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Lateral shift in THz beam at 45 • rotation angle versus refractive index for different values of sample thickness d.
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We report a simple method to measure the thickness and refractive index of parallel-plane samples simultaneously using femtosecond and terahertz pulses. The time-of-flight measurements of the pulses with and without a sample are exploited to determine the thickness and refractive index of the sample. The accuracy in thickness measurement using femt...
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... the overall intensity of the beam profile reduces due to lateral shift, the temporal position of the THz pulse does not change. Figure 9 shows simulation results using equation (2) for lateral shift versus refractive index of the sample for different values of thickness. It can be noted that the lateral shift is considerably less than the beam diameter (25 mm in our case) for samples having a thickness of less than 20 mm. ...
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Citations
... Various methods aimed toward this goal have been proposed previously [1]. These include methods based on time-of-flight measurements [2,3], Fourier transform spectroscopy [4,5], tomographic measurements [6,7], interferometric techniques [8][9][10], a combination of spatial and temporal measurements [11,12], spatial beam profiling and beam propagation analysis techniques [13], and combinations of spatial refractometry and ellipsometry methods [14][15][16]. Most of these methods involve bulk motion of mechanical components, are computationally intense, or involve expensive hardware preventing a cost-effective solution. ...
In this paper, we present a scheme to simultaneously measure the thickness and refractive index of parallel plate samples, involving no bulk mechanical motion, by deploying an electronically tunable Twyman–Green interferometer configuration. The active electronic control with no bulk mechanical motion is realized via the introduction of a tunable focus lens within the classical motion-based Twyman–Green interferometer configuration. The resulting interferometer is repeatable and delivers accurate estimates of the thickness and refractive index of a sample under test. Elimination of bulk motion also promises a potential for miniaturization. We develop a theoretical model for estimating sample thickness and index values using this reconfigurable interferometer setup and present detailed experimental results that demonstrate the working principle of the proposed interferometer.
... Employing intensity stable lasers, cooled detectors and better rotation stage could further improve the noise-limited resolution. One more aspect of this interferometer is worth discussing which relates to a transverse displacement of a light beam after passing through a tilted plate given as 34 , δ(θ) = [(n − 1)t/n] × θ . For single pass, the maximum lateral shift is δ = 72 μm for θ = 60 • which is negligible compared to the beam size ∼ 5 mm. ...
White light interferometry is a well established technique with diverse precision applications, however, the conventional interferometers such as Michelson, Mach-Zehnder or Linnik are large in size, demand tedious alignment for obtaining white light fringes, require noise-isolation techniques to achieve sub-nanometric stability and importantly, exhibit unbalanced dispersion causing uncertainty in absolute zero delay reference. Here, we demonstrate an ultrathin white light interferometer enabling picometer resolution by exploiting the wavefront division of a broadband incoherent light beam after transmission through a pair of micrometer thin identical glass plates. Spatial overlap between the two diffracted split wavefronts readily produce high-contrast and stable white light fringes, with unambiguous reference to absolute zero path-delay position. The colored fringes evolve when one of the ultrathin plates is rotated to tune the interferometer with picometric resolution over tens of μm range. Our theoretical analysis validates formation of fringes and highlights self-calibration of the interferometer for picoscale measurements. We demonstrate measurement of coherence length of several broadband incoherent sources as small as a few micrometer with picoscale resolution. Furthermore, we propose a versatile double-pass configuration using the ultrathin interferometer enabling a sample cavity for additional applications in probing dynamical properties of matter.
... Furthermore, to achieve the high fringe contrast, the gap between the glass plates u should be minimum to ensure the maximum spatial overlap between the two split wavefronts of the transmitted light because the light passing through the central gap adds a non-interfering background contribution. One more aspect of this interferometer is worth discussing which relates to a transverse displacement of a light beam after passing through a tilted plate given as 34 , δ (θ ) = [(n − 1)t/n] × θ . For single pass, the maximum lateral shift is δ = 72 µm for θ = 60 • which is negligible compared to the beam size ∼ 5 mm. ...
White light interferometry is a well established technique with diverse precision applications, however, the conventional interferometers such as Michelson, Mach-Zehnder or Linnik are large in size, demand tedious alignment for obtaining white light fringes, require noise-isolation to achieve sub-nanometric stability and importantly, exhibit unbalanced dispersion causing uncertainty in absolute zero delay reference. Here, we demonstrate an ultrathin white light interferometer enabling picometer resolution by exploiting the wavefront division of a broadband incoherent light beam after transmission through a pair of micrometer thin identical glass plates. Spatial overlap between the two diffracted split wavefronts readily produce high-contrast and stable white light fringes, with unambiguous reference to absolute zero path-delay position. The colored fringes evolve when one of the ultrathin plates is rotated to tune the interferometer with picometric precision over tens of µm range. Our theoretical analysis validates formation of fringes and highlights self-calibration of the interferometer for picoscale measurements. We demonstrate measurement of coherence lengths of several broadband incoherent sources as small as a few micrometer with picoscale precision. Furthermore, we propose a versatile double-pass configuration using the ultrathin interferometer enabling a sample cavity for additional applications in probing dynamical properties of matter.
... Some of these approaches are briefly described in the following. In Ref. 1 5 applies the time-of-flight method to two transmission measurements from different angles. This approach is similar to our approach: We also use the time-of-flight method for two different measurements. ...
Investigating layer thicknesses with terahertz time-domain spectroscopy usually requires the refractive index of the layer of interest. The refractive index is either investigated beforehand by independent measurements or it is measured simultaneously. The challenge of a simultaneous determination is that measuring two unknown values requires the use of two linearly independent equations. To address this challenge, our approach is to combine transmission and re ection time-of-flight measurements. By combining these measurements, we obtain a system of two linearly independent equations for the two unknown values: thickness and average refractive index. We experimentally validate our simultaneous estimation approach by measuring calibration foils of different thicknesses.
... Especially when using an optical coherence tomography system, it will affect the contrast of the image and the subsequent calculation of tissue thickness [22][23][24]. As a result, there are various measurement techniques proposed over the last decade for the simultaneous measurement of the refractive index and thickness of samples [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. However, to the best of our knowledge, there is not any existing method to simultaneously measure the thickness and refractive index for each layer of multilayered samples. ...
In order to simultaneously obtain the thickness and refractive index for each layer of multilayered samples, this paper proposes a novel measurement system with a simple structure by using geometric optics. The key point of the overall structure is that the binary linear equation relation between the upper and lower planes of the layer can be known by the distances between laser spots reflected from the layer. Using two optical paths with different incident angles of laser beams, the CCD sensors capture the spot signal for image processing binarization and data sampling. After the obtained laser spots’ spacing is substituted into the equations, the thicknesses and refractive indexes of the multilayered samples can be calculated and measured. The proposed measurement system is characterized numerically using simulations on the commercial software program Zemax and then experimentally tested using a laboratory-built prototype. The experiment results show that the refractive indexes and the thicknesses of three-layer samples were measured with high accuracy (with maximum measurement errors of 2.4% and 2% for a refractive index n and thickness d, respectively).
... As a further demonstration of their functionality, the n + -i-n + nanowire detectors were used to measure the refractive index (figure 4(g)) and absorption coefficient ( figure 4(h)) of a piece of paper cards over a THz range from 0.5 to 1.5 THz in the THz-TDS system. The measured values using the n + -i-n + nanowire detector are in good agreements with the literature [44,45] as well as those obtained using a conventional (bulk ion-implanted InP) THz detector [19], demonstrating the practical value of the n + -i-n + InP single-nanowire THz detectors. Table 1 summarizes a detailed comparison of device performance among the undoped InP single-nanowire THz detector, n + -i-n + InP single-nanowire THz detector and traditional ion-implanted InP THz detector. ...
Developing single-nanowire terahertz (THz) electronics and employing them as sub-wavelength components for highly-integrated THz time-domain spectroscopy (THz-TDS) applications are a promising approach to achieve future low-cost, highly integrable and high-resolution THz tools, which are desirable in many areas spanning from security, industry, environmental monitoring and medical diagnostics to fundamental science. In this work, we present the design and growth of n<sup>+</sup>-i-n<sup>+</sup> InP nanowires. The axial doping profile of the n<sup>+</sup>-i-n<sup>+</sup> InP nanowires has been calibrated and characterized using combined optical and electrical approaches to achieve nanowire devices with low contact resistances, on which the highly-sensitive InP single-nanowire photoconductive THz detectors have been demonstrated. While the n<sup>+</sup>-i-n<sup>+</sup> InP nanowire detector has a only pA-level response current, it has a 2.5 times improved signal-to-noise ratio compared with the undoped InP nanowire detector and is comparable to traditional bulk THz detectors. This performance indicates a promising path to nanowire-based THz electronics for future commercial applications.
... The techniques for simultaneous measurement of thickness and RI may be classified as: geometrical optics [3], polarization [4][5][6], ellipsometry [7,8], spectrogoniometry [9], using terahertz pulses [10,11] and interferometry [12][13][14][15]. The measurement based on geometrical optics offers advantages of low cost and non-contact measurement but the measurement accuracy and sensitivity of measurement is limited. ...
We report simultaneous measurement of thickness and refractive index (RI) using Coherent Gradient Sensing (CGS) based interferometric setup. The collimated light beam transmitted through the specimen is analyzed using two gratings in tandem and a spatial filtering arrangement. At the imaging plane, the desired orders superpose yielding interference fringes. Phase shifting interferometric (PSI) technique has been used to obtain the phase information directly. For decoding the information of thickness and RI, the specimen is rotated and the variation in interferometric phase before and after the specimen rotation determined. Simultaneous measurement on a glass plate of thickness 6.019 mm and refractive index of 1.4570 has been demonstrated. Detailed uncertainty analysis is also reported. The expanded uncertainty for thickness and RI in the measurement process was determined to be ±0.005 mm and ±0.00008 respectively.
... Applications of optical interferometry include interference lithography for the fabrication of large area variable period gratings [2], surface metrology [3], vibration analysis [4][5][6], astronomical optical interferometry e.g. space interferometry mission, 2009 [7] etc. Thickness and refractive index of thin films can also be measured by interferometry with higher accuracy than that in other optical techniques [8,9]. The important part in all the interferometric setups is the analysis of fringe data which is mainly done through fringe analysis software. ...
We present a comparison of different techniques for the analysis of the shift and tilt in optical interference fringes. Fringe center, Radon transform, and Gaussian approximation methods are used for fringe analysis. We have measured the tilt and shift between two relevant fringe patterns. The error in tilt measurement was about 2%, and the displacement of the order of few nanometers was measured by the fringe shift analysis. The comparison between the techniques is analyzed with respect to percentage error.
... Based on the above results, a moderate angle of NR can be obtained for some certain conditions. Then, we can consider a slab of material and measure the lateral shift (LS) in the transmitted wave [30]. The measure LS can then be used to determined the angle of refraction. ...
The phenomenon of negative refraction at terahertz (THz) frequency for a high-temperature superconducting material is theoretically investigated based on the inhomogeneous wave theory. Negative refraction, which arises from the negative refractive angle, can occur when a -polarized wave is incident from a dielectric to a superconductor. The dependences of negative refractive angle on the angle of incidence, the wave frequency, and the temperature have been analytically studied. The conditions for obtaining a pronounced result in negative refraction are elucidated. The study of negative refraction is of potential use in designing a tunable polarizer. In addition, it is an extended study of a transmission problem that is not treatable from the Fresnel equations in electromagnetics and optics.
... Perhaps the best known example is the widely cited series of treatments by Duvillaret, Garet, and Coutaz who concluded that sample thickness is frequently the "main source of error in terahertz time-domain spectroscopy" [2][3][4]. This is also the conclusion of others [5,6]. Duvillaret et al. proposed two strategies for reducing this error. ...
The accuracy of any measurement with terahertz time-domain spectroscopy (THz-TDS) depends strongly on knowing and supplying the precise sample thickness when processing the raw terahertz data. Sample thickness usually is estimated using other (non-THz) metrology and invariably involves some degree of uncertainty. It turns out that the terahertz data itself typically also contains information regarding sample thickness. However, there is limited systematic work addressing the following questions: What is the best method for extracting sample thickness from THz-TDS data? And, what is the thickness resolution obtainable with THz-TDS? In this study we demonstrate how these questions can be answered in general by answering them for a specific example: undoped silicon wafers. We determine the accuracy with which the exact thickness of nominally 500 mu m silicon wafers can be measured using transmission mode THz-TDS. We analyze and compare the resolution of 5 different approaches for determining sample thickness using THz-TDS data, including two new methods and three methods proposed in the literature. The quantitative results and analyses of methods we present will be useful in developing far-infrared optical metrology. Conversely the quantitative results presented can be used to relate uncertainty in sample thickness to uncertainty in the measured terahertz data both in the time domain and frequency domain (phase and amplitude). Finally, a precise understanding of the relationship between sample thickness and THz-TDS data can be used to formulate superior THz-TDS data work-up methodologies.
