent in this radiation regime. Terahertz time-domain spectroscopy (THz-TDS) has been used extensively for noncontact measurement of thickness and other optical parameters of
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materials opaque at other wavelengths [7–18]. These techniques are based on recording the temporal profiles of the THz pulse with and without sample. The ratio of the Fourier transforms of the temporal profile with and without sample yields the complex transmission coefficient of the sample. Assuming few hypotheses, solution of the inverse electromagnetic problem leads to the complex refractive index of the sample material [7,10–13]. The measurements obtained by this method are extremely sensitive to the absorption and scattering coefficients of the sample that causes discrepancies in measurements. Dai et al. [12] claim that absorption coefficients of silicon measured by some researchers have been up to 5 times measured by others. In this paper, we present a simple technique for the determination of thickness and refractive index of sample plates employing only one measurement method. It is based on the relative delay produced in the time of flight of the THz pulse at normal incidence and at a known angle (40° in our case). Our technique is equally good for other wavelengths depending upon the transparency of the material. In addition, this technique is more robust because only
the temporal position of the THz pulse is important and no spectroscopy is involved. As compared to THzTDS, our technique is less sensitive to absorption and scattering induced by the sample because the temporal position or time of flight of THz pulse does not change due to absorption. We have measured thickness and group index of refraction in the frequency range from 0.1 to 1 THz of nine different polymers and semiconductor materials with thickness ranging from 0.5 to 13 mm. The results are found to be in good agreement with those measured by other techniques. 2. Experimental Setup
The schematic of the experimental setup is shown in Fig. 1. The THz pulses are generated and detected by photoconductive switching of the THz emitter and detector (TERAVIL) having a built-in silicon lens for the collimation of the THz beam. The THz pulses are pumped and probed by 100 fs laser pulses having 3 mm beam diameter from an erbium-doped fiber laser centered at 780 nm (TOPTICA Photonics). The average powers of the pump and probe beams are 40 and 35 mW, respectively. The distance between the THz emitter and detector is around 30 cm. Bias supply voltage applied to the THz emitter by the electrical chopper is 40 V at 60 kHz. On illumination, the resistances of the THz emitter and detector are around 1 MΩ and 2.5 MΩ, respectively. The sample is mounted on a rotation stage with resolution of 0.1°. The experiments are performed in free space at normal room temperature (∼20 °C).
Fig. 2. Schematic showing relative delays in the THz pulse. S: sample. d: thickness of the sample. LS: lateral shift. Δtn : time delay due to sample inserted at normal incidence to the THz radiation beam. Δtθ : time delay due to sample inserted at an angle θ to the THz radiation beam.
3. Modeling and Measurements
For simultaneous measurement of thickness and refractive index, first the sample is placed normal to the THz beam, and the time delay in the THz pulse is measured. Then it is rotated with a known angle θ with respect to its previous position as shown in Fig. 2, and the resultant delay is measured. We developed a MATLAB code that automatically determines the delay using an autocorrelation function between the two THz pulses. Modeling the optical path difference (OPD) corresponding to the two measurements yields two independent equations. These two equations allow the determination of the two unknown parameters, thickness and refractive index. The OPD introduced due to the
Fig. 1. Experimental setup. 20 July 2012 / Vol. 51, No. 21 / APPLIED OPTICS
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insertion of the sample at normal incidence of the THz radiation beam is given by ΔPn n − n0 d;
(1)
where n and n0 are the refractive indices of the sample and air, respectively; d is the thickness of the sample; and ΔPn is the OPD corresponding to Δtn (the time delay in the THz pulse in the case when sample is inserted normal to the THz beam path as shown in Fig. 2). Modeling the OPD by ray tracing when the sample is rotated with an angle θ, we get ΔPθ
d n − n0 cos β; cosθ − β
(2)
where ΔPθ is the OPD corresponding to Δtθ (the time delay in the THz radiation pulse when the sample is placed at an angle θ as shown in Fig. 2). When the sample is rotated with the angle θ, the THz pulse experiences further change in OPD as well as a lateral shift. The effects of the lateral shift on measurement are discussed in detail in Section 4. By Snell’s law, the angle of refraction β of the transmitted THz radiation pulse is β
sin−1
n0 sin θ : n
(3)
Using this value of β and n from Eq. (1) in Eq. (2), and assuming n0 1, gives an equation independent of n as ΔPθ
d cosθ − sin d sin θ ∕ ΔPn d ΔPn d d sin θ −1 : − cos sin × d ΔPn d −1
(4)
This equation can be solved numerically for d after getting ΔPn and ΔPθ experimentally. Subsequently, Eq. (1) can be solved for refractive index n. Figure 3 shows relative time delay measurements of the THz pulse to determine the OPD for a Teflon strip having thickness 6.15 mm as measured by micrometer. In this case, Δtn is 8.76 ps at normal Table 1.
No. 1 2 3 4 5 6 7 8 9
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Fig. 3. Relative delays of THz pulse for Teflon of thickness 6.15 mm.
incidence and Δtθ is 11.25 ps with θ 40°. We used the MATLAB routine to solve Eq. (4) to determine the thickness of the samples. The results for thickness d and refractive index n measured for different materials using time-of-flight measurements of the THz pulse are shown in Table 1. In our case, accuracy in the measurement of refractive index depends mostly on the accuracy in thickness measurement. We have verified the results of thickness by comparing with the micrometer reading as shown in Table 1. Results for refractive index measurement of silicon, Teflon, polycarbonate, Perspex, and quartz were verified by the available literature [12,17,18]. To verify the refractive indices of all nine materials, we used the thickness measured with the micrometer combined with the time delay at normal incidence [Eq. (1)] as given in the last column of Table 1. To confirm the reliability of the technique, we have measured change in OPD with respect to change in angle of incidence of the THz beam as shown in Fig. 4. A quartz strip having 5.70 mm thickness (measured by micrometer) was used in this experiment. For theoretical plot, the value of refractive index was selected as 1.904, which has been calculated using the time delay in the THz pulse at normal incidence. 4. Discussion
Our results for refractive index of silicon, Teflon, polycarbonate, Perspex, and quartz are in good
Experimental Results for Different Materials
Material
d (mm) Micrometer
d (mm) THz
n THz
n Literature
n Using Δtn and Micrometer Eq. (1)
Teflon Perspex Bakelite silicon quartz glass SZNF-9 polyurethane polycarbonate ceramic
6.15 5.08 6.65 0.54 5.70 3.00 12.40 9.80 2.70
6.136 5.068 6.679 0.557 5.686 3.024 12.406 9.827 2.659
1.428 1.613 1.846 3.404 1.905 3.504 1.503 1.656 2.332
1.4 (quoted by Thorlabs for Teflon lenses) 1.62 [18] — 3.4 [12,16] 1.96 [18] — — 1.63 [18] —
1.427 1.611 1.849 3.408 1.904 3.509 1.505 1.657 2.329
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12
1.4 Theory Experiment
10
Lateral Shift (mm)
Change in OPD (mm)
1.2 1 0.8 0.6 0.4
8 6 4 2
0.2
0
-40
-30
-20
-10
0
10
20
30
40
50
Fig. 4. Comparison of experimental and theoretical results of change in OPD versus angle for quartz of thickness 5.7 mm.
agreement with experimental results reported earlier [12,17,18]. Furthermore, accuracy in the thickness measurement is comparable to some previous demonstrations [3,5–8]. To check the consistency of the technique, we have measured change in OPD with respect to change in angle of incidence of the THz beam as shown in Fig. 4. It can be noticed that experimental results overlap well with the theoretical plot. The technique is not sensitive to the absorption of the sample because when absorption occurs, the signal does change (in amplitude or even shape), but the temporal position of the THz pulse stays unaltered. Beside the instability of the THz emitter and detector, there are two main sources of error in the measurement of thickness and refractive index. These are accuracy of the rotation stage used to rotate the sample (0.1° in our case) and accuracy in translation of the retroreflector used in the path of the pump beam to scan the THz pulse (3 μm, or 0.01 ps, in our case). Incorporating these errors, simulation results show that accumulative error in thickness measurement can be around 30 μm in the worst condition, and refractive index can be measured with accuracy up to two decimal points for samples around 5 mm thick. However, accuracy in refractive index measurement increases with increase in sample thickness. It is important to note that the THz beam transmitted through the sample will be shifted from its original position (lateral shift) with increase in the angle of incidence (Fig. 2). This lateral shift LS depends on the thickness of the sample and actually defines the upper limit of thickness measurement. Although, overall intensity of the beam profile reduces due to lateral shift, but the temporal position of the THz pulse does not change. Figure 5 illustrates simulation results for lateral shift versus refractive index of the sample for different values of thickness. The equation used to calculate the lateral shift is LS
d sinθ − β : cos β
1
1.5
2
2.5
3
3.5
4
n
Angle (Degrees)
Fig. 5. Lateral shift at 40° rotation angle versus refractive index for different values of sample thickness d.
It can be noted that the lateral shift is considerably smaller than the beam diameter (25 mm) in the case of samples having thickness less than 20 mm. Therefore, the technique can be used to measure thickness of the samples up to 20 mm with refractive index up to 4. Decrease in refractive index increases the upper limit of thickness measurement. Also, the agreement in theoretical and experimental values of Fig. 4 proves that lateral shift causes no error as long as a detectable intensity of the THz beam is received by the silicon lens before the THz detector. The upper limit of thickness measurement can be increased to some extent by making the THz detector (along with the probe pulse) laterally movable, but it will increase the complexity of the experimental setup. Another way to increase the upper limit of thickness measurement (imposed by the lateral shift) is by rotating the sample with an angle smaller than 40°, but it will decrease the sensitivity in the pulse delay with rotation. The sensitivity in delay with rotation reduces with decrease in thickness of the sample, which defines the lower limit of thickness measurement. Therefore, the technique is not so effective for samples having thickness below 0.5 mm unless the accuracies in rotation of the sample and translation of the retroreflector are improved. Use of a more 0.5
Change in OPD (mm)
-50
d=1mm d=10mm d=20mm
n=1.5 n=2.0 n=2.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
d (mm)
(5)
Fig. 6. Change in OPD at 40° rotation angle versus sample thickness for different values of n. 20 July 2012 / Vol. 51, No. 21 / APPLIED OPTICS
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stable THz emitter and detector can also improve the precision and lower limit of the measurement. Figure 6 shows change in OPD versus sample thickness for different values of refractive index. It can be noted that change in OPD is very small for sample thickness below 0.5 mm. The diameter of the THz beam is another limitation on the size of sample to be measured (25 mm diameter in our case). The beam diameter can be reduced simply by using a small circular aperture, but it will reduce the upper limit of thickness measurement imposed by the lateral shift. Although our technique is simpler and more robust as compared to THz-TDS, unfortunately it cannot measure complex refractive index. But unlike THz-TDS, it can easily be employed using IR or visible wavelengths. 5. Conclusion
We have presented a simple technique for simultaneous determination of thickness and refractive index of a material sample having parallel plane sides using time-of-flight measurements of a THz radiation pulse. Further work is in progress to improve the accuracy and range of thickness measurement. Application of the technique for determination of refractive index of biological tissues is under study. We gratefully acknowledge Dr. D. Grischkowsky from Oklahoma State University, USA, for suggesting suitable references for this paper and Dr. A. Davies from University of North Carolina Charlotte, USA, for answering our questions about a THz measurement system. We also acknowledge all the reviewers of this paper for their valuable comments. References 1. P. H. Tomlines, P. Wolliams, C. Hart, A. Beaumont, and M. Tedaldi, “Optical coherence refractometry,” Opt. Lett. 33, 2272–2274 (2008). 2. S. Kim, J. Na, M. J. Kim, and B. H. Lee, “Simultaneous measurement of refractive index and thickness by combining low-coherence interferometry and confocal optics,” Opt. Express 16, 5516–5526 (2008). 3. G. D. Gillen and S. Guha, “Use of Michelson and Fabry–Perot interferometry for independent determination of the refractive index and physical thickness of wafers,” Appl. Opt. 44, 344–347 (2005).
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4. P. D. Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000). 5. M. T. Fathi and S. Donati, “Thickness measurement of transparent plates by a self-mixing interferometer,” Opt. Lett. 35, 1844–1846 (2010). 6. X. Wang, C. Zhang, L. Zhang, L. Xue, and J. Tian, “Simultaneous refractive index and thickness measurements of bio tissue by optical coherence tomography,” J. Biomed. Opt. 7, 628–632 (2002). 7. 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). 8. R. Wilk, F. Breitfeld, M. Mikulics, and M. Koch, “Continuous wave terahertz spectrometer as a noncontact thickness measuring device,” Appl. Opt. 47, 3023–3026 (2008). 9. S. Krishnamurthy, M. T. Reiten, S. A. Harmon, and R. A. Cheville, “Characterization of thin polymer films using terahertz time-domain interferometry,” Appl. Phys. Lett. 79, 875–877 (2001). 10. J. Zhang and D. Grischkowsky, “Waveguide terahertz timedomain spectroscopy of nanometer water layers,” Opt. Lett. 29, 1617–1619 (2004). 11. M. Theuer, R. Beigang, and D. Grischkowsky, “Sensitivity increase for coating thickness determination using THz waveguides,” Opt. Express 18, 11456–11463 (2010). 12. J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, “Terahertz time-domain spectroscopy characterization of the farinfrared absorption and index of refraction of high-resistivity, float-zone silicon,” J. Opt. Soc. Am. B 21, 1379–1386 (2004). 13. P. Mousavi, F. Haran, D. Jez, F. Santosa, and J. S. Dodge, “Simultaneous composition and thickness measurement of paper using terahertz time-domain spectroscopy,” Appl. Opt. 48, 6541–6546 (2009). 14. T. Yasui, T. Yasuda, K. I. Sawanaka, and T. Araki, “Terahertz paintmeter for noncontact monitoring of thickness and drying progress in paint film,” Appl. Opt. 44, 6849–6856 (2005). 15. R. K. May, M. J. Evans, S. Zhong, I. Warr, L. F. Gladden, Y. Shen, and J. A. Zeitler, “Terahertz in-line sensor for direct coating thickness measurement of individual tablets during film coating in real-time,” J. Pharm. Sci. 100, 1535–1544 (2011). 16. J. L. Johnson, T. D. Dorney, and D. M. Mittleman, “Enhanced depth resolution in terahertz imaging using phase-shift interferometry,” Appl. Phys. Lett. 78, 835–837 (2001). 17. C. C. Homes, G. L. Carr, R. P. S. M. Lobo, J. D. Laveigne, and D. B. Tanner, “Silicon beam splitter for far-infrared and terahertz spectroscopy,” Appl. Opt. 46, 7884–7888 (2007). 18. M. Naftaly and R. E. Miles, “Terahertz time-domain spectroscopy for material characterization,” Proc. IEEE 95, 1658–1665 (2007).
