Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46:59 https://doi.org/10.1007/s10762-025-01070-8 RESEARCH Tutorial: Accurate Determination of Refractive Index and Absorption Coefficient in Terahertz Time-Domain Spectroscopy Chi Ki Leung1 · Jasper N. Ward-Berry1 · Elena Wanvig i Dot1 · Jongmin Lee1 · J. Axel Zeitler1 Received: 22 May 2025 / Accepted: 18 July 2025 / Published online: 18 August 2025 © The Author(s) 2025 Abstract This tutorial presents an up-to-date methodology primarily for determining the refrac- tive index and absorption coefficient of strongly terahertz-absorbing solid materials, with principles extended to other sample types. The accurate and straightforward methodology requires three terahertz time-domain spectroscopy (THz-TDS) mea- surements: baseline, reference, and sample. The baseline is a measurement of the terahertz time-domain spectrometer’s empty beam path. The reference consists of a weakly terahertz-absorbing material, and the sample is a well-mixed binary mixture of a weakly and a strongly terahertz-absorbing material. During THz-TDS data pro- cessing, the concept of half-width is introduced, defining the time span over which half of a symmetric apodisation function is applied. The half-width ensures consis- tent application of the apodisation function across all THz-TDS measurements and facilitates the automatic specification of a uniform time-delay range for fast Fourier transform. Complex transfer functions for the reference and sample are derived with respect to the baseline, enabling the extraction of their respective refractive index and absorption coefficient spectra. The reference provides the closest possible exper- imental estimate of the weakly terahertz-absorbing material’s actual optical constants within the sample and, where applicable, also incorporates the effects of sample poros- ity by approximation. Utilising effective medium theories, the refractive index and absorption coefficient spectra of the strongly terahertz-absorbingmaterial can be accu- rately determined. This tutorial additionally discusses the conventional approaches, addresses cases with limited time-delay ranges and different experimental configura- tions, identifies simplification strategies, highlights potential pitfalls in the derivation process, and discusses their implications, ensuring robust analysis of THz-TDS data. Keywords Refractive index · Absorption coefficient · Terahertz · Spectroscopy · Processing · Tutorial Extended author information available on the last page of the article 123 http://crossmark.crossref.org/dialog/?doi=10.1007/s10762-025-01070-8&domain=pdf http://orcid.org/0000-0002-0399-2784 http://orcid.org/0009-0006-3472-4631 http://orcid.org/0000-0003-0272-0274 http://orcid.org/0000-0002-3476-2922 http://orcid.org/0000-0002-4958-0582 59 Page 2 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 1 Introduction Terahertz time-domain spectroscopy (THz-TDS) leverages the broadband frequency range of 0.1 to 10THz to reveal spectral features and properties of different materials. By employing femtosecond laser pulses to generate broadband terahertz radiation, THz-TDS enables the non-destructive and simultaneous measurement of both ampli- tude and phase information, facilitating the direct determination of a material’s refractive index n and absorption coefficient α [1]. Over the past few decades, THz- TDS has evolved significantly, from its initial demonstration in the late 1980s [2, 3], to custom-built spectrometers in research laboratories, and now to state-of-the-art, compact turn-key commercial instruments offering extensive time-delay range and frequency bandwidth. The intrinsic strengths of THz-TDS and the advancements in instrumentation have driven its application across a diverse range of areas, including novel materials, phar- maceuticals, medicine, biology, and energy [4, 5]. Central to these applications is the ability of THz-TDS to directly and non-invasively probe optical and electronic prop- erties such as the refractive index n, absorption coefficient α, dielectric constant ε, and conductivity σ . In the study of novel materials, THz-TDS has been employed to investigate the dielectric properties and conductivity of semiconductors [6, 7], low- dimensional materials [8] and metamaterials [9], as well as the optical properties and conductivity of metal-organic frameworks [10–12]. The charge carrier dynamics in perovskites can also be understood via conductivity measurements by THz-TDS [13– 15]. All these novel materials demonstrate strong potential for terahertz photonics. In pharmaceuticals, THz-TDS is used to characterise polymorphs and the degree of crys- tallinity in drugs and drug products via the absorption coefficient [16–19], while the refractive index is used to quantify the porosity of pharmaceutical tablets at manufac- turing [16, 20–23]. For biomedical applications, THz-TDS can diagnose skin [24–26] and colon cancers [27–29] by measuring the changes in refractive index, absorption coefficient, and dielectric constant. The hydration dynamics of biomolecules can be investigated by THz-TDS based on similar principles [30–33]. On energy applications, THz-TDS has been utilised to study charge carrier dynamics in solar cells [34–37]. The refractive index acquired by THz-TDS is also used to evaluate the porosity and density of battery electrodes and solid-state electrolytes [38–40], as well as to assess the structural integrity of wind turbine composite materials [41]. In tandem with these expanding applications, significant efforts have been made to enhance the accessibility of THz-TDS. Extensive high-quality literature has been pub- lished to provide comprehensive overviews of THz-TDS [1, 42], alongside detailed tutorials on common experimental methods [43, 44] and robust data processing tech- niques [45–55]. The dotTHz project offers a suite of open-access resources which further facilitate THz-TDS data processing, analysis, and collaboration [56, 57]. All these lower the barrier to the adoption of THz-TDS. This tutorial complements conventional approaches and existing resources by pro- viding an up-to-date, accurate, and straightforward methodology for determining refractive index n and absorption coefficient α, which are essential to THz-TDS applications. The presented methodology, primarily tailored for strongly terahertz- absorbing materials, is enabled by the advances in terahertz time-domain spectrom- 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 3 of 37 59 eters, particularly the extended time-delay range. A basic familiarity with complex numbers and the concept of Fourier transform is assumed in this tutorial. The con- ventional approaches to THz-TDS measurements are also covered in this tutorial and compared with the presented updated methodology. The optical constant deriva- tions for solid materials, from both the conventional and the updated comprehensive methodology, are implemented in the THzPy Python package [58] and CaTSper, the code-free graphical user interface THz-TDS analysis tool [59]. Both THzPy and CaTSper are part of the open-source tools provided by the dotTHz project [56, 57]. Additionally, this tutorial addresses cases involving limited time-delay ranges and different experimental configurations, identifies simplification strategies, highlights potential pitfalls in the derivation process, and discusses their implications. For fur- ther in-depth treatments of specific topics such as phase unwrapping algorithms, the tutorial directs readers to the cited specialist literature. By offering a comprehensive and systematic approach to determine n and α, this tutorial ensures accurate THz-TDS characterisation and robust data analysis. 2 Terahertz Time-Domain SpectroscopyMeasurements This tutorial is based on a simple experiment setup (Fig. 1) suitable for determining the optical constants of a weakly terahertz-absorbing material, a strongly terahertz- absorbing material, and for the specific case of a thick sample comprising a strongly terahertz-absorbing material. Two pellets, the reference and sample pellets, and three measurements, consisting the baseline, reference, and sample, were involved in the tutorial. The reference pellet solely consists of a weakly terahertz-absorbing material, while the sample pellet ismade from a homogeneous binarymixture of aweakly and strongly terahertz-absorbing material. Homogeneous mixing in the sample pellet is crucial to obtaining reliable and reproducible optical constants. The pellet diameter and the mass of the weakly terahertz-absorbing material should remain constant between the reference and sample pellets. The weakly terahertz-absorbing material acts as a dilu- ent in the sample pellet and ensures that the strongly terahertz-absorbing material’s absorption characteristics does not exceed the dynamic range of the spectrometer. Otherwise, accurate determination of optical constants would be infeasible. Typical weakly terahertz-absorbing diluents include polyethylene (PE) or polytetrafluoroethy- lene (PTFE). When preparing the pellets, flat and parallel pellet faces should be aimed for to minimise possible thickness variations. A suitable and precise tool such as a micrometer should be used for accurate measurement of the reference and sample pellet thickness, dr and ds, since these values directly impact the calculated optical constants. A forthcoming companion manuscript will provide a comprehensive step- by-step guide covering these practical experimental aspects. Once the pellets are prepared, THz-TDS measurements at normal incidence can be performed. The baseline measurement is established with no material placed in the terahertz beam path between the emitter and detector (Fig. 1a). The reference measurement is then conducted with the reference pellet (Fig. 1b) and similarly for the sample measurement (Fig. 1c). 123 59 Page 4 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 (a) Baseline (b) Reference (c) Sample ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂̂ ̂ ̂ ( − ) Weakly terahertz-absorbing material Well-mixed binary mixture of a weakly and a strongly terahertz- absorbing material Positive direction Fig. 1 Experimental setup of the a baseline, b reference, and c sample measurements in terahertz time- domain spectroscopy For the optical constants of a weakly terahertz-absorbingmaterial, only the baseline and reference measurement are necessary. As for the optical constants of a strongly terahertz-absorbing material, all three baseline, reference, and sample measurements are required. If the reference and sample pellets are very thick, some terahertz spec- trometers may not have the necessary time-delay range to consistently acquire all of the three measurements. In such cases, it is only practical to acquire the reference and sample measurements. By carefully selecting the relevant measurement approach and systematically implementing it across sample sets, optical constants can be accurately determined with experimental error minimised. 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 5 of 37 59 Polyethylene and α-lactose monohydrate were chosen as the weakly and strongly terahertz-absorbing material for demonstration in this tutorial. The powder masses were weighed by a mass balance (PAS214, Fischer Scientific, France). A total of 285mg polyethylene powder was used in both the reference and sample pellet. The sample pellet, a well-mixed binary mixture, also contained 15mg α-lactose monohy- drate powder. Homogeneous mixing was achieved using a pestle and mortar. The two pellets weremade using amanual hydraulic press (GS15011, Specac Ltd., UK)with 13 mm diameter. The resultant thickness of the reference and sample pellets are 2.56mm and 2.65mm, respectively, as measured by a micrometer (AK9635D, Sealey, UK). All terahertz measurements were conducted at normal incidence and were acquired from averaging 1000 acquisitions using a commercial terahertz spectrometer (TeraSmart, Menlo Systems GmbH, Germany) under a nitrogen-purged environment. Whilst powdered materials are used for demonstration in this tutorial, the presented methodologies are applicable to different solid forms, for example, solid samples prepared from moulding or extrusion. 3 Data Processing from Time Domain to Frequency Domain THz-TDS measurements must first be processed and then transformed into the fre- quency domain via a fast Fourier transform (FFT), before data analysis takes place. This section outlines the key steps involved in processing time-domain terahertz data into the frequency domain. For further details, readers are encouraged to reference the excellent literature resources available [43, 44, 50, 54, 56, 57]. The time-domain terahertz measurement is apodised to increase the signal-to-noise ratio of the primary transmitted pulse, which contains critical information (Fig. 2). Apodisation also suppresses noise sources, such as background noise and Fabry-Pérot reflections, which are multiple internal reflections arising from media boundaries and are also known as etalon effects. Fabry-Pérot reflections can only be effectively sup- pressed by apodisation when they are well separated from the primary transmitted pulse in time domain. The magnitude of separation can be controlled by the pellet thickness, based on the refractive index of the material. A 20ps separation provides a comfortable room for effective apodisation, whilst maintaining a suitable spectral resolution to observe typical spectral features in solids aswell as liquids, which are dis- cussed in Sect. 8. For solids, the suggested separation can be achieved by a 2mm pellet with a refractive index of 1.50 in the terahertz region. Detailed considerations of these practical experimental aspects will be further addressed in an upcoming companion manuscript. The apodisation function should be positioned such that its maximum aligns with the primary peak of the given terahertz pulse (Fig. 2). Sincemost of the commonly used apodisation functions are symmetric, it is convenient to define themby their half-width. The half-width specifies the time span over which half of the symmetric apodisation function is applied.The chosenhalf-width should bewide enough to encompass the key features of the primary terahertz pulse, while remaining sufficiently narrow to exclude any Fabry-Pérot reflections. This eliminates the complications arising from Fabry- 123 59 Page 6 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Pérot reflections in downstream optical constant extraction, which is comprehensively addressed in the tutorial by Neu and Schmuttenmaer [44]. While excluding Fabry-Pérot reflections via apodisation significantly simplifies the analysis and is often sufficient, especially when the primary and reflected pulses are well-separated in time, alternative methods exist that explicitly account for these multiple internal reflections [45, 46]. These approaches model the full measured time- domain signal or the corresponding complex frequency-domain transfer function, including the oscillations caused by Fabry-Pérot resonances within the sample and any containing windows, such as in a cuvette (Sect. 8), for more complex experi- mental configurations. By fitting the experimental data to a theoretical model that incorporates these multiple reflections, it is possible to extract the optical constants, potentially with higher accuracy, particularly for optically thin samples or materials where Fabry-Pérot reflections are pronounced and overlap with the primary pulse. However, these methods are generally more complex, often requiring accurate knowl- edge of sample thickness and iterative numerical algorithms to solve for the optical properties, such as the Nelly package [60, 61]. The apodisation-based approach pre- sented in this tutorial provides a robust and widely applicable starting point, balancing accuracy with analytical simplicity. The same apodisation function with the specified half-width is systematically applied to each THz-TDS measurement. On each application, the maximum of the apodisation function and the primary peak of the specific terahertz pulse should be aligned (Fig. 2). The global minimum and maximum time values across the applied apodisation functions automatically define auniform time-delay range for FFT (Fig. 2). In each measurement, electric field values that lie outside the apodisation function but within the time-delay range are set to zero. This process ensures consistency and preserves phase differences between measurements. Moreover, since the apodisation function smoothly tapers to zero at both ends, there are no discontinuities between the apodised electric field and zero-padding, as well as between repeated electric field pulses under periodic extension. Otherwise, the presence of discontinuities violates the assumption in FFT that the time-domain signal is periodic, leading to spectral energy spreading artefactually, which is an undesirable phenomenon known as spec- tral leakage. Apodisation thus prevents spectral leakage to occur. All these considerations in terahertz time-domain processing inform the selection of suitable apodisation functions. The Hann apodisation function is widely adopted in terahertz time-domain processing. Compared to the Gaussian apodisation func- tion, which is a general-purpose smoothing function for signal processing, the Hann apodisation function fully tapers to zero at both ends and is thus more effective in suppressing spectral leakage. The boxcar apodisation function, also known as the rectangular apodisation function, is suboptimal as it simply truncates the signal with- out any smoothing. This contributes to sharp discontinuities and does not improve the signal-to-noise ratio, resulting in spectral leakage and ultimately inaccurate opti- cal properties. An appropriate choice of apodisation function is thus imperative to terahertz time-domain processing. Subsequently, the time-domain terahertzmeasurements are fast Fourier transformed into the frequency domain with an appropriate upsampling factor, usually in the range of 21 to 24, to enhance spectral resolution where necessary. Choosing an excessively 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 7 of 37 59 −30 −10 10 30 50 70 Time (ps) −2 0 2 4 6 E le ct ri c fie ld am pl itu de (a .u .) Terahertz electric fields for Baseline Reference Sample Scaled Hann apodisation functions for Baseline Reference Sample 4 5 6 Time (ps) −2 0 2 4 6 A m pl itu de (a .u .) Fig. 2 Raw data of the measured baseline, reference, and sample terahertz time-domain electric fields. Hann apodisation functions with a uniform half-width of 15ps were applied during data processing and are scaled here for visualisation. The white region indicates the uniform time-delay range used for the fast Fourier transform (FFT). The inset highlights the separation between the primary transmitted pulses of the reference and sample terahertz electric fields high up sampling factor reduces computational efficiency and may yield a spectral resolution that no longer reflects the fidelity or meaningful resolution of the raw data. In the frequency domain, the terahertz electric field is characterised by its amplitude and phase. The phase difference between terahertz measurements, �φ, is unwrapped and calculated by referencing Jepsen’s method [54]. Careful phase unwrapping is crucial as noise, especially at frequencies with low signal amplitude, can introduce spurious large phase jumps (� π ) between adjacent points, potentially leading to 2π ambiguities in the accumulated phase if not handled correctly, complicating the recovery of the true phase evolution. In addition, a phase offset is usually introduced by instrumental noise and can be determined and compensated by extrapolating phase values from a low-frequency region with a high signal-to-noise ratio. It is important to recognise that the extracted optical constants derived from these processed data are only reliable over the frequency range where the measured terahertz signal, especially 123 59 Page 8 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 the sample signal, remains significantly above the system’s noise floor, i.e., within the valid dynamic range of the measurement, the limits of which are discussed by Jepsen and Fischer [47]. In this tutorial, the Hann apodisation function with a half-width of 15ps (Fig. 2) and an upsampling factor of 23 is employed for FFT. The phase difference from 0.2THz to 0.4THz is extrapolated for determining the phase offset. 4 Basic Principles of Terahertz Electric Field in Frequency Domain To extract optical constants from the processed frequency-domain terahertz electric fields, the basic principles must be understood and leveraged. Consider the frequency-dependent emitted terahertz electric field, Ê0(ω). As Ê0(ω) propagates throughmedium1 (Fig. 3), it undergoes attenuation. The direction inwhich Ê0(ω) propagates is always defined as positive in this tutorial (Fig. 3). The resultant terahertz electric field after propagation, Êprop(ω), can be written as Êprop(ω) = Ê0(ω) p̂1(ω) (1) where the propagation factor p̂1(ω) is described by p̂1(ω) = exp ( iωn̂1(ω)d1 c0 ) (2) ̂ ̂ Medium 1 Medium 2 ̂ Positive direction Fig. 3 Diagram showing the emitted Ê0(ω), propagated Êprop(ω), transmitted Êtrans(ω), and reflected Êrefl(ω) terahertz electric fields across media 1 and 2. For demonstration purposes, the illustration of Êrefl(ω) is exaggerated and is not geometrically accurate 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 9 of 37 59 with c0 as the speed of light in a vacuum, n̂1(ω) as the complex refractive index, and d1 as the thickness of the medium 1. Note thatω is the angular frequency and is related to the frequency ν by ω = 2πν. n̂1(ω) can be expressed as n̂1(ω) = n1(ω) + iκ1(ω) (3) n̂1(ω) = n1(ω) + i α1(ω)c0 2ω (4) where n1(ω) is the real refractive index, κ1(ω) = α1(ω)c0/2ω is the extinction coef- ficient, and α1(ω) is the absorption coefficient of medium 1. It should be noted that the sign convention for the imaginary part of the complex refractive index can differ in literature (i.e., n̂ = n + iκ versus n̂ = n − iκ). This choice is typically linked to the assumed time dependence of the electromagnetic field (e.g., e−iωt versus e+iωt ) in the underlying wave equation. This tutorial adopts the convention of n̂ = n + iκ , consistent with a time dependence of e−iωt . It is important to ensure that the definition is used consistently throughout any derivation; both conventions correctly describe wave attenuation (positive κ or α) for passive media (i.e., media that do not generate energy at the frequency of interest), when paired with the corresponding propagation factor definition (e.g., Eq. 1). The equations presented herein are internally consistent based on the stated convention. When the terahertz electric field Êprop(ω) encounters a boundary between two media with differing refractive indices, n̂1(ω) and n̂2(ω), at normal incidence, part of the field is transmitted and part is reflected (Fig. 3). The transmitted Êtrans(ω) and reflected Êrefl(ω) terahertz electric fields can be expressed as Êtrans(ω) = Êprop(ω)t̂12(ω) (5) Êrefl(ω) = Êprop(ω)r̂12(ω) (6) which are governed by the normal-incidence Fresnel coefficients. The complex trans- mission t̂12 and reflection r̂12 coefficients are given by t̂12(ω) = 2n̂1(ω) n̂1(ω) + n̂2(ω) (7) r̂12(ω) = n̂1(ω) − n̂2(ω) n̂1(ω) + n̂2(ω) (8) where the subscripts indicate the terahertz electric field is traversing from medium 1 to medium 2. Note that these coefficients are related by t̂12(ω) = 1 + r̂12(ω). While this tutorial focuses on terahertz transmission, the reflected field is included for completeness. 123 59 Page 10 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Based on the principles of terahertz electric fields and the experimental setup (Fig. 1), the baseline Êb(ω) (Eq. 9), reference Êr(ω) (Eq. 10), and sample Ês(ω) (Eq. 11) electric fields can be derived using a modular approach. For low-loss (i.e., low absorption and scattering) materials or mixtures, the phase change due to the media boundary is much less than that due to propagation through a medium. The phase changes of the terahertz electric field at the boundaries can thus be neglected. Consequently, the Fresnel transmission coefficients can be approximated as real val- ues, t , with only the real refractive indices n considered [42, 44]. However, the phase change due to propagation in a medium cannot be ignored. Thus, complex refractive indices n̂ are continued to be implemented in the propagation factors p̂. Êb(ω) = Ê0(ω) p̂a(ω) (9) Êr(ω) = Ê0(ω) p̂a1(ω)tar(ω) p̂r(ω)tra(ω) p̂a2(ω) (10) Ês(ω) = Ê0(ω) p̂a1(ω)tas(ω) p̂s(ω)tsa(ω) p̂a3(ω) (11) For alternative experimental configurations, the expressions for Êb(ω), Êr(ω), and Ês(ω) can be derived in a similar manner. The same modular approach applies, using the emitted terahertz electric field Ê0(ω) together with the factors p̂1(ω), t̂12(ω), and r̂12(ω). The resultant expressions for Êb(ω), Êr(ω) and Ês(ω) will be different, and thus subsequent derivations of the optical constants will vary and will often be more complex. A brief example on the implications of alternative experimental setup on the optical constant derivations is provided in Sect. 8. It is therefore essential to ensure the equivalence of experimental setups before referencing and applying optical constant expressions from literature. Otherwise, erroneous results and analysis may arise. 5 Optical Constants of aWeakly Terahertz-AbsorbingMaterial Only the terahertz electric fields of the measured baseline Êb(ω) (Eq. 9) and refer- ence Êr(ω) (Eq. 10) are required for determining the optical constants of a weakly terahertz-absorbing material. The complex transfer function of the reference, T̂r/b, can be expressed in terms of its magnitude |Tr/b(ω)| and phase�φr/b(ω) (Eq. 12), or as the ratio of Êr(ω)/Êb(ω) (Eq. 13). In this tutorial, the refractive index of air na is approxi- mated as 1. This approximation assumes that the ambientmedium (e.g., nitrogen purge or dry air) has negligible absorption and dispersion in the terahertz range of interest compared to the sample. The assumption is generally valid for common experimental conditions, but should be considered if high accuracy phase measurements are critical or if significant atmospheric absorption lines are present. T̂r/b(ω) = |Tr/b(ω)| exp (i�φr/b(ω)) (12) T̂r/b(ω) = Êr(ω) Êb(ω) (13) 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 11 of 37 59 = Ê0(ω) p̂a1(ω)tar(ω) p̂r(ω)tra(ω) p̂a2(ω) Ê0(ω) p̂a(ω) (14) = tar(ω)tra(ω) p̂a1(ω) p̂a2(ω) p̂a(ω) p̂r(ω) (15) = 2na na + nr(ω) × 2nr(ω) nr(ω) + na × exp ( iωna (−dr) c0 ) × exp ( iωn̂r(ω)dr c0 ) (16) = 4nanr(ω) (na + nr(ω))2 exp (−iωnadr c0 ) exp ⎛ ⎝ iω ( nr(ω) + i αr(ω)c0 2ω ) dr c0 ⎞ ⎠ (17) = 4nanr(ω) (na + nr(ω))2 exp ( −αr(ω)dr 2 ) exp ( iω (nr(ω) − na) dr c0 ) (18) In the derivation above, it is assumed that the baseline air path da physically corresponds to the total path length occupied by the reference setup, i.e., da = da1 + dr + da2 (Fig. 1). This assumption enables the ratio of air propagation fac- tors p̂a1(ω) p̂a2(ω)/ p̂a(ω) in Eq. 15 to simplify to exp (iωna (−dr) /c0) in Eq. 16. In Eq. 17, substituting n̂r(ω) = nr(ω)+ iαr(ω)c0/2ω into the exponent yields a real part, −αrdr/2, which is responsible for exponential amplitude decay and thus absorption, and an imaginary part, iωnrdr/c0, which represents the phase shift due to propagation. The real and imaginary parts of the exponent are then combined with the air path term exp (−iωnadr/c0) to reach Eq. 18. By comparing the terms contributing to the phase and magnitude of the reference complex transfer function (Eqs. 12 and 18), the following expressions can be obtained: �φr/b(ω) = ω (nr(ω) − na) dr c0 (19) |Tr/b(ω)| = 4nanr(ω) (na + nr(ω))2 exp ( −αr(ω)dr 2 ) (20) Rearranging Eqs. 19 and 20 results in the refractive index nr(ω) and absorption coef- ficient αr(ω) of the reference, which is the weakly terahertz-absorbing material: nr(ω) = c0�φr/b(ω) drω + na (21) αr(ω) = 2 dr ln ( 4nanr(ω) |Tr/b(ω)| (na + nr(ω))2 ) (22) The optical constants discussed in this tutorial are consistently expressed in terms of their respective conventional units adopted in the scientific community. Since nr(ω) is defined as the ratio of the speed of light in a vacuum to that in a medium, which is the weakly terahertz-absorbing material in this scenario, nr(ω) is unitless (Eq. 21). The units of αr(ω) are contributed solely by the thickness term 1/dr, given logarithms must be unitless (Eq. 22). In the physics community, dr is conventionally measured 123 59 Page 12 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 in cm and so αr(ω) is correspondingly expressed in cm−1. In chemistry, the molar absorption coefficient εr(ω) is more widely adopted and is defined by dividing αr(ω) by the molar concentration of the medium, which is the weakly terahertz-absorbing material in this case. The typical units for εr(ω) are cm−1 M−1 = cm−1 mol−1 dm3. Whilst the molar absorption coefficient ε(ω) is particularly useful for liquid mixtures (Sect. 8), it is less applicable to solids, which are the main focus of this tutorial. For solidmixtures or porous solidmedia, porosity (i.e., volume fraction of air in a solid) and effectivemedium effects need to be considered.Appropriate effectivemedium theories (EMTs), as discussed in Sect. 6, should be employed to extract the optical constants of the individual solid components. Since solid materials, including the individual solid components and pure solids, are generally incompressible, their densities and hence molar concentrations are effectively constant. Consequently, the information conveyed by ε(ω) is largely equivalent to that of the absorption coefficient α(ω), but additional concentration-based normalisation is required. Nevertheless, ε(ω) is reported throughout this tutorial for completeness and for ease of comparison with related chemical literature. The expressions for the optical constants of a weakly terahertz-absorbing material are often referenced in other high-quality THz-TDS tutorials in literature. For example, Eq. 21 for nr(ω) aligns with Eq. 1 in Jepsen and Fischer [47] and Eq. 22 in Jepsen et al. [42], and Eq. 22 for αr(ω) aligns with Eq. 2 in Jepsen and Fischer [47] and Eq. 24 in Jepsen et al. [42]. This tutorial, along with the aforementioned references, uses natural logarithms for all logarithmic operations due to their convenience in manipulating exponential terms. Some communities, especially in chemistry, commonly use the decadic logarithmby convention [62, 63]. Readers are advised to verify the logarithmic base assumed in absorption coefficient derivations, as it may not always be explicitly stated in literature. The optical constant derivations presented in this section are equivalent to the conventional approach for determining the optical properties of a pure material from a single component pellet, or the optical properties of amixture, typically a binary system consisting a weakly terahertz-absorbing diluent and a strongly terahertz-absorbing inclusion [42]. 6 Optical Constants of a Strongly Terahertz-AbsorbingMaterial To determine the optical constants of a strongly terahertz-absorbingmaterial, the base- line Êb(ω), reference Êr(ω), and sample Ês(ω) electric fields are all required. As the sample pellet comprises a well-mixed binary mixture of a weakly terahertz-absorbing diluent and a strongly terahertz-absorbing inclusion, the optical constants of theweakly terahertz-absorbing diluent should first be extracted using the methodology described in Sect. 5. This enables the actual properties of the weakly terahertz-absorbing diluent to be accurately accounted for. If the weakly terahertz-absorbing diluent is in powder form, the reference pellet will be porous, resulting in a nr(ω) spectrum lower than that of the corresponding solidmaterial. Since the sameweakly terahertz-absorbing diluent is used in the sample pellet to homogeneously dilute the strongly terahertz-absorbing inclusion, the sample will similarly be porous. Despite the sample pellet predomi- 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 13 of 37 59 nantly consists of the weakly terahertz-absorbing diluent, the properties of the major component in the solid mixture may differ from its properties as a pure material. The extracted optical constants of the weakly terahertz-absorbing porous diluent there- fore provide the closest possible experimental estimate of its actual behaviour within the sample. This estimation assumes that the well-mixed binary sample pellet can be approximated as a physical mixture containing strongly terahertz-absorbing solid inclusions dispersed in a weakly terahertz-absorbing porous diluent, while neglecting any porosity changes arising from solid-state mixing. If both the reference and sample pellets are non-porous, the optical properties of the weakly terahertz-absorbing dilu- ent in its pure solid form, as determined in Sect. 5, are assumed to be equivalent to its properties in the binary sample mixture. By similarly applying the methodology from Sect. 5 but with the sample and baseline measurements considered instead, the complex transfer function and opti- cal constants of the sample can be derived as T̂s/b(ω) = 4nans(ω) (na + ns(ω))2 exp ( −αs(ω)ds 2 ) exp ( iω (ns(ω) − na) ds c0 ) (23) ns(ω) = c0�φs/b(ω) dsω + na (24) αs(ω) = 2 ds ln ( 4nans(ω) |Ts/b(ω)| (na + ns(ω))2 ) (25) For completeness, the molar absorption coefficient of the solid sample εs(ω) can be calculated from dividing αs(ω) by the molar concentration of the sample mixture. However, the implications of εs(ω) in solid-state systems, as discussed in Sect. 5, should be examined. Effective medium theories (EMTs) are then leveraged to determine the optical con- stants of the strongly terahertz-absorbing solid inclusion. EMTs are originally aimed to predict the macroscopic optical properties, such as dielectric constant, of a composite material, based on those of the constituent components and the volume fractions. It is generally assumed that the wavelength of the terahertz radiation is much greater than the size of the inclusions, and scattering can thus be neglected in EMTs. Examples of common EMTs include Maxwell-Garnett, which typically assumes spherical inclu- sions and works well for low-to-moderate inclusion concentrations, and Bruggeman, which is often preferred for higher concentrations or more symmetric mixtures [49, 55, 64, 65]. This tutorial focuses on the application of the Maxwell-Garnett equation, which is expressed in terms of the dielectric constant, ε̂(ω): ε̂(ω) = n̂(ω)2 = ( n(ω) + i α(ω)c0 2ω )2 (26) and the volumetric fraction of the strongly terahertz-absorbing inclusion ηi. Since both the reference and sample pellets share the same diameter and contain identical 123 59 Page 14 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 masses of the weakly terahertz-absorbing material, ηi can be expressed in terms of the thickness difference of the pellets (Fig. 1b and c): ηi = ds − dr ds (27) This implicitly assumes that the sample binary mixture is geometrically equivalent to two adjacent, single-component media with matching volume fractions (Fig. 4a). Whilst such geometric equivalence is valid for volumetric fraction calculations, it does not hold in EMTs. The optical properties determined from EMTs depend on the microscopic arrangement of components, which is clearly different between the two geometric configurations (Fig. 4b). The Maxwell-Garnett equation states that ε̂s(ω) − ε̂r(ω) ε̂s(ω) + 2ε̂r(ω) = ηi ε̂i(ω) − ε̂r(ω) ε̂i(ω) + 2ε̂r(ω) (28) where ε̂r, ε̂s, and ε̂i are the dielectric constants of the resulting reference pellet, sam- ple pellet, and the strongly terahertz-absorbing inclusion, respectively. The refractive index ni(ω) and absorption coefficient αi(ω) of the inclusion can then be determined by extracting the real and imaginary parts of +√ ε̂i(ω) using Eq. 26. Since the alge- braic expressions for ε̂i(ω), ni(ω), and αi(ω) are complicated, these constants are best computed numerically. If needed, the molar absorption coefficient of the solid inclu- sion εi(ω) can be similarly calculated from αi(ω) following the procedure outlined in Sect. 5, though the additional significance εi(ω) provides in solid-state systems should be evaluated (Sect. 5). The application of the Maxwell-Garnett equation accounts for dilution effects and is most appropriate when both the reference and sample pellets are non-porous. In cases where the pellets are porous, the porosity can be accounted for by applying the optical constants of the porous weakly terahertz-absorbing diluent in the Maxwell- Garnett equation, as discussed earlier in this section. Approximating the porosity of the sample pellet based on that of the diluent provides a practical solution when the true densities of the constituent materials, and thus the sample porosity, are not readily measurable. For amore detailed treatment of porosity in optical constant determination from THz-TDS measurements, readers are referred to the work of Parrott et al. [49]. The example n(ω) and α(ω) spectra of the weakly terahertz-absorbing porous polyethylene reference, the strongly terahertz-absorbing solid α-lactose monohydrate inclusion, and the well-mixed porous sample binary mixture with polyethylene and α- lactose monohydrate are shown in Fig. 5. The ε(ω) of the strongly terahertz-absorbing α-lactose monohydrate inclusion is also displayed in Fig. 5c [66]. Since the exact average molecular weight of the polyethylene is unknown, ε(ω) for the sample and reference is not appended. Whilst all derivations and optical constants in this tutorial are expressed in terms of the angular frequency ω in line with physics conventions, all experimental results are displayed in terms of the frequency ν according to common data presentation practices in the community. The resulting nr(ω) spectrum of porous polyethylene derived from Eq. 21 (Fig. 5) is lower than the 1.53 to 1.54 range of solid polyethylene reported in literature [67, 68]. This is within expectation since 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 15 of 37 59 Weakly terahertz-absorbing material Strongly terahertz-absorbing material (a) Equivalence in volumetric fraction (b) Inequivalence in effective medium theories and optical properties = ≠ Host matrix/diluentInclusion Two distinct mediaMixture Fig. 4 a Binary systems with different geometric configurations but with identical component volume fractions; b the binary systems are inequivalent in effective medium theories and have different optical properties polyethylene powder was used and the pores (i.e., air) between the powder particles contribute to a lower effective nr(ω). As discussed earlier, the experimentally derived optical constants of the porous polyethylene provide a useful source for the sample pellet porosity to be considered and approximated, enabling a more accurate optical constant extraction of the strongly terahertz-absorbing α-lactose monohydrate. The ni(ω) and αi(ω) spectra for α-lactose monohydrate (Fig. 5) align with the results presented by Parrott et al. [49], demonstrating the validity of the methodology and simplifying assumptions. Furthermore, these experimentally determined spectra of pure components, particularly the characteristic absorption peaks (Table 1), provide valuable data for material characterisation and for validating theoretical predictions of optical constants. For instance, the absorption spectra of powdered crystals can be computed from density functional theory using the PDielec Python package [69, 70]. Reporting both the full spectral data and a tabulated summary of key characteristic absorption peaks in literature, as demonstrated in this tutorial, is good practice and facilitates reproducibility and comparison with future studies. An existing approach for determining the optical constants of a strongly terahertz- absorbing inclusion involves the baseline and the sample measurements only. In this approach, ns(ω) and αs(ω) of the binary sample mixture are analogously determined by Eqs. 24 and 25. To apply EMT, approximated optical properties of the weakly terahertz-absorbing diluent are required. Since such diluents often exhibit relatively 123 59 Page 16 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 1.4 1.5 1.6 1.7 1.8 1.9 R ef ra ct iv e in de x (- ) (a) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 0 40 80 120 160 A bs or pt io n co ef fi ci en t (c m − 1 ) (b) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 0 10 20 30 40 50 M ol ar ab so rp tio n co ef fi ci en t (c m − 1 M − 1 ) (c) Strongly absorbing inclusion Sample Weakly absorbing reference Fig. 5 a Refractive index n and b absorption coefficient α spectra of the strongly terahertz-absorbing solid inclusion (α-lactosemonohydrate), the sample (porouswell-mixedbinarymixture ofα-lactosemonohydrate and polyethylene), and the weakly terahertz-absorbing reference (porous polyethylene). cMolar absorption coefficient ε spectrum of α-lactose monohydrate, the strongly terahertz-absorbing inclusion. The presented spectra were derived using T̂r/b (Eq. 18) and T̂s/b (Eq. 23). The shaded regions indicate frequencies that are beyond the dynamic range. The purple box highlights the range of nr(ω) values which are averaged to obtain the approximated nr(ω) = 1.46 Table 1 Characteristic absorption peak frequencies and corresponding optical constants of α-lactose mono- hydrate Frequency (THz) Refractive index ni(ω) (-) Absorption coefficient αi(ω) (cm−1) Molar absorption coeffi- cient εi(ω) (cm−1 M−1) 0.529 1.64 27.4 8.02 1.20 1.67 20.0 5.86 1.38 1.64 92.7 27.1 1.82 1.64 35.7 10.4 2.55 1.66 136 39.7 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 17 of 37 59 constant nr(ω) spectra (e.g., Fig. 5a) and negligible absorption (e.g., Fig. 5b), nr(ω) can be treated as a constant and αr(ω) can be reasonably approximated as 0cm−1 [44]. This allows the optical constants of the strongly terahertz-absorbing inclusion to be extracted via EMT. In this example, the experimentally averaged nr = 1.46 for porous polyethylene (Fig. 5) and different nr estimates are applied to evaluate their impact on the derivedoptical constantsni(ω),αi(ω), and εi(ω)of the strongly terahertz-absorbing α-lactose monohydrate inclusion (Fig. 6). When the experimentally averaged nr = 1.46 is used, the resultant ni(ω) spectrawithin the dynamic range generally agreeswith that obtained from the direct comprehensive method presented earlier in this section (Fig. 6). A slight 0.04 to 0.06 deviation (i.e., nr = 1.40, 1.50) from the experimental nr average already yields ni(ω) spectra that are beyond reasonable and lie outside the limits of Fig. 6a, not to mention the results from the estimates of nr = 1.30 and 1.60, which are aimed to examine sensitivity and limits. For αi(ω) and εi(ω), the spectra derived using the experimentally averaged nr = 1.46 generally agree with those from the direct comprehensive method up to approximately 1.5 THz, beyond which they begin to diverge. A slightly higher estimate of nr = 1.50 results in considerable overestimation of both αi(ω) and εi(ω), while all other nr estimates yield unreasonable spectra. These results indicate that when the diluent is porous, as in the example of powdered polyethylene with nr = 1.46, using nr estimates based on the solid material (e.g., nr = 1.53 to 1.54 for solid polyethylene [67, 68]) can lead to inaccurate optical properties of the inclusion. With an accurate and reliable nr estimate, the baseline and sample measurement approach remains useful in providing low-frequency spectral information with adequate precision. However, for highly accurate optical properties, the presented direct comprehensive approach using baseline, reference, and sample measurements is recommended. 7 Optical Constants of a Thick Sample When the reference and sample pellets are very thick, some terahertz spectrometers may not have the necessary time-delay range to consistently acquire all the baseline, reference, and sample measurements. In such cases, it is only feasible to acquire the reference and sample measurements. These two measurements should have a more comparable time delay (Fig. 2) and should fit within the limited time-delay range. The complex transfer function of the sample is thus alternatively defined with respect to the reference. Again, based on the experimental setup in Fig. 1 and applying the methodology in Sect. 5, the complex transfer function is now expressed as T̂s/r(ω) = |Ts/r(ω)| exp (i�φs/r(ω)) (29) T̂s/r(ω) = Ês(ω) Êr(ω) (30) = tas(ω)tsa(ω) tar(ω)tra(ω) × p̂a3(ω) p̂a2(ω) × p̂s(ω) × 1 p̂r(ω) (31) = ns(ω)(na + nr(ω))2 nr(ω)(na + ns(ω))2 exp ( iωna (dr − ds) c0 ) exp ( iωn̂sds c0 ) exp (−iωn̂rdr c0 ) (32) 123 59 Page 18 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 1.4 1.5 1.6 1.7 1.8 1.9 R ef ra ct iv e in de x i (- ) (a) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 0 40 80 120 160 A bs or pt io n co ef fi ci en t i (c m − 1 ) (b) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 0 10 20 30 40 50 M ol ar ab so rp tio n co ef fi ci en t i (c m − 1 M − 1 ) (c) Strongly terahertz-absorbing inclusion, derived from ŝ/b( ), r̂/b( ) ŝ/b( ), r = 0 cm−1, r = 1.46 Strongly terahertz-absorbing inclusion, derived from ŝ/b( ), r = 0 cm−1 and r = 1.30 r = 1.40 r = 1.50 r = 1.60 Strongly terahertz-absorbing inclusion, derived from ŝ/b( ), r̂/b( ) ŝ/b( ), r = 0 cm−1, r = 1.46 Strongly terahertz-absorbing inclusion, derived from ŝ/b( ), r = 0 cm−1 and r = 1.30 r = 1.40 r = 1.50 r = 1.60 Fig. 6 a Refractive index ni, b absorption coefficient αi, and c molar absorption coefficient εi spectra of the strongly terahertz-absorbing inclusion, α-lactose monohydrate, determined from baseline and sample measurements only using different nr estimates (including the experimentally averaged nr = 1.46 of porous polyethylene (Fig. 5)) and αr = 0cm−1. The results are compared against the spectra derived from T̂s/b(ω) and T̂r/b(ω) using experimental data only (light turquoise solid line). The shaded regions indicate frequencies that are beyond the dynamic range 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 19 of 37 59 = ns(ω)(na + nr(ω))2 nr(ω)(na + ns(ω))2 exp ( αr(ω)dr − αs(ω)ds 2 ) exp ( iω dr(na − nr(ω)) + ds(ns(ω) − na) c0 ) (33) It should be noted that the simplification of the air path propagation terms p̂a3(ω)/ p̂a2(ω) (Eqs. 31 to 32) relies on geometrical assumptions about the exper- imental setup, typically implying da3 − da2 = dr − ds (Fig. 1). Similarly, by comparing like terms in Eqs. 29 and 33, ns(ω) and αs(ω) of the sample can be derived as ns(ω) = c0�φs/r(ω) dsω + dr(nr(ω) − na) ds + na (34) αs(ω) = αr(ω)dr ds + 2 ds ln ( ns(ω)(na + nr(ω))2 |Ts/r(ω)| nr(ω)(na + ns(ω))2 ) (35) Again, themolar absorption coefficient of the sample εs(ω) can be obtained by dividing αs(ω) by the molar concentration of the sample mixture, although its relevance to solid-state analysis is often limited (Sect. 5). With the same rationale as Sect. 6, nr(ω) and αr(ω) of the weakly terahertz- absorbing diluent are approximated as a constant and as 0cm−1, respectively. The Maxwell-Garnett effective medium theory can then be similarly applied as in Eqs. 27 to 28 (Sect. 6) to determine ni(ω) and αi(ω). The molar absorption coefficient of the solid inclusion εi(ω) can then be similarly determined according to Sect. 5; however, its applicability to solid-state systems should be carefully considered (Sect. 5). In this example, the nr(ω) spectrum of the weakly terahertz-absorbing porous polyethylene in Fig. 5a is again averaged to obtain the nr(ω) = 1.46 approxima- tion. The value accounts for the porosity of the reference polyethylene pellet, which is also taken as an approximation of the sample pellet porosity, as discussed in Sect. 6. With accurate nr and αr approximations of the weakly terahertz-absorbing polyethy- lene, the resultant optical constants of the sample and the strongly terahertz-absorbing α-lactose monohydrate inclusion, ns(ω), ni(ω), αi(ω), and εi(ω), are nearly identical to those derived from the direct comprehensive method using experimentally deter- mined T̂r/b(ω) and T̂s/b(ω) in Sect. 6 (Figs. 7 and 8). An exception arises in the αs(ω) spectrum, where the αr = 0 cm−1 approximation leads to the omission of the term αr(ω)dr/ds in Eq. 35, resulting in the observed discrepancy in Fig. 7. The discrepancy becomes more pronounced beyond 1.75 THz, where baseline absorption in the exper- imental αr(ω) spectra increase. Nevertheless, the deviations are low in magnitude and have negligible effects on the optical constants of the inclusion, which is of main interest. The impact of inaccurate nr estimations on the resultant spectra is most evident in ns(ω), which closely follows the estimated nr (Fig. 9a). This is because the sample pellet predominantly consists of the same weakly terahertz-absorbing material as the reference pellet. The similarity between nr and ns(ω) enables the inaccuracies in nr estimation to effectively cancel out in Eq. 35, rendering αs(ω) largely invariant to different nr estimations. However, a minor discrepancy in αs(ω) remains due to the αr = 0 cm−1 approximation (Fig. 9b), as discussed earlier in this section. For the 123 59 Page 20 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 1.3 1.4 1.5 1.6 1.7 R ef ra ct iv e in de x s (- ) (a) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) −10 −5 0 5 10 15 20 A bs or pt io n co ef fi ci en t s (c m − 1 ) (b) Sample, derived from ŝ/r ( ), r = 1.46, r = 0 cm−1 ŝ/r ( ), r̂/b( ) ŝ/b( ), r̂/b( ) Fig. 7 aRefractive indexns andb absorption coefficientαs spectra, calculated via three differentmethods, of the porous well-mixed sample consisting weakly terahertz-absorbing polythethylene and strongly terahertz- absorbing α-lactose monohydrate. The spectra derived from T̂s/r(ω) and T̂r/b(ω) (orange dashed line) are the same as that derived from T̂s/b(ω) and T̂r/b(ω) (dark turquoise dotted line), which is also presented in Fig. 5. The shaded regions indicate frequencies that are beyond the dynamic range strongly terahertz-absorbing inclusion’s ni(ω) (Fig. 10a), αi(ω) (Fig. 10b), and εi(ω) (Fig. 10c), the influence of an inaccurate nr estimation is mitigated by the Maxwell- Garnett equation (Eq. 28), which comprises a ratio involving the inaccurate reference optical constants and the correspondingly offset sample spectra. An underestimation of nr causes a slightlymore pronounced deviation in ni(ω) compared to an overestima- tion of a similar magnitude (Fig. 10a). A similar trend is observed for the characteristic absorption peaks in αi(ω) (Fig. 10b) and εi(ω) (Fig. 10c). Deviations in the absorp- tion baselines are relatively minor in comparison to the shifts observed in the ni(ω) spectra (Fig. 10a), particularly when evaluated on the scale of αi(ω) and εi(ω). In contrast to the approach involving only baseline and sample measurements (Fig. 6), the method presented in this section, relying solely on the reference and sample mea- surements, exhibits significantly greater robustness against inaccurate nr estimates. Only minor or negligible systematic offsets are observed in the resulting ni(ω), αi(ω), 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 21 of 37 59 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 1.4 1.5 1.6 1.7 1.8 1.9 R ef ra ct iv e in de x i (- ) (a) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 0 40 80 120 160 A bs or pt io n co ef fi ci en t i (c m − 1 ) (b) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 0 10 20 30 40 50 M ol ar ab so rp tio n co ef fi ci en t i (c m − 1 M − 1 ) (c) Strongly terahertz-absorbing inclusion, derived from ŝ/r ( ), r = 1.46, r = 0 cm−1 ŝ/r ( ), r̂/b( ) ŝ/b( ), r̂/b( ) Fig. 8 a Refractive index ni, b absorption coefficient αi, and c molar absorption coefficient εi spectra, calculated via three different methods, of the strongly terahertz-absorbing inclusion α-lactose monohydrate. The spectra derived from T̂s/r(ω) and T̂r/b(ω) (light orange dashed line) are the same as that derived from T̂s/b(ω) and T̂r/b(ω) (light turquoise dotted line), which is also presented in Fig. 5. The shaded regions indicate frequencies that are beyond the dynamic range and εi(ω) spectra (Fig. 10). This demonstrates that performing reference and sample measurements, instead of the baseline and sample measurements, is a more reliable and optimal approach when the optical properties of the weakly terahertz-absorbing diluent need to be estimated. Nevertheless, the actual optical constants of the weakly terahertz-absorbing material and the effects of dilution are best addressed when an accurate estimate of nr, or ideally, the nr(ω) and αr(ω) spectra are used (Sect. 6). The nr(ω) and αr(ω) spectra should therefore be prioritised when determining the optical constants of the strongly terahertz-absorbing inclusion, particularly in applications where accuracy is critical. 123 59 Page 22 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 1.3 1.4 1.5 1.6 1.7 R ef ra ct iv e in de x s (- ) (a) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) −10 −5 0 5 10 15 20 A bs or pt io n co ef fi ci en t s (c m − 1 ) (b) Sample, derived from ŝ/b( ), r̂/b( ) ŝ/r ( ), r = 0 cm−1, r = 1.46 Sample, derived from ŝ/r ( ), r = 0 cm−1 and r = 1.30 r = 1.40 r = 1.50 r = 1.60 Sample, derived from ŝ/b( ), r̂/b( ) ŝ/r ( ), r = 0 cm−1, r = 1.46 Sample, derived from ŝ/r ( ), r = 0 cm−1 and r = 1.30 r = 1.40 r = 1.50 r = 1.60 Fig. 9 a Refractive index ns and b absorption coefficient αs spectra of the porous sample mixture resulted from different nr estimates and αr = 0 cm−1. The porous sample contains a well-mixed binary mixture of α-lactose monohydrate and polyethylene. The results are compared against the spectra derived from T̂s/b(ω) and T̂r/b(ω) using experimental data only (dark turquoise solid line), and another spectra derived from T̂s/r(ω), the experimentally averaged nr = 1.46 of porous polyethylene (Fig. 5) and αr = 0 cm−1 (dark purple dashed line). The shaded regions indicate frequencies that are beyond the dynamic range When the actual reference spectra nr(ω) (Eq. 21) and αr(ω) (Eq. 22) derived from T̂r/b are instead used in Eqs. 34 and 35, the calculated sample ns(ω) and αs(ω) are equivalent to those derived using Eqs. 24 and 25 (Fig. 7). The same would apply to εs(ω). Logically, the equivalence extends to the optical constants of the inclusion, ni(ω), αi(ω), and εi(ω) (Fig. 8). Similarly, irrespective of the sample complex transfer function derivations, T̂s/b(ω) (Eq. 24) or T̂s/r(ω) (Eq. 33), the resultant optical constants of the sample (Fig. 7) and inclusion (Fig. 8) remain the same. 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 23 of 37 59 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 1.4 1.5 1.6 1.7 1.8 1.9 R ef ra ct iv e in de x i (- ) (a) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 0 40 80 120 160 A bs or pt io n co ef fi ci en t i (c m − 1 ) (b) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Frequency (THz) 0 10 20 30 40 50 M ol ar ab so rp tio n co ef fi ci en t i (c m − 1 M − 1 ) (c) Strongly terahertz-absorbing inclusion, derived from ŝ/b( ), r̂/b( ) ŝ/r ( ), r = 0 cm−1, r = 1.46 Strongly terahertz-absorbing inclusion, derived from ŝ/r ( ), r = 0 cm−1 and r = 1.30 r = 1.40 r = 1.50 r = 1.60 Strongly terahertz-absorbing inclusion, derived from ŝ/b( ), r̂/b( ) ŝ/r ( ), r = 0 cm−1, r = 1.46 Strongly terahertz-absorbing inclusion, derived from ŝ/r ( ), r = 0 cm−1 and r = 1.30 r = 1.40 r = 1.50 r = 1.60 Fig. 10 aRefractive index ni, b absorption coefficient αi, and cmolar absorption coefficient εi spectra of the strongly terahertz-absorbing inclusion,α-lactosemonohydrate, resulted fromdifferentnr estimates andαr = 0 cm−1. The results are compared against the spectra derived from T̂s/b(ω) and T̂r/b(ω) using experimental data only (light turquoise solid line), and another spectra derived from T̂s/r(ω), the experimentally averaged nr = 1.46 of porous polyethylene (Fig. 5) and αr = 0 cm−1 (light purple dashed line). The shaded regions indicate frequencies that are beyond the dynamic range 8 Optical Constants Derived from Alternative Experimental Configurations The modular approach for deriving terahertz electric field expressions, detailed in Sect. 4, can be readily adapted to alternative experimental configurations. Conse- quently, the resulting optical constant derivations will also vary with the experimental setup and often involve increased complexity. 123 59 Page 24 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 This is demonstrated through a common experimental configuration for a liquid sample held in a cuvette made of quartz or weakly terahertz-absorbing polymer (Fig. 11). The baseline measurement remains to involve an empty terahertz beam path (Fig. 11a), so the baseline terahertz electric field Êb(ω) is the same as Eq. 9: Êb(ω) = Ê0(ω) p̂a(ω) (36) The reference measurement consists of the empty cuvette, and so the terahertz beam passes through the two parallel windows of the cuvette, separated by an air gap (Fig. 11b). By applying the modular approach (Sect. 4), the reference terahertz electric field Êr,liq(ω) can be derived. Similar to Sect. 4, the Fresnel transmission coefficients are continued to be approximated as real, t , due to the phase change at media boundary is much less than that in the propagation through a medium. Êr,liq(ω) = Ê0(ω) p̂a4(ω)taw(ω) p̂w(ω)twa(ω) p̂a5(ω)taw(ω) p̂w(ω)twa(ω) p̂a6(ω) (37) It is assumed here that the two parallel windows of the cuvette are of equal thickness (Fig. 11). It is additionally assumed that the thickness of the cuvette windows dw and cuvette path length da5 = ds,liq are large (Fig. 11), so any Fabry-Pérot reflections can be excluded by apodisation during data processing (Sect. 3). The liquid sample is then placed in the cuvette for the sample measurement (Fig. 11c), and the sample terahertz electric field Ês,liq(ω) can be similarly derived with the same assumptions which are made in Êr,liq(ω) (Eq. 37): Ês,liq(ω) = Ê0(ω) p̂a4(ω)taw(ω) p̂w(ω)tws(ω) p̂s,liq(ω)tsw(ω) p̂w(ω)twa(ω) p̂a6(ω) (38) Next, the complex transfer function of the reference T̂r,liq/b is derived following the same methodology outlined in Sect. 5: T̂r,liq/b(ω) = ∣∣Tr,liq/b(ω) ∣∣ exp ( i�φr,liq/b(ω) ) (39) T̂r,liq/b(ω) = Êr,liq(ω) Êb(ω) (40) = taw(ω)2twa(ω)2 p̂a4(ω) p̂a5(ω) p̂a6(ω) p̂a(ω) p̂w(ω)2 (41) = 16n2anw(ω)2 (na + nw(ω))4 exp (−i2ωnadw c0 ) exp ⎛ ⎝ i2ω ( nw(ω) + i αw(ω)c0 2ω ) dw c0 ⎞ ⎠ (42) = 16n2anw(ω)2 (na + nw(ω))4 exp (−αw(ω)dw) exp ( i2ω (nw(ω) − na) dw c0 ) (43) The simplification of the air path terms p̂a4(ω) p̂a5(ω) p̂a6(ω)/ p̂a(ω) (Eqs. 41 to 42) is similarly based on the geometrical assumption about the experimental setup, where da = da4 + da5 + da6 + 2dw (Fig. 11). 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 25 of 37 59 (a) Baseline (b) Reference (c) Sample , ̂ ̂ ̂̂ , ̂ ̂ , Weakly-terahertz absorbing window Strongly terahertz-absorbing liquid Positive direction ̂ ̂ ̂ ̂ , ̂ ̂̂ ̂ ̂ ̂ ̂ ̂ ̂ Fabry-Pérot reflections Fabry-Pérot reflections Fig. 11 Experimental setup of the a baseline, b reference, and c sample measurements for liquid sample analysis in terahertz time-domain spectroscopy. Examples of possible Fabry-Pérot reflections are indicated in b and c. For display purposes, the illustrated Fabry-Pérot reflections are not geometrically accurate 123 59 Page 26 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 The refractive index nw(ω) and absorption coefficient αw(ω) of the cuvette window can be derived by comparing like terms in Eqs. 39 and 43: nw(ω) = c0�φr,liq/b(ω) 2dwω + na (44) αw(ω) = 1 dw ln ( 16n2anw(ω)2∣∣Tr,liq/b(ω) ∣∣ (na + nw(ω))4 ) (45) For the sample complex transfer function, the geometry of the experimental config- uration (Fig. 11) and the terahertz electric field expressions (Eqs. 36 to 38) suggest it can be more simply derived when defined with respect to the reference, again applying the methodology demonstrated in Sect. 5: T̂s,liq/r,liq(ω) = ∣∣Ts,liq/r,liq(ω) ∣∣ exp ( i�φs,liq/r,liq(ω) ) (46) T̂s,liq/r,liq(ω) = Ês,liq(ω) Êr,liq(ω) (47) = tws(ω)tsw(ω) twa(ω)taw(ω) × 1 p̂a5(ω) × p̂s,liq(ω) (48) = ns,liq(ω)(na + nw(ω))2 na(nw(ω) + ns,liq(ω))2 exp (−iωnads,liq c0 ) exp ⎛ ⎜⎝ iω ( ns,liq(ω) + i αs,liq(ω)c0 2ω ) ds,liq c0 ⎞ ⎟⎠ (49) = ns,liq(ω)(na + nw(ω))2 na(nw(ω) + ns,liq(ω))2 exp ( −αs,liq(ω)ds,liq 2 ) exp ( iω ( ns,liq(ω) − na ) ds,liq c0 ) (50) The air path propagation term 1/ p̂a5(ω) leverages the geometrical assumption of da5 = ds,liq from the experimental setup (Fig. 11, Eqs. 48 to 49). The refractive index ns,liq(ω) and absorption coefficient αs,liq(ω) of the liquid sam- ple are then derived by comparing like terms in Eqs. 46 and 50: ns,liq(ω) = c0�φs,liq/r,liq(ω) ds,liqω + na (51) αs,liq(ω) = 2 ds,liq ln ( ns,liq(ω)(na + nw(ω))2∣∣Ts,liq/r,liq(ω) ∣∣ na(nw(ω) + ns,liq(ω))2 ) (52) The molar absorption coefficient εs,liq(ω) can be similarly determined by dividing αs,liq(ω)by themolar concentration of the terahertz-absorbing solute of interest. Liquid samples typically form a single homogeneous phase, where the solute is dissolved in a solvent or a multi-component liquid system. Since the absorption characteristics of the solutewithin a specific liquid environment are commonlyof interest, the concentration- 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 27 of 37 59 independent quantity εs,liq(ω) serves as a valuable parameter for characterisation, analysis, and comparison across different systems or studies. As demonstrated, when the experimental configurations are altered, the resultant optical constant derivations of the respective weakly terahertz-absorbing reference (Eq. 21 versus Eq. 44; Eq. 22 versus Eq. 45) and strongly terahertz-absorbing sample (Eq. 24 or Eq. 34 versus Eq. 51; Eq. 25 or Eq. 35 versus Eq. 52) may no longer share the same mathematical form. The approach to derivations, for example, the sample complex transfer function (Eq. 47), should also be adjusted for simplicity. Despite the derivation process for the liquid sample experimental configuration is slightly more complex than that of the solid sample (Sects. 5 and 6), relatively simple optical constant expressions result (Eqs. 44, 45, 51, and 52) when Fabry-Pérot reflections can be neglected. However, when the cuvette windows and/or the liquid sample are thin (i.e., low dw and/or ds,liq), significant Fabry-Pérot reflections can occur (Fig. 11b and c). These reflections may not be well separated from the primary pulse in the time domain and may not be effectively suppressed or excluded by the apodisation function during data processing. Consequently, the Fabry-Pérot reflections must be explicitly accounted for in the derivations. Whilst the derivations will still follow the same principle of relating the complex transfer function’s magnitude and phase to the material properties, the resultant equations are often more complex and may require numerical methods or iterative fitting for accurate solutions [42, 60, 61]. Software packages such as THzPy [58] may, in future, include pre-built models for standard geometries like cuvettes. 9 Conclusion Terahertz time-domain spectroscopy (THz-TDS) is a non-destructive technique that simultaneously measures the amplitude and phase of electromagnetic waves in the 0.1THz to 10THz frequency range, enabling the direct determination of a mate- rial’s frequency-dependent refractive index n(ω) and absorption coefficient α(ω) [1]. Accelerated by advancements in instrumentation, THz-TDS demonstrates strong capability in a wide range of applications, including in novel material characteri- sation [6–15], pharmaceutical manufacturing [16, 20, 21, 24–26], and biomedical diagnostics [27–29]. Leveraging the extended time-delay range provided by state-of- the-art spectrometers, this tutorial presents an up-to-date, accurate, and straightforward methodology for determining n(ω) and α(ω) for strongly terahertz-absorbing mate- rials. The methodology complements conventional approaches, which are compared to and discussed in this tutorial. Both the conventional and updated comprehensive methodologies for deriving optical constants of solid materials are implemented in the publicly available THzPy Python package [58] and CaTSper, the code-free graphi- cal user interface THz-TDS analysis tool [59]. This tutorial, THzPy and CaTSper, complements the existing literature collection of excellent THz-TDS reviews [1, 42], tutorials [43–52, 54, 55], and open-access resources available through the dotTHz project [56]. To accurately determine the optical constants of a strongly terahertz-absorbing material, baseline, reference, and sample THz-TDSmeasurements should be acquired. 123 59 Page 28 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 The baseline is simply an empty THz-TDS measurement. The reference should only contain a weakly terahertz-absorbing material, while the sample should consist of a well-mixed binarymixture of the same amount of weakly terahertz-absorbingmaterial and a small amount of strongly terahertz-absorbing material. This tutorial introduces the concept of half-width, which defines the time span over which half of the sym- metric apodisation function is applied, to enable simple and consistent THz-TDS data processing and fast Fourier transform into the frequency domain. The half-width enables the systematic application of the apodisation function to each THz-TDS mea- surement and the automatic specification of a consistent time-delay range for the fast Fourier transform. Based on the principles of terahertz electric field propagation and transmission, complex transfer functions for the reference and sample with respect to the baseline are derived, enabling the extraction of their n(ω) and α(ω). The optical constants of the weakly terahertz-absorbingmedium, deduced from the reference, pro- vide the closest experimental approximation of its properties within the sample and, if relevant, also account for the effects of sample porosity by approximation. Effec- tive medium theories, such as the Maxwell-Garnett equation, can then be utilised to accurately determine the optical constants, ni(ω) and αi(ω), of the strongly terahertz- absorbing material from the binary sample mixture. For thick samples or spectrometers with limited time-delay ranges, THz-TDSmea- surements can be reduced to include only the reference and sample. An accurate approximation of the weakly terahertz-absorbing reference material’s refractive index nr is crucial to reflect its actual optical properties and the effects of dilution when the strongly terahertz-absorbing material’s refractive index ni(ω) and absorption coeffi- cient αi(ω) are determined from the binary sample mixture. If an accurate value of nr is unavailable, a reasonable estimate will suffice, but a slight systematic offset may be introduced in the ni(ω) and αi(ω) spectra. This offset is more pronounced with underestimations of nr than overestimations of similar magnitude. Nonetheless, the magnitude of the resultant offsets is significantly lower than those from an alternative approach using baseline and sample measurements only, when the same inaccurate nr estimate is applied. This demonstrates the significantly greater robustness of the pre- sented updated methodology. It should also be noted that for quantitative applications, beyond applying the methodology described here, a further important step is to assess uncertainty in the derived optical constants by propagating errors from parameters such as thickness, measurement noise levels, and phase. The experimental setup demonstrated in this tutorial is particularly suited for strongly terahertz-absorbingmaterials. Alternative setups will result in different math- ematical descriptions of the acquired terahertz electric field, but the same systematic approach can be used to derive a material’s optical constants. Supplementary Information This section outlines the practical implementation of the optical constant determination workflow described in this tutorial using software tools from the dotTHz project. Example scripts and data, including minimal examples, are available in the software 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 29 of 37 59 repositories. The specific raw data and source code, which are used for data processing and analysis in this tutorial, are available in a separate repository (please see details in Data and Code Availability section). Workflow Using the THzPy Python Library The analysis can be performedprogrammatically using theTHzPyPython library [58], available from the dotTHzTAG GitHub repository [57]. Ensure you have a working Python environment with THzPy installed (e.g., via pip install thzpy). The typical workflow mirrors the sections of this tutorial. 1. Load data Load the time-domain data for the baseline, reference, and sample measurements using appropriate THzPy functions. Data held in the .thz format may be loaded using the built-in DotthzFile object and accessed by name in the followingmanner: Data may also be loaded from other formats (e.g., a .csv file). In these cases, the data should be formatted as a 2d array of shape (2, n) with field data in the first row and time data in the second row. The library will attempt to fix incorrectly formatted data, but this may not always work. 2. Apply apodisation Apodisation can be done using several common window functions. For single waveforms, a window function is applied as follows: waveform_windowed = window(waveform, half_width=15, win_func="hanning"). Where the sameapodisation is needed formultiplewaveforms, the common_window function is provided for convenience: Currently, the following window functions are supported: boxcar, bartlett, black- man, hamming, hanning. 3. Calculate optical constants • ThzPyprovides functions for different sample geometries. Thedocumentation should be consulted to identify the appropriate function for a given use case. • The functions perform phase unwrapping using Jepsen’s method [54] before applying the appropriate transfer function. 123 59 Page 30 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Fig. 12 Time domain tab in CaTSper graphical user interface. The highlighted regions are numbered according to the steps outlined in the CaTSper workflow • The functions simplify the analysis process to the input of a few key function parameters, an example for a binary mixture is presented here: 4. Visualise results Results are output as either a 2d array of the complex refractive index as a function of frequency or as a dictionary of a full selection of optical constants, depending on the value of the all_optical_constants parameter. These can then be easily plotted using a plotting library such as Matplotlib. Please refer to the THzPy documentation for specific function names, parameters, and detailed examples. 123 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 31 of 37 59 Fig. 13 Frequency domain tab in CaTSper graphical user interface. The highlighted regions are numbered according to the steps outlined in the CaTSper workflow Workflow Using the CaTSper Graphical User Interface For users preferring a graphical user interface (GUI), the CaTSper tool [59] pro- vides access to the underlying THzPy [58] functionalities. The steps generally involve (Figs. 12 and 13): 1. Load data Click the “Import THz Files” button to launch a file dialog where .thz files can be selected, then press deploy to load the data from the selected file (Fig. 12). Clear memory will clear the program, removing the data from previously loaded files. 2. Select data Measurements loaded from selected files will be displayed in the measurements list and can be added to the selection list for plotting and processing. 3. View waveforms Two plot windows are available in the “Time Domain” tab. The measurements in the selection list will be displayed on each plot when the relevant button is clicked. The selection of waveforms to be plotted from eachmeasurement can be controlled with the checkbox in GUI region 2 (Fig. 12). The colour map and legend can be changed, and the plot can be exported either as an image or as data. 123 59 Page 32 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 4. View/edit metadata Themetadata of the measurement currently highlighted in the measurements list is displayed. The software will attempt to identify thickness metadata automatically, but this can be adjusted manually. Metadata values can be edited if required. 5. Specify datasets The software will attempt to determine which datasets in the loaded measure- ments correspond to sample/reference/baseline; these can be adjusted manually if required. 6. Define window function The half-width and window function are specified here. 7. Specify transform settings The transfer function and FFT settings are specified here. The “Transform” buttons will then apply the settings to the selectedmeasurements and calculate their optical properties, which will be added to the frequency domain tab (Fig. 13). 8. Select data Measurements transformed in the previous stepwill be shown in themeasurements list in the frequency domain tab (Fig. 13). They can then be added to the selection list for plotting. 9. Specify constant A range of optical constants are calculated during the transform process. The optical constant to be plotted can be selected by clicking the relevant button. 10. Set plot settings Set plot options including waveform selection (FFT Amplitude only), linear/log y-axis, and real/imaginary values. The colour map can also be selected, and the legend toggled. 11. View data Similar to Step 3, data is displayed in two plotting environments with multiple options for export. Acknowledgements CKLwould like to acknowledgeResearchCenter Pharmaceutical EngineeringGmbH, Austria for PhD funding. JNW-Bwould like to acknowledgeAstraZeneca,UK for PhD funding. EWDwould like to acknowledge the Cambridge Trust and the Swiss Benevolent Society for PhD funding. JL would like to acknowledge GSK, UK for PhD funding. The authors would like to thank Prof. Timothy M. Korter and Salvatore Zarrella for their valuable feedback. Author Contributions CKL: conceptualization, methodology, software, formal analysis, investigation, data curation, validation, writing—original draft, writing—review and editing, visualisation. JNW-B: data cura- tion, software, validation, writing—review and editing. EWD: methodology, validation, writing—review and editing. JL: software, writing—review and editing. JAZ: writing—review and editing, supervision, project administration, funding acquisition. Funding CKL acknowledges Research Center Pharmaceutical Engineering GmbH, Austria for PhD fund- ing. JNW-B acknowledges AstraZeneca, UK for PhD funding. EWD acknowledges the Cambridge Trust and the Swiss Benevolent Society for PhD funding. JL acknowledges GSK, UK for PhD funding. Data Availability The raw data involved in this tutorial is available here as a ∗.thz file: https://doi.org/10. 17863/CAM.120263.2. 123 https://doi.org/10.17863/CAM.120263.2 https://doi.org/10.17863/CAM.120263.2 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 33 of 37 59 Code Availability The methodology presented in this tutorial is available in the open-source THzPy Python package and CaTSper graphical user interface, as part of the dotTHz project. The source code written specifically for this tutorial is available here: https://doi.org/10.17863/CAM.120263.2. Declarations Ethics Approval Not applicable. Consent to Participate Not applicable. Consent for Publication All the authors agreed to publish. Conflict of Interest JAZ is a member of the editorial board of JIMT. OpenAccess This article is licensedunder aCreativeCommonsAttribution 4.0 InternationalLicense,which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Koch, M., Mittleman, D.M., Ornik, J., Castro-Camus, E.: Terahertz time-domain spectroscopy. Nature Reviews Methods Primers 3(1), 48 (2023) https://doi.org/10.1038/s43586-023-00232-z 2. Smith, P.R., Auston, D.H., Nuss,M.C.: Subpicosecond photoconducting dipole antennas. IEEE Journal of Quantum Electronics 24(2), 255–260 (1988) https://doi.org/10.1109/3.121 3. Van Exter, M., Fattinger, C., Grischkowsky, D.: Terahertz time-domain spectroscopy of water vapor. Optics letters 14(20), 1128–1130 (1989) https://doi.org/10.1364/OL.14.001128 4. Naftaly, M., Vieweg, N., Deninger, A.: Industrial applications of terahertz sensing: State of play. Sensors 19(19), 4203 (2019) https://doi.org/10.3390/s19194203 5. Leitenstorfer, A., Moskalenko, A.S., Kampfrath, T., Kono, J., Castro-Camus, E., Peng, K., Qureshi, N., Turchinovich, D., Tanaka, K., Markelz, A.G., et al: The 2023 terahertz science and technology roadmap. Journal of Physics D: Applied Physics 56(22), 223001 (2023) https://doi.org/10.1088/1361- 6463/acbe4c 6. Ulbricht, R., Hendry, E., Shan, J., Heinz, T.F., Bonn, M.: Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy. Reviews of Modern Physics 83(2), 543–586 (2011) https:// doi.org/10.1103/RevModPhys.83.543 7. Joyce, H.J., Boland, J.L., Davies, C.L., Baig, S.A., Johnston,M.B.: A review of the electrical properties of semiconductor nanowires: insights gained from terahertz conductivity spectroscopy. Semiconductor Science and Technology 31(10), 103003 (2016) https://doi.org/10.1088/0268-1242/31/10/103003 8. Wang, Y., Zhang, T., Ma, K., Bin, Z., Zhang, X., Tang, F., Xu, X., Yin, T., Hu,M.: Terahertz Nanoscopy on Low-Dimensional Materials: Toward Ultrafast Physical Phenomena. ACS Applied Materials & Interfaces (2025) https://doi.org/10.1021/acsami.4c14419 9. Émond, N., Torriss, B., Morris, D., Chaker, M.: Natural metamaterial behavior across the phase tran- sition for WxV(1−x)O2 films revealed by terahertz spectroscopy. Acta Materialia 140, 20–30 (2017) https://doi.org/10.1016/j.actamat.2017.08.029 10. Ryder, M.R., Voorde, B., Civalleri, B., Bennett, T.D., Mukhopadhyay, S., Cinque, G., Fernandez- Alonso, F., De Vos, D., Rudić, S., Tan, J.-C.: Detecting molecular rotational dynamics complementing the low-frequency terahertz vibrations in a zirconium-basedmetal-organic framework. PhysicalReview Letters 118(25), 255502 (2017) https://doi.org/10.1103/PhysRevLett.118.255502 123 https://doi.org/10.17863/CAM.120263.2 http://creativecommons.org/licenses/by/4.0/ https://doi.org/10.1038/s43586-023-00232-z https://doi.org/10.1109/3.121 https://doi.org/10.1364/OL.14.001128 https://doi.org/10.3390/s19194203 https://doi.org/10.1088/1361-6463/acbe4c https://doi.org/10.1088/1361-6463/acbe4c https://doi.org/10.1103/RevModPhys.83.543 https://doi.org/10.1103/RevModPhys.83.543 https://doi.org/10.1088/0268-1242/31/10/103003 https://doi.org/10.1021/acsami.4c14419 https://doi.org/10.1016/j.actamat.2017.08.029 https://doi.org/10.1103/PhysRevLett.118.255502 59 Page 34 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 11. Pattengale, B., Neu, J., Ostresh, S., Hu, G., Spies, J.A., Okabe, R., Brudvig, G.W., Schmuttenmaer, C.A.: Metal–organic framework photoconductivity via time-resolved terahertz spectroscopy. Journal of the American Chemical Society 141(25), 9793–9797 (2019) https://doi.org/10.1021/jacs.9b04338 12. Wahaia, F., Kašalynas, I., Pashnev, D., Valušis, G., Urbanowicz, A., Karaliunas, M., Singh, D.P., Wallentowitz, S., Seifert, B.: Optical properties of millimeter-size metal-organic framework single crystals using THz techniques. Journal of Molecular Structure 1322, 140612 (2025) https://doi.org/ 10.1016/j.molstruc.2024.140612 13. Zhao, D., Chia, E.E.: Free carrier, exciton, and phonon dynamics in lead-halide perovskites studied with ultrafast terahertz spectroscopy. Advanced Optical Materials 8(3), 1900783 (2020) https://doi. org/10.1002/adom.201900783 14. Ulatowski, A.M., Farrar, M.D., Snaith, H.J., Johnston, M.B., Herz, L.M.: Revealing ultrafast charge-carrier thermalization in tin-iodide perovskites through novel pump–push–probe terahertz spec- troscopy. ACS photonics 8(8), 2509–2518 (2021) https://doi.org/10.1021/acsphotonics.1c00763 15. Kim, R.H., Liu, Z., Huang, C., Park, J.-M., Haeuser, S.J., Song, Z., Yan, Y., Yao, Y., Luo, L., Wang, J.: Terahertz nanoimaging of perovskite solar cell materials. ACS Photonics 9(11), 3550–3556 (2022) https://doi.org/10.1021/acsphotonics.2c00861 16. Bawuah, P., Zeitler, J.A.: Advances in terahertz time-domain spectroscopy of pharmaceutical solids: A review. TrAC Trends in Analytical Chemistry 139, 116272 (2021) https://doi.org/10.1016/j.trac.2021. 116272 17. Sibik, J., Zeitler, J.A.: Direct measurement of molecular mobility and crystallisation of amorphous pharmaceuticals using terahertz spectroscopy. Advanced drug delivery reviews 100, 147–157 (2016) https://doi.org/10.1016/j.addr.2015.12.021 18. Santitewagun, S., Thakkar, R., Zeitler, J.A., Maniruzzaman, M.: Detecting crystallinity using terahertz spectroscopy in 3D printed amorphous solid dispersions. Molecular pharmaceutics 19(7), 2380–2389 (2022) https://doi.org/10.1021/acs.molpharmaceut.2c00163 19. Ornik, J., Knoth, D., Koch, M., Keck, C.M.: Terahertz-spectroscopy for non-destructive determination of crystallinity of L-tartaric acid in smartFilms® and tablets made from paper. International journal of pharmaceutics 581, 119253 (2020) https://doi.org/10.1016/j.ijpharm.2020.119253 20. Bawuah, P., Markl, D., Turner, A., Evans, M., Portieri, A., Farrell, D., Lucas, R., Anderson, A., Goodwin, D.J., Zeitler, J.A.: A fast and non-destructive terahertz dissolution assay for immediate release tablets. Journal of Pharmaceutical Sciences 110(5), 2083–2092 (2021) https://doi.org/10.1016/ j.xphs.2020.11.041 21. Bawuah, P., Evans, M., Lura, A., Farrell, D.J., Barrie, P.J., Kleinebudde, P., Markl, D., Zeitler, J.A.: At-line porosity sensing for non-destructive disintegration testing in immediate release tablets. Inter- national Journal of Pharmaceutics: X 5, 100186 (2023) https://doi.org/10.1016/j.ijpx.2023.100186 22. Anuschek, M., Vilhelmsen, T.K., Zeitler, J.A., Rantanen, J.: Thz-tds transflection measurements as a process analyser for tablet mass. International Journal of Pharmaceutics 666, 124750 (2024) 23. Lee, J., Leung, C.K., Gingras, L., Holzwarth, R., Goodwin, D.J., Dhenge, R.M., Nassar, J., Bawuah, P., Zeitler, J.A.: Improved robustness by concurrent rapid and non-destructive terahertz sensing of pharmaceutical tablet thickness and porosity. International Journal of Pharmaceutics, 125773 (2025) 24. Pickwell, E., Wallace, V.: Biomedical applications of terahertz technology. Journal of Physics D: Applied Physics 39(17), 301 (2006) https://doi.org/10.1088/0022-3727/39/17/R01 25. Lindley-Hatcher, H., Stantchev, R., Chen, X., Hernandez-Serrano, A.I., Hardwicke, J., Pickwell- MacPherson, E.: Real time THz imaging—opportunities and challenges for skin cancer detection. Applied Physics Letters 118(23) (2021) https://doi.org/10.1063/5.0055259 26. Hernandez-Serrano, A.I., Ding, X., Young, J., Costa, G., Dogra, A., Hardwicke, J., Pickwell- MacPherson, E.: Terahertz probe for real time in vivo skin hydration evaluation. Advanced Photonics Nexus 3(1), 016012–016012 (2024) https://doi.org/10.1117/1.APN.3.1.016012 27. Wahaia, F., Valušis, G., Bernardo, L.M., Almeida, A., Moreira, J.A., Lopes, P.C., Macutkevic, J., Kasalynas, I., Seliuta, D., Adomavicius, R., Henrique, R., Lopes, M.: Detection of colon cancer by terahertz techniques. Journal of Molecular Structure 1006(1-3), 77–82 (2011) https://doi.org/10.1016/ j.molstruc.2011.05.049 28. Cao, Y., Chen, J., Zhang, G., Fan, S., Ge, W., Hu, W., Huang, P., Hou, D., Zheng, S.: Characterization and discrimination of human colorectal cancer cells using terahertz spectroscopy. Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 256, 119713 (2021) https://doi.org/10.1016/j.saa. 2021.119713 123 https://doi.org/10.1021/jacs.9b04338 https://doi.org/10.1016/j.molstruc.2024.140612 https://doi.org/10.1016/j.molstruc.2024.140612 https://doi.org/10.1002/adom.201900783 https://doi.org/10.1002/adom.201900783 https://doi.org/10.1021/acsphotonics.1c00763 https://doi.org/10.1021/acsphotonics.2c00861 https://doi.org/10.1016/j.trac.2021.116272 https://doi.org/10.1016/j.trac.2021.116272 https://doi.org/10.1016/j.addr.2015.12.021 https://doi.org/10.1021/acs.molpharmaceut.2c00163 https://doi.org/10.1016/j.ijpharm.2020.119253 https://doi.org/10.1016/j.xphs.2020.11.041 https://doi.org/10.1016/j.xphs.2020.11.041 https://doi.org/10.1016/j.ijpx.2023.100186 https://doi.org/10.1088/0022-3727/39/17/R01 https://doi.org/10.1063/5.0055259 https://doi.org/10.1117/1.APN.3.1.016012 https://doi.org/10.1016/j.molstruc.2011.05.049 https://doi.org/10.1016/j.molstruc.2011.05.049 https://doi.org/10.1016/j.saa.2021.119713 https://doi.org/10.1016/j.saa.2021.119713 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 35 of 37 59 29. Vafapour, Z., Troy, W., Rashidi, A.: Colon cancer detection by designing and analytical evaluation of a water-based THz metamaterial perfect absorber. IEEE Sensors Journal 21(17), 19307–19313 (2021) https://doi.org/10.1109/JSEN.2021.3087953 30. Fischer, B.M., Walther, M., Jepsen, P.U.: Far-infrared vibrational modes of DNA components studied by terahertz time-domain spectroscopy. Physics in Medicine & Biology 47(21), 3807 (2002) https:// doi.org/10.1088/0031-9155/47/21/319 31. Born, B., Havenith, M.: Terahertz dance of proteins and sugars with water. Journal of Infrared, Mil- limeter, and Terahertz Waves 30, 1245–1254 (2009) https://doi.org/10.1007/s10762-009-9514-6 32. Conti Nibali, V., Havenith, M.: New insights into the role of water in biological function: studying solvated biomolecules using terahertz absorption spectroscopy in conjunctionwithmolecular dynamics simulations. Journal of the American Chemical Society 136(37), 12800–12807 (2014) https://doi.org/ 10.1021/ja504441h 33. Markelz,A.G.,Mittleman,D.M.: Perspective on terahertz applications in bioscience andbiotechnology. Acs Photonics 9(4), 1117–1126 (2022) https://doi.org/10.1021/acsphotonics.2c00228 34. Tiwana, P., Parkinson, P., Johnston, M.B., Snaith, H.J., Herz, L.M.: Ultrafast terahertz conductivity dynamics in mesoporous TiO2: influence of dye sensitization and surface treatment in solid-state dye- sensitized solar cells. The Journal of Physical Chemistry C 114(2), 1365–1371 (2010) https://doi.org/ 10.1021/jp908760r 35. Ponseca Jr, C., Sundström, V.: Revealing the ultrafast charge carrier dynamics in organo metal halide perovskite solar cell materials using time resolved THz spectroscopy. Nanoscale 8(12), 6249–6257 (2016) https://doi.org/10.1039/C5NR08622A 36. Hempel, H., Savenjie, T.J., Stolterfoht, M., Neu, J., Failla, M., Paingad, V.C., Kužel, P., Heilweil, E.J., Spies, J.A., Schleuning, M., Zhao, J., Friedrich, D., Schwarzburg, K., Siebbeles, L.D.A., Dörflinger, P., Dyakonov, V., Katoh, R., Hong, M.J., Labram, J.G., Monti, M., Butler-Caddle, E., Lloyd-Hughes, J., Taheri, M.M., Baxter, J.B., Magnanelli, T.J., Luo, S., Cardon, J.M., Ardo, S., Unold, T.: Predicting solar cell performance from terahertz andmicrowave spectroscopy.AdvancedEnergyMaterials 12(13), 2102776 (2022) https://doi.org/10.1002/aenm.202102776 37. Neu, J.: Optical pump terahertz probe (OPTP) and time resolved terahertz spectroscopy (TRTS) of emerging solar materials. APL Photonics 8(7) (2023) https://doi.org/10.1063/5.0152726 38. Leung, C.K., Raju, K., Lee, J., De Volder, M., Zeitler, J.A.: Battery electrode characterisation based on reflection coefficient and time delay of terahertz time-domain reflectometry. In: 2024 49th International Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz), pp. 1–2 (2024). https://doi. org/10.1109/IRMMW-THz60956.2024.10697545 . IEEE 39. Zarrinkhat, F., Hernandez-Serrano, A.I., Pentland, A., Taday, P.F., Arnone, D.D., Pepper,M.: Exploring porosity in battery electrodes: terahertz technology unveiling remote sensing. In: 2024 49th Interna- tional Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz), pp. 1–2 (2024). https://doi.org/10.1109/IRMMW-THz60956.2024.10697663 . IEEE 40. Kiritharan, S., Lucas, S., Deglìnnocenti, R., Hua, X., Dawson, R., Lin, H.: Porosity characterisation of solid-state battery electrolyte with terahertz time-domain spectroscopy. Journal of Power Sources 595, 234050 (2024) https://doi.org/10.1016/j.jpowsour.2024.234050 41. Hsu, D.K., Lee, K.-S., Park, J.-W., Woo, Y.-D., Im, K.-H.: NDE inspection of terahertz waves in wind turbine composites. International journal of precision engineering and manufacturing 13, 1183–1189 (2012) https://doi.org/10.1007/s12541-012-0157-5 42. Jepsen, P.U., Cooke, D.G., Koch, M.: Terahertz spectroscopy and imaging–Modern techniques and applications. Laser & Photonics Reviews 5(1), 124–166 (2011) https://doi.org/10.1002/lpor. 201000011 43. Withayachumnankul, W., Naftaly, M.: Fundamentals of measurement in terahertz time-domain spec- troscopy. Journal of Infrared, Millimeter, and Terahertz Waves 35, 610–637 (2014) https://doi.org/10. 1007/s10762-013-0042-z 44. Neu, J., Schmuttenmaer, C.A.: Tutorial: An introduction to terahertz time domain spectroscopy (THz- TDS). Journal of Applied Physics 124(23) (2018) https://doi.org/10.1063/1.5047659 45. Duvillaret, L., Garet, F., Coutaz, J.-L.: A reliable method for extraction of material parameters in terahertz time-domain spectroscopy. IEEE Journal of selected topics in quantum electronics 2(3), 739–746 (1996) https://doi.org/10.1109/2944.571775 46. Duvillaret, L., Garet, F., Coutaz, J.-L.: Highly precise determination of optical constants and sample thickness in terahertz time-domain spectroscopy. Applied optics 38(2), 409–415 (1999) https://doi. org/10.1364/AO.38.000409 123 https://doi.org/10.1109/JSEN.2021.3087953 https://doi.org/10.1088/0031-9155/47/21/319 https://doi.org/10.1088/0031-9155/47/21/319 https://doi.org/10.1007/s10762-009-9514-6 https://doi.org/10.1021/ja504441h https://doi.org/10.1021/ja504441h https://doi.org/10.1021/acsphotonics.2c00228 https://doi.org/10.1021/jp908760r https://doi.org/10.1021/jp908760r https://doi.org/10.1039/C5NR08622A https://doi.org/10.1002/aenm.202102776 https://doi.org/10.1063/5.0152726 https://doi.org/10.1109/IRMMW-THz60956.2024.10697545 https://doi.org/10.1109/IRMMW-THz60956.2024.10697545 https://doi.org/10.1109/IRMMW-THz60956.2024.10697663 https://doi.org/10.1016/j.jpowsour.2024.234050 https://doi.org/10.1007/s12541-012-0157-5 https://doi.org/10.1002/lpor.201000011 https://doi.org/10.1002/lpor.201000011 https://doi.org/10.1007/s10762-013-0042-z https://doi.org/10.1007/s10762-013-0042-z https://doi.org/10.1063/1.5047659 https://doi.org/10.1109/2944.571775 https://doi.org/10.1364/AO.38.000409 https://doi.org/10.1364/AO.38.000409 59 Page 36 of 37 Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 47. Jepsen, P.U., Fischer, B.M.: Dynamic range in terahertz time-domain transmission and reflection spec- troscopy. Optics letters 30(1), 29–31 (2005) https://doi.org/10.1364/OL.30.000029 48. Pupeza, I., Wilk, R., Koch, M.: Highly accurate optical material parameter determination with THz time-domain spectroscopy. Optics express 15(7), 4335–4350 (2007) https://doi.org/10.1364/OE.15. 004335 49. Parrott, E.P., Zeitler, J.A., Gladden, L.F.: Accurate determination of optical coefficients from chemical samples using terahertz time-domain spectroscopy and effective medium theory. Optics Letters 34(23), 3722–3724 (2009) https://doi.org/10.1364/ol.34.003722 50. Scheller, M., Jansen, C., Koch, M.: Analyzing sub-100-μm samples with transmission terahertz time domain spectroscopy. Optics Communications 282(7), 1304–1306 (2009) https://doi.org/10.1016/j. optcom.2008.12.061 51. Naftaly, M., Dudley, R.: Methodologies for determining the dynamic ranges and signal-to-noise ratios of terahertz time-domain spectrometers. Optics letters 34(8), 1213–1215 (2009) https://doi.org/10. 1364/OL.34.001213 52. Krüger, M., Funkner, S., Bründermann, E., Havenith, M.: Uncertainty and ambiguity in terahertz parameter extraction and data analysis. Journal of Infrared, Millimeter, and Terahertz Waves 32, 699– 715 (2011) https://doi.org/10.1007/s10762-010-9669-1 53. Scheller, M.: Data extraction from terahertz time domain spectroscopy measurements. Journal of Infrared, Millimeter, and Terahertz Waves 35, 638–648 (2014) https://doi.org/10.1007/s10762-014- 0053-4 54. Jepsen, P.U.: Phase retrieval in terahertz time-domain measurements: a “how to” tutorial. Journal of Infrared, Millimeter, and Terahertz Waves 40, 395–411 (2019) https://doi.org/10.1007/s10762-019- 00578-0 55. Bawuah, P., Markl, D., Farrell, D., Evans, M., Portieri, A., Anderson, A., Goodwin, D., Lucas, R., Zeitler, J.A.: Terahertz-based porosity measurement of pharmaceutical tablets: a tutorial. Journal of Infrared, Millimeter, and Terahertz Waves 41, 450–469 (2020) https://doi.org/10.1007/s10762-019- 00659-0 56. Lee, J., Leung, C.K., Ma, M., Ward-Berry, J., Santitewagun, S., Zeitler, J.A.: The dotthz project: A standard data format for terahertz time-domain data. Journal of Infrared, Millimeter, and Terahertz Waves 44(11), 795–813 (2023) https://doi.org/10.1007/s10762-023-00947-w 57. Terahertz Applications Group, University of Cambridge: dotTHzTAG GitHub. https://github.com/ dotTHzTAG. Accessed 2025-04-22 (2025) 58. Terahertz Applications Group, University of Cambridge: THzPy. https://github.com/dotTHzTAG/ thzpy. Accessed 2025-05-04 (2025) 59. Terahertz Applications Group, University of Cambridge: CaTSper. https://github.com/dotTHzTAG/ CaTSper. Accessed 2025-07-08 (2025) 60. Tayvah, U., Spies, J.A., Neu, J., Schmuttenmaer, C.A.: Nelly: A user-friendly and open-source imple- mentation of tree-based complex refractive index analysis for terahertz spectroscopy. Analytical Chemistry 93(32), 11243–11250 (2021) https://doi.org/10.1021/acs.analchem.1c02132 61. Tayvah, U., Spies, J.A., Neu, J., Schmuttenmaer, C.A.: Nelly. https://github.com/YaleTHz/nelly. Accessed 2025-05-04 (2021) 62. Sheppard, N., Willis, H., Rigg, J.: Names, symbols, definitions and units of quantities in optical spectroscopy (Recommendations 1984). Pure and Applied Chemistry 57(1), 105–120 (1985) https:// doi.org/10.1351/pac198557010105 63. International Union of Pure and Applied Chemistry (IUPAC): decadic absorbance. IUPAC Com- pendium of Chemical Terminology – The Gold Book. Version 5.0.0, accessed 2025-04-22 (2025). https://doi.org/10.1351/goldbook.D01536. https://doi.org/10.1351/goldbook.D01536 64. Scheller, M., Jansen, C., , Koch, M.: Applications of effective medium theories in the terahertz regime. In: Kim, K.Y. (ed.) Recent Optical and Photonic Technologies. IntechOpen, Rijeka (2010). Chap. 12. https://doi.org/10.5772/6915 65. Markel, V.A.: Introduction to the maxwell garnett approximation: tutorial. Journal of the Optical Society of America A 33(7), 1244–1256 (2016) https://doi.org/10.1364/JOSAA.33.001244 66. National Center for Biotechnology Information: PubChem Compound Summary for CID 104938, Lactose monohydrate. https://pubchem.ncbi.nlm.nih.gov/compound/alpha-Lactose-monohydrate. Accessed 2025-05-20 (2025) 67. Naftaly, M., Miles, R., Greenslade, P.: THz transmission in polymer materials—a data library. In: 2007 Joint 32nd International Conference on Infrared andMillimeter Waves and the 15th International Con- 123 https://doi.org/10.1364/OL.30.000029 https://doi.org/10.1364/OE.15.004335 https://doi.org/10.1364/OE.15.004335 https://doi.org/10.1364/ol.34.003722 https://doi.org/10.1016/j.optcom.2008.12.061 https://doi.org/10.1016/j.optcom.2008.12.061 https://doi.org/10.1364/OL.34.001213 https://doi.org/10.1364/OL.34.001213 https://doi.org/10.1007/s10762-010-9669-1 https://doi.org/10.1007/s10762-014-0053-4 https://doi.org/10.1007/s10762-014-0053-4 https://doi.org/10.1007/s10762-019-00578-0 https://doi.org/10.1007/s10762-019-00578-0 https://doi.org/10.1007/s10762-019-00659-0 https://doi.org/10.1007/s10762-019-00659-0 https://doi.org/10.1007/s10762-023-00947-w https://github.com/dotTHzTAG https://github.com/dotTHzTAG https://github.com/dotTHzTAG/thzpy https://github.com/dotTHzTAG/thzpy https://github.com/dotTHzTAG/CaTSper https://github.com/dotTHzTAG/CaTSper https://doi.org/10.1021/acs.analchem.1c02132 https://github.com/YaleTHz/nelly https://doi.org/10.1351/pac198557010105 https://doi.org/10.1351/pac198557010105 https://doi.org/10.1351/goldbook.D01536 https://doi.org/10.1351/goldbook.D01536 https://doi.org/10.5772/6915 https://doi.org/10.1364/JOSAA.33.001244 https://pubchem.ncbi.nlm.nih.gov/compound/alpha-Lactose-monohydrate Journal of Infrared, Millimeter, and Terahertz Waves (2025) 46 :59 Page 37 of 37 59 ference on Terahertz Electronics, pp. 819–820 (2007). https://doi.org/10.1109/ICIMW.2007.4516747 . IEEE 68. Vieweg, N., Rettich, F., Deninger, A., Roehle, H., Dietz, R., Göbel, T., Schell, M.: Terahertz-time domain spectrometer with 90 dB peak dynamic range. Journal of Infrared, Millimeter, and Terahertz Waves 35, 823–832 (2014) https://doi.org/10.1007/s10762-014-0085-9 69. Kendrick, J., Burnett, A.D.: Pdielec: The calculation of infrared and terahertz absorption for powdered crystals. Journal of Computational Chemistry 37(16), 1491–1504 (2016) https://doi.org/10.1002/jcc. 24344 70. Kendrick, J., Burnett, A.D.: PDielec Package. http://www.github.com/JohnKendrick/PDielec. Accessed 2025-05-04 (2024) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Authors and Affiliations Chi Ki Leung1 · Jasper N. Ward-Berry1 · Elena Wanvig i Dot1 · Jongmin Lee1 · J. Axel Zeitler1 B J. Axel Zeitler jaz22@cam.ac.uk Chi Ki Leung ckl46@cam.ac.uk Jasper N. Ward-Berry jnw35@cam.ac.uk Elena Wanvig i Dot ew655@cam.ac.uk Jongmin Lee jl2112@cam.ac.uk 1 Department of Chemical Engineering and Biotechnology, University of Cambridge, Philippa Fawcett Drive, Cambridge CB30AS, UK 123 https://doi.org/10.1109/ICIMW.2007.4516747 https://doi.org/10.1007/s10762-014-0085-9 https://doi.org/10.1002/jcc.24344 https://doi.org/10.1002/jcc.24344 http://www.github.com/JohnKendrick/PDielec http://orcid.org/0000-0002-0399-2784 http://orcid.org/0009-0006-3472-4631 http://orcid.org/0000-0003-0272-0274 http://orcid.org/0000-0002-3476-2922 http://orcid.org/0000-0002-4958-0582 Tutorial: Accurate Determination of Refractive Index and Absorption Coefficient in Terahertz Time-Domain Spectroscopy Abstract 1 Introduction 2 Terahertz Time-Domain Spectroscopy Measurements 3 Data Processing from Time Domain to Frequency Domain 4 Basic Principles of Terahertz Electric Field in Frequency Domain 5 Optical Constants of a Weakly Terahertz-Absorbing Material 6 Optical Constants of a Strongly Terahertz-Absorbing Material 7 Optical Constants of a Thick Sample 8 Optical Constants Derived from Alternative Experimental Configurations 9 Conclusion Supplementary Information Workflow Using the THzPy Python Library Workflow Using the CaTSper Graphical User Interface Acknowledgements References