Attenuated total reflection terahertz time ... - thz.org.mx THz-TDS is the most sensitive method for measuring highly absorbing liquid in the terahertz range [15] and since its first

This is a preprint. The final version can be found in:

IEEE Transactions on Terahertz Science and Technology. Vol 6, p. 32-39 (2016).

http://dx.doi.org/10.1109/TTHZ.2015.2505909

1

Attenuated total reflection terahertz time-domain spectroscopy:

measurement uncertainty and a new scheme for its reduction

Amin Soltani1, David Jahn

1, Lennart Duschek

1, Enrique Castro-Camus

2,

Martin Koch1, Withawat Withayachumnankul

3,4,

1Faculty of Physics & Materials Science Center Structure & Technology Research Laboratory Philipps

University Marburg, Hans Meerweinstr., 35032, Marburg, Germany

2 Centro de Investigaciones en ´Optica A.C., Loma del Bosque 115, Lomas del Campestre, Leon,

Guanajuato 37150, Mexico 3School of Electrical & Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia

4Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology,

Ookayama, Meguro-ku, Tokyo, Japan

Abstract

Terahertz time-domain spectroscopy (THz TDS) in the attenuated total reflection (ATR)

configuration is ideally suited to characterize highly absorptive media. In this article, we analyze

the impact of random phase and amplitude errors and prism misalignment on the optical

constants extracted from the ATR THz measurements. Analytical models together with empirical

data suggest that, among those factors, small prism misalignment has a relatively strong impact

on the optical constants. We propose an alternative configuration for ATR THz TDS that

significantly reduces the impact of prism misalignment between the reference and sample

measurements. As a result, this configuration facilitates repetitive reference and sample scans to

mitigate the effect from long-term signal drifts. Based on this proposed method, the measured

optical constants of distilled water and pure ethanol have a significantly reduced uncertainty. The

presented analysis and method can be used to improve the measurement accuracy in standard

ATR THz TDS setups.




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1. Introduction

Terahertz time-domain spectroscopy (THz-TDS) is one promising spectroscopic modality

developed in the last 25 years for characterization of many materials in the far-infrared band [1],

[2]. Compared to traditional spectroscopic methods, THz-TDS has an advantage in that it

acquires both the phase and amplitude information, which allows direct determination of the

complex refractive index of the sample [3], [4]. To date, several types of samples have been

investigated using THz-TDS, ranging from semiconductors [5], [6] to polymers [7]–[10] and

liquids [10]–[13]. However, the investigation of samples dissolved in polar liquids, particularly

water, has proven challenging given the exceptionally high absorption of those liquids in this

frequency range. A standard transmission THz TDS setup is not ideal since the maximum

measurable absorption is limited by the system dynamic range [14]. Yet, the study of water

dissolved biomolecules such as proteins and nucleic acids with THz radiation can provide

important information about both their vibrational and solvation dynamics [15]–[17], [10].

Therefore alternative approaches are desired. In particular, reflection type THz spectroscopy

have been considered for these high absorption samples [17].

Attenuated total reflection (ATR) spectroscopy is one of the reflection techniques that have been

explored. ATR THz-TDS is the most sensitive method for measuring highly absorbing liquid in

the terahertz range [15] and since its first implementation with THz-TDS in 2004 by Tanaka et

al. [18], a number of investigations using ATR have been performed in the terahertz band. The

samples which have been studied using ATR THz TDS include water [15], [19], heavy water

[20], intermolecular stretching and hydration processes [21], [22], monolayer cells [23]. Despite

its prevalence, important metrological aspects of ATR THz TDS have not been discussed in

detail. In particular, error sources and their impact on ATR measurements have only been partly

investigated [24]. In this article the amplitude and phase instability as well as angular deviations

in ATR spectroscopy systems will be characterized in the context of their impact on the

measureable sample parameters. In addition, an improvement scheme for ATR THz TDS will be

proposed in order to increase the measurement accuracy of this technique. Experimental

demonstrations with water and ethanol as samples will be discussed.

2. Error sources in ATR THz time-domain spectroscopy




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2.1 Standard ATR spectroscopy

As shown in Fig.1, the core element of an ATR THz TDS setup is a prism. In most cases, the

prism is made of high-resistivity silicon (>10 kΩ-cm) that is almost perfectly transparent and

non-dispersive across the terahertz band [4]. In order to perform an ATR measurement, a sample

is deposited onto the prism base. The prism is inserted into the terahertz path to allow in- and

out-coupling of a terahertz beam either with TE or with TM polarization. With a proper prism

design, the terahertz beam experiences total internal reflection at the prism-sample interface to

create an evanescent field inside the sample close to the reflecting boundary. The minimum

thickness of the sample is required to be greater than the penetration depth of the evanescent

field [25] to record the spectrum that corresponds to the bulk sample [26]. The method also

involves the measurement of a reference pulse in the absence of the sample. In connection with

total internal reflection the so-called Goos-Hänchen effect occurs [27]–[29]. This effect will lead

to a shift of the beam along the reflecting boundary parallel to the plane of incidence. The

incident beam could be assumed as a superposition of plane waves with different incident angles.

Each plane wave experiences a different phase shift upon reflection from the boundary. The

difference of the phase shift in each reflected plane wave is the origin of the Goos-Hänchen shift.

This shift can be estimated by the stationary phase method. This Goos-Hänchen shift is common

to both the reference and sample measurements and the path difference is only a few microns,

which is much smaller than the wavelength and also much smaller than the silicon lens size of

the detector antenna [30]. Hence, it does not have a considerable effect on the accuracy of the

measurement or data analysis.




4

Fig.1 Terahertz ATR spectroscopy setup. The angle �� = �� + is required to be larger than the critical angle. The dimension of our prism is mentioned the sketch.

After being recorded, the terahertz waveforms of the reference and sample measurements are

Fourier transformed. The frequency-dependent ratio of the two spectra for the p or TM

polarization, defined as the transfer function, is given as

�� = �� = �� .

( 1 )

where E��ω and E� !"#��ω are the Fourier transform of the electric field for the reference

and sample measurements, and r"�%�!& %�" andr"�%�!&� !"#�"

are the Fresnel reflection

coefficients for the reference and sample measurements, respectively.

As demonstrated in [24], the complex refractive index of the sample, retrieved from this

transfer function, is given by

()*+,-./ = 0120341 ± 71 − 92;(-�<*, sin��+@� 3AB . ( 2 )

where ; is ; = CD��E�F� G�� 40&�∙��

0I�∙�� B . ( 3 )

In order to comply with energy conservation, the sign of Eq. 2 should be chosen such that the

imaginary part of the refractive index remains negative. The same expression for the complex

refractive index is applicable to the s or TE polarization.

2.2 Random errors

For the commonly used transmission geometry, the influence of errors arising from laser power

fluctuations [31], electrical noise in the electronic circuitry [32], and delay jitter [33] on the

quality of the measured spectra, have been investigated in the past [34]. In this section, the effect




5

of those error sources on ATR THz TDS will be discussed. In order to study the effect of random

fluctuations on the extracted optical parameters, the transfer function in Eq. 1 is re-expressed as

J�� = KL<�M . ( 4 )

where K and N are the amplitude and phase components of the transfer function with the standard

deviations O�and OM, respectively. From the Gaussian error propagation formalism, for the real and imaginary parts of the

complex refractive index, i.e.

() = ( + PQ. ( 5 )

it is possible to show that, to the first order, the standard deviation of the real part is

OG�ω = 7RSGSMT3 OM3 + RSGS�T3 O�3 ( 6 )

and the standard deviation for the imaginary part is

OU�ω = 7RSUSMT3 OM3 + RSUS�T3 O�3. ( 7 )

Given that

VWVX = Re RVW[VXT VWV\ = Re RVW[V\T V]VX = Im RVW[VXT V]V\ = Im RVW[V\T ( 7.1-4 )

and using the chain rule

∂([aN = a([a; a;ab abaN , ∂([aK = a([a; a;ab abaK . ( 8.1-2 )

The first two terms in Eqs. 8.1 and 8.2 are identical, and are equal to

∂([a; = − 1; c([ ± 1([ (dePfg2 sin��hie 271−92;(dePfg sin��hie 2Aj ,

( 9 )

and

a;ab = − cos��hie (dePfg2edePfg−hPed91+ebA2 .

( 10 )

The sensitivity of the refractive index to the phase is




6

VW[SM = 01 c( ± 0G G��m �%W�E m70&931G�� %W�E Amj CD��E�F� G�� 3��0I��J m JP� .

( 11 )

and the sensitivity to the amplitude component is

VW[S� = 01 c( ± 0G G��m �%W�E m70&931G�� %W�E Amj CD��E�F� G�� 3��0I��J m �J� .

( 12 )

The numerical models for propagation of the standard deviation in Eqs. 6 and 7 are fully

validated with the Monte Carlo method. This numerical models are then adopted in order to

quantify the effect of the phase and amplitude error. For this purpose we vary the standard

deviations of σX between 0 and 15 fs (15 steps) and σ\ between 0 to 0.05 (5 steps), for the phase and amplitude components, respectively. FigureFig. 2 demonstrates the effect of the phase and

amplitude errors on the modelled complex refractive index of distilled water [17]. Based on

experimental observations, we estimate typical values σX of 10 fs and σ\ of 0.01. We assume that

similar values hold for most THz TDS systems [24]. From these empirical values, the results

clearly demonstrate that the phase deviation has a greater impact than the amplitude deviation on

both the extracted refractive index and extinction coefficient. It is noted that although the laser

stability can be improved to be in the order of sub-femtosecond, the thermal drift of the fibers

remains a major factor for the phase instability in the fiber coupled setup.

(a) (b)




7

(c) (d)

Fig. 2 Standard deviations of refractive index (a, c) and extinction coefficient (b, d) for distilled water. The standard

deviations are derived from Eqs. 6 and 7 with the standard deviation in the amplitude component, 0 p O� p0.05and the standard deviation in the phase component 0fs p σM p 15 fs.

2.3 Systematic error

In addition to random error which can occur in the amplitude and phase, a parameter that can

potentially have a significant impact on the results from ATR spectroscopy is the incidence angle

of the terahertz radiation on the prism. The typical measurement procedure runs as follows: first,

a reference pulse is recorded. Subsequently the sample, usually a liquid, is placed directly on the

top surface of the prism. If the prism is not firmly fixed onto the platform, small mechanical

stress induced by sample deposition and cleaning can introduce a very small yet non-negligible

variation in the position or angle of the prism. Importantly, this small deviation leads to a change

in the system response for the sample and reference measurements. This systematic error can

propagate to and adversely affect the estimated complex optical properties of the sample. Let t be the angular displacement of the prism. In this case, the propagation path lengths of the THz

beam for the sample measurement (u0v + u3v and for the reference measurement (u0 + u3 are not equal as depicted in Fig. 3. Additionally, the incidence and total-reflection angles for

reference and sample beams are different. By accounting for these factors, the transfer function

can be rewritten as

�� = �� = @wx@w /�yz{wx/�yz{w ��

�� /�yz{mx/�yz{m @mx@m . ( 13 )




8

where i0v , i3v are the transmission coefficients of the air-prism and prism-air interfaces for the

sample measurement, and i0, i3 are the transmission coefficients of the two interfaces for the

reference case. In addition, the output THz sample beam from the prism is not parallel to the

reference. This implies that the detector might also be imperfectly aligned. Depending on the

deviation angle this particular factor might or might not be relevant. Our measurement setup

contains four high-density polyethylene (HDPE) 60 mm lenses and the whole terahertz path is 50

cm. The terahertz beam travels 5 cm from the emitter to the first lens, then 14 cm in collimated

version to the second lens. The distance between two middle lenses are 13 cm and the prism is

located in the middle. The beam travels the last two lenses before being collected by a 5-mm

silicon lens into the detector antenna. For 0.05 degree angular deviation of the prism, the

maximum beam misalignment on the detector is 150 microns, which is negligible for the 5 mm

silicon lens attached to the detector. Yet, for the sake of simplicity, the misalignment between

the output beam and the receiver is neglected in the following analysis.

Fig. 3 Propagation of the THz beam through a tilted prism. The reference measurement is denoted by the black

line, and the sample measurement by the blue dotted line. The dimension of the prism is mentioned the sketch.

We numerically estimate the transfer function that accounts for this prism tilting effect by using

Eq. 13, and then extract the complex refractive index from this transfer function by using a

standard extraction model described in Section 2.1. In Fig. 4 the real and imaginary parts of the

refractive index of distilled water are shown as a function of angular deviations t between −0.05 and +0.05 degrees with a step size of 0.01 degree. Interestingly, such a relatively small angular

deviation can have an enormous impact on both the real and imaginary parts of the extracted

refractive index, and the effect is enhanced at high frequencies. Hence, we conclude that such a




9

misalignment can have a significant effect, exceeding effects from phase and amplitude

fluctuations. A larger contrast between the refractive indices of the air and the sample (water in

this work) enhances the effect of the angular deviation of the prism.

Fig. 4 Refractive index (a) and extinction coefficient (b) of distilled water under the influence of the prism

misalignment. The angular deviation of the prism between the reference and sample measurements is varied

between -0.05º and 0.05º with a step size of 0.01º.

It should be noted that a large prism size enhances the error from angular deviation.

Mathematically, an increment in the size of the prism L in Fig. 3 increases the difference

between u and uv, and thus enhances the systematic error. Fig. 5 presents the refractive index

and extinction coefficient of distilled water for t=0.05 degree and a range of the prism size, }, between 260 and 380 mm. The results confirms that by increasing the prism size the error from

angular deviation is enhanced.




10

Fig. 5 Refractive index (a) and extinction coefficient (b) of distilled water for angular deviation t=0.05 degree and a variation of the prism size from 260 mm to 380 mm with a step size of 10 mm. The black lines show the

error-free refractive index and extinction coefficient.

3. Alternative configuration for ATR THz spectroscopy

3.1 Method description

Fig. 6 Schematic of the proposed configuration for ATR spectroscopy. The prism is divided into two sections and

can be translated orthogonally to the beam axis for reference and sample measurements.

A possible approach to the reliability improvement of ATR THz TDS is to use a long prism

fixed to an optics translation stage as shown in Fig. 6. The top surface is divided into two

sections, one for the sample measurement and the other for the reference measurement. A lateral




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shift of the prism transversal to the beam axis allows alternating measurements between the

sample and reference chambers. Since the sample is pre-deposited on the prism, this setup can

mitigate the effects from arbitrary prism tilting upon sample loading. Additionally, without

accentuating the effect of prism misalignment, it facilitates alternating reference and sample

scans during a long course of measurement in order to reduce the effect of long-term signal drifts

[35]. Of course, the translation of the prism is expected to introduce angular misalignment

which, as discussed earlier, could be a major source of error for the measurements. However, in

this case the angular misalignment is known in priori and can be calibrated via a post-processing

step. In order to reduce this effect, spectral calibration curves can be obtained for both the phase

and amplitude components in order to compensate the misalignment [36]. The calibration

function

~�+.<��+@<�G�� = �� ( 14 )

can be obtained by dividing the sample spectrum in the absence of a sample by the reference

spectrum on both prism positions. An example of such a function is presented in Fig. 7.

Fig. 7 Amplitude (left) and phase (right) profiles of the calibration function in Eq. 14. This empirical function can

be used to correct the effect from misalignment of the prism between the reference and sample measurements.

The correction to the reference measurement is therefore given by

��/��/�@/� = ��/�,/+*��/�� ∗ ~�+.<�+�@<�G�� . . ( 15 )




12

By loading the sample onto one side of the prism, the relative misalignment between the sample

and reference sides is maintained, and this correction factor remains valid. Additionally, the

calibration function can account for the angular tolerance of the prism (±0.5 degree for our

prism).

3.2 Experimental results

In order to verify the validity of the method proposed in Section 3.1, we performed ATR

measurements on distilled water and ethanol. The spectrometer we used is described in Ref. [24]

as system B. Firstly, we performed a measurement on distilled water at room temperature. The

resulting of refractive index and absorption coefficient are shown in Fig. 8 along with the values

published by two other independent research groups. The results are in general agreement with a

difference under 5% across the frequency range of interest. Additionally, two series of

measurements on distilled water and ethanol were carried out with the traditional ATR method

and the proposed method. Each liquid, interlaced with the reference scanning, was measured 20

times using both methods. The measurable data were used to generate the curves and standard

deviations shown in Fig. 9. It is obvious that the standard deviation is in both cases significantly

smaller for the proposed method. The uncertainty reduction is as a result of the shortened

duration between the reference and sample measurements, and the correction to the angular

deviation.

Fig. 8 Measured refractive index (n) and absorption coefficient ( of distilled water with the




13

proposed ATR THz method in comparison to the published data.




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4. Conclusion

We discussed the impact of amplitude and phase errors and angular misalignment of the prism

on the data extracted from ATR THz-TDS measurements on liquids. A careful analysis shows

that a small prism misalignment in the order of a thousandth of a radian can produce a significant

deviation in the extracted complex refractive index. This effect is much larger than the deviation

introduced by the amplitude and phase errors in typical THz-TDS systems. In addition we

proposed an alternative measurement setup in which the prism is translated along its axis

between each sample and reference measurement. The method accounts for possible angular

deviation of the prism and other alignment imperfections via the frequency-dependent correction

factor. This method allows alternate acquisitions of sample and reference pulses with a minimal

effect from the prism misalignment. This alternating measurement approach is important for

observation of sample progress over a long period of time, for example to monitor chemical

processes in aqueous solutions over a few days, without suffering from the effect of slow drifts

in temperature, laser power, or other environmental conditions that could affect the measurement

results.

Fig. 9 Comparison of the uncertainty for the properties of distilled water (a) and ethanol (b) measured with the

proposed method (red) and conventional ATR THz spectroscopy (black).




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Amin Soltani was born in 1984, in Iran. In 2007, he received his B.Sc. in Electrical

Engineering from Iran University of Science and Technology (IUST), Iran and

received his M.Sc. in Electrical Engineering in 2010 from School of Electrical and

Computer Engineering University of Tehran, Iran. Since 2012, he is working toward

the Ph.D. degree in the field of THz attenuated total reflection (ATR) spectroscopy

under the supervision of Prof. M. Koch at the Philipps University of Marburg,

Germany.

David Jahn received his B.Sc. in physics from the Philipps-Universität Marburg,

Germany in 2009 and his M.Sc. in physics at the Ludwig-Maximilians Universität

München, Germany in 2012. He is currently pursuing the Ph.D. degree with the

Department of Physics at the Philipps-Universität Marburg. During his Ph.D. he spent

six months in a collaborative project with INRIM, Italy. His current research interests

range from THz metrology to THz surface plasmons and their use for flat beamforming devices.

Lennart Duschek was born in 1989, in Fulda, Germany. In 2014, he received his

master of science in physics from the Philipps University of Marburg, Germany.

Since 2014, he is working toward the Ph.D. degree in the structure and technology

research laboratory group of Prof. Dr. Kerstin Volz at the Philipps-Universität

Marburg in the field of transmission electron microscopy.




19

Enrique Castro-Camus obtained a degree in Physics from Universidad Nacional

Autónoma de Mexico in 2002 and a D.Phil (PhD) in Condensed Matter Physics from

the University of Oxford (UK) in 2006 in the area of terahertz spectroscopy.

Subsequently he was appointed as a research fellow at the Materials Department of

the University of Oxford working on nanomaterials. In 2009 he joined the Photonics

Department at Centro de Investigaciones en Optica (Leon, Mexico) as a research

professor. He is a National Researcher level II and a Fellow of the Mexican Academy of Sciences. He is

an associate editor of the Journal of Infrared Millimeter and Terahertz Waves and has served as a chair

and committee member of multiple scientific meetings in the area of terahertz and optics. Since mid-2015,

he becomes the head of the spectroscopy and imaging division of the "Laboratorio Nacional de Ciencia y

Tecnologia de Terahertz".

Martin Koch was born in Marburg, Germany, in 1963. He received the Diploma and

Ph.D. degree from the University of Marburg in 1991 and 1995, respectively. From 1995

to 1996, he was a post-doctorate at Bell Labs/Lucent Technologies, Holmdel, NJ, USA.

From 1996 to 1998, he worked in the photonics and opto-electronics group at the

University of Munich. From 1998 to 2009, he was associate professor at the Technical

University of Braunschweigin the EE department. In 2003, he spent a three-month

sabbatical at the University of California in Santa Barbara. Since 2009, he is professor for experimental

semiconductor physics at the University of Marburg. Marburg, Germany. Prof. Koch was awarded the

Kaiser-Friedrich Research Prize (2003), the IPB Patent Award (2009), and is editor-in-chief of the Journal

of Infrared, Millimeter and Terahertz Waves.




20

Withawat Withayachumnankul received the B.Eng. and M.Eng degrees in electronic

engineering from King Mongkut’s Institute of Technology Ladkrabang (KMITL),

Bangkok, Thailand, in 2001 and 2003, and the Ph.D. degree in electrical engineering

(commendation) from the University of Adelaide, Australia, in 2010. From 2003-2012,

he served as a Lecturer at KMITL with the Faculty of Engineering. From 2010-2013,

he has held an ARC Australian Postdoctoral Fellowship with the University of Adelaide. Since 2014, he

has become a lecturer at the University of Adelaide. Currently, he is also a JSPS Visiting Fellow at Tokyo

Institute of Technology, Japan, and an Associate of RMIT University, Melbourne, Australia. His research

interests include terahertz technology, metamaterials, plasmonics, and optical antennas. Dr.

Withayachumnankul has authored and co-authored 50 journal publications. He serves as a grant assessor

for Swiss National Science Foundation (SNSF), German Academic Exchange Service (DAAD), French

National Research Agency (ANR), and Australian Research Council (ARC). He is a recipient of the

IEEE/LEOS Graduate Student Fellowship (2008) and the SPIE Scholarship in Optical Science and

Engineering (2008).

Documents

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