Design Considerations for Integration of Terahertz Time-Domain Spectroscopy in Microfluidic Platforms

Abstract

Microfluidic platforms have received much attention in recent years. In particular, there is interest in combining spectroscopy with microfluidic platforms. This work investigates the integration of microfluidic platforms and terahertz time-domain spectroscopy (THz-TDS) systems. A semiclassical computational model is used to simulate the emission of THz radiation from a GaAs photoconductive THz emitter. This model incorporates white noise with increasing noise amplitude (corresponding to decreasing dynamic range values). White noise is selected over other noise due to its contributions in THz-TDS systems. The results from this semiclassical computational model, in combination with defined sample thicknesses, can provide the maximum measurable absorption coefficient for a microfluidic-based THz-TDS system. The maximum measurable frequencies for such systems can be extracted through the relationship between the maximum measurable absorption coefficient and the absorption coefficient for representative biofluids. The sample thickness of the microfluidic platform and the dynamic range of the THz-TDS system play a role in defining the maximum measurable frequency for microfluidic-based THz-TDS systems. The results of this work serve as a design tool for the development of such systems.

1. Introduction

The development of ultrafast pulsed lasers (pulse duration <1 ps) was a significant scientific achievement, with a plethora of applications including semiconductor characterization [1,2], optical interferometric studies [3,4], and material analyses [5], and was a key advancement to allow electromagnetic measurements over the terahertz (THz) spectrum [6,7,8,9]. The THz spectrum contains electromagnetic radiation in a frequency interval from 0.1 to 10 THz (corresponding to wavelengths from 3 mm to 30 μm, respectively) and is of great interest for science and technology. The THz spectrum is situated between the infrared spectrum (used in photonics [10]) and the microwave spectrum (used in electronics [11]). Since initial advancements in ultrafast pulsed lasers, the THz spectrum has been employed in many contemporary applications. These contemporary applications include communications [12], chemical and biological sensing [13], security [14], quality control [15], and biomedical spectroscopic devices [16].

Biomedical spectroscopic devices is a challenging and emerging application of the THz spectrum. The THz spectrum is valuable for biomedical spectroscopic devices for several reasons: the THz spectrum is strongly attenuated by water, and therefore is very sensitive to heightened moisture content associated with disease [17]; the THz spectrum has low photon energies, and therefore has little to no ionization hazard for biological tissues [18]; and the THz spectrum has a close match between its photon energies and the conformational modes of biomolecules [19]. Because the THz spectrum possesses these characteristic properties, there has been an increasing interest in terahertz imaging and spectroscopy for biomedical spectroscopic devices over the last years [16,18].

Spectroscopy over the THz spectrum is either implemented through continuous-wave THz spectroscopy systems using continuous-wave (single-frequency) THz radiation [20] or through THz time-domain spectroscopy (THz-TDS) systems using pulsed (multiple-frequency) THz radiation [9]. The latter technique is particularly appealing, with both the amplitude and phase being measured, and allows extensive characterization of biomedical samples in biomedical spectroscopic devices.

Terahertz time-domain spectroscopy systems are currently being explored for biomedical applications such as oncological diagnostics [18] and protein analyses [21]. However, in these THz-TDS systems, the high absorption of liquid places restrictions on the volume size of the biomedical samples, as the dynamic range must be large enough that the THz radiation at each frequency is not attenuated below the noise floor. In response, microfluidic platforms [22,23,24] have been integrated with THz-TDS systems and these studies have begun to appear in the literature. For example, Tang et al. [25] and George et al. [26] have both initiated work integrating such microfluidic-based THz-TDS systems. These microfluidic-based THz-TDS systems require careful design considerations due to significant and fundamental challenges. These challenges are caused by a twofold effect: the maximum measurable absorption coefficient of a THz-TDS system monotonically decreases from its maximum (scaling with the logarithm of its dynamic range function); while simultaneously, the absorption coefficient of most liquids monotonically increases over the THz spectrum [27] (as resonance peaks typically seen in vapour THz-TDS measurements blur together). These twofold challenges force the maximum measurable frequency of a THz-TDS system to be much less than the bandwidth of the THz-TDS system. This challenge is further exacerbated by the fact that the maximum measurable absorption coefficient scales with the reciprocal of the thickness of the biomedical sample in the microfluidic platform. As such, thick microfluidic platforms with integrated THz-TDS systems possessing low dynamic range will have a low maximum measurable frequency (much less than the bandwidth of the THz-TDS system), whereas thin microfluidic platforms with integrated THz-TDS systems possessing high dynamic range will have a high maximum measureable frequency (approaching the bandwidth of the THz-TDS system). This is a significant issue as many applications require absorption coefficient measurements up to high frequencies within the THz spectrum; e.g., to measure cytidine deaminase absorption near 2.14 THz for oncology [28]; or to measure L-glucose absorption near 2.12 THz for DNA analyses [29]. At the same time, it is difficult to scale microfluidic platforms down to micron-scale thicknesses and it is difficult to produce high dynamic range THz-TDS systems. The dynamic range can be quite low for such THz-TDS systems as the data acquisition must take place quickly, given the dynamic nature of microfluidic platforms [30]. (Achieving a large dynamic range value can require minutes for data acquisition [31].) With this in mind, such microfluidic platforms with integrated THz-TDS systems must be carefully designed in terms of sample thickness and dynamic range.

This work provides an investigation into the design of a microfluidic-based THz-TDS system. The effects of dynamic range (in the THz-TDS system) and sample thickness (in the microfluidic platform) are considered. The analysis begins by defining a semiclassical computational model to simulate emission from a photoconductive THz emitter (as these are the most common THz emitters in modern THz-TDS systems [32]). The semiclassical computational model is based on the work of Rodriquez and Taylor [33]. The analysis continues by superimposing white noise with increasing noise amplitude into the results of the semiclassical computational model, which corresponds to a decreasing dynamic range. Finally, the analysis solves for the maximum measurable frequency, with input parameters of dynamic range and sample thickness, for a nominal fluidic absorption coefficient. (It is assumed that the absorption coefficient of biofluids possessing high water content have minimal deviation from the absorption coefficient of water.) Such results can be used to design microfluidic-based THz-TDS systems.

It should be noted that the connection between sample thickness and maximum measureable frequency for THz-TDS systems was first noted in the seminal work of Jepsen and Fischer [34]. In Jepsen and Fischer’s work, a THz-TDS system (with a single dynamic range value) was presented and tested with two polylactide samples with sample thicknesses of d = 0.58 mm and 2.79 mm. As a result of this increased sample thickness, the maximum measurable frequency of the THz-TDS system falls from 1.9 THz to 1.2 THz. In contrast, our work leverages these initial (and fundamental) findings to provide a thorough analysis for the design of microfluidic-based THz-TDS systems, both in terms of sample thickness and dynamic range value. This is accomplished through extensive simulation and analysis with sample thicknesses spanning 30–480 μm (applicable to modern microfluidic platforms [35]) and dynamic range values spanning 2.8 × 10⁰–5.8 × 10³ (applicable to modern THz-TDS systems [8]).

2. Design and Simulation

An exemplary microfluidic-based THz-TDS system is shown in Figure 1. Such a system has a photoconductive THz emitter based on a GaAs substrate, which is illuminated by a pump laser pulse. This pump laser pulse will generate a THz reference pulse in the photoconductive THz emitter. The electric field associated with the THz reference pulse is represented as E_ref(t) in the time domain and E_ref(f) in the frequency domain. The THz reference pulse passes through a microfluidic platform with sample thickness of d, where it undergoes a transformation into the THz sample pulse, which is represented as E_sam(t) in the time domain and E_sam(f) in the frequency domain. Data representing this THz reference pulse is generated with a semiclassical computational model.

The semiclassical computational model is based on the work of Rodriquez and Taylor [33] and solves semiconductor and optical (electromagnetic) equations for a photoconductive THz emitter. Here, the photoconductive THz emitter has a gap between electrodes that is illuminated with an optical pulse which is Gaussian, both temporally and spatially. This optical pulse is represented as

$$I(x,t)=\frac{\mathsf{\Phi }}{{\tau }_p}\exp [-4\ln 2t^2/{\tau }_p^2]\exp [-4\ln 2x^2/x_p^2]$$

(1)

where the temporal full-width-at-half-maximum is τ_p = 1 ps (being equal to the semiconductor response time of GaAs), the transverse spatial dimension is x, the spatial full-width-at-half-maximum is x_p = 150 μm, and the fluence of the illumination pulse is Φ. The semiclassical computational model also requires the transport equation for the surface electron current, which is

$$\frac{\partial n_s(x,t)}{\partial t}=\frac{(1-R)}{h\nu }I(x,t)+\frac{1}{q}\frac{\partial k_n(x,t)}{\partial x}$$

(2)

and the transport equation for the surface hole current, which is

$$\frac{\partial p_s(x,t)}{\partial t}=\frac{(1-R)}{h\nu }I(x,t)-\frac{1}{q}\frac{\partial k_p(x,t)}{\partial x}$$

(3)

where the surface density of electrons and holes are n_s(x,t) and p_s(x,t), respectively; the surface currents of electrons and holes are k_n(x,t) and k_p(x,t), respectively; the elementary charge is q; Planck’s constant is h; the optical frequency of the ultrafast pulsed laser is ν = 365 THz (corresponding to a wavelength of 800 nm); and the GaAs reflectivity is R = 0.3. These equations can be solved with a finite difference method along with Poisson’s Equation, which is

$$\frac{\partial E(x,t)}{\partial x}=\frac{q}{\delta {\epsilon }_r{\epsilon }_o}[p_s(x,t)-n_s(x,t)]$$

(4)

where the optical penetration depth is δ = 700 nm, the permittivity of free space is ε₀, and the relative permittivity of GaAs is ε_r = 13. Ultimately, an approximation to a Hertzian dipole antenna can be used, and the electric field THz pulse in the far-field can be expressed as

$$E(z,t)=\frac{L_y}{8\pi {\epsilon }_o{c}^{2}z}\frac{d}{dt}\underset{0}{\overset{L_x}{\int }}{k}_n(x,t)dx$$

(5)

where the y-direction length of the gap is L_y = 300 μm and the x-direction length of the gap is L_x = 300 μm. This radiation simulates noise-free emission of radiation from a photoconductive THz emitter. It should be noted that the semiclassical computational model provides a simulation for one pulse. As such, the repetition rate of the ultrafast pulsed laser is not considered. (Repetition rates of 90 MHz are standard for titanium sapphire or erbium-doped fibre ultrafast pulsed lasers.)

To incorporate noise into the semiclassical computational model of a photoconductive THz emitter and subsequent THz-TDS system, various noise sources must be considered. These noise sources include white noise, 1/f noise (i.e., pink noise), mechanical noise from the pellicle beamsplitter, and others. Of these, the most significant noise processes are 1/f noise associated with the THz emitter and white noise associated with the THz detector [36,37]. In a THz-TDS system with a sample that has low absorption (e.g., quartz with a thickness of microns), the 1/f noise associated with the THz emitter can be larger than or similar to the white noise associated with the THz detector. However, the 1/f noise drops off considerably at higher frequencies (due to the reciprocal relationship to frequency). Therefore, for a low absorption sample, 1/f noise need not be considered for the high-frequency regime, but still could be considered for the low-frequency regime. On the other hand, in a THz-TDS system with high absorption (e.g., water or biofluids with thickness of tens of microns or greater, as in the microfluidic platforms considered in this work), the 1/f noise associated with the THz emitter will be substantially attenuated due to the high absorption when passing through the sample. Here, the white noise associated with the THz detector, which is not affected by the absorption, takes a prominent role over both low- and high-frequency regimes. The microfluidic platforms described in our work represent a high-absorption THz-TDS system. Therefore, white noise associated with the THz detector is considered in the analysis and 1/f noise associated with the THz emitter is not considered in the analysis [36].

White noise associated with the THz detector can be incorporated into the semiclassical computational model through an additive white noise signal, n(t), defined as a uniform random noise signal that is centered about zero with a peak-to-peak spread equal to the amplitude of the electric field THz pulse. This noise signal is superimposed onto the electric field THz pulse at an arbitrary point in space (i.e., an arbitrary z value) in the far-field to form the reference THz pulse

$$E_{ref}(t)=E(z,t)+A_{n}n(t)$$

(6)

where the noise amplitude is A_n.

3. Results and Discussion

For a representative photoconductive THz emitter with an incident average power of 50 mW and an external biasing field of 25 V/150 μm, Figure 2a shows the output of the semiclassical computational model in the time domain, E_ref(t), with increasing noise amplitudes of A_n = 0, 10⁻³, 10⁻², 10⁻¹, 10⁰, and 10¹. Figure 2b shows the equivalent output in the frequency domain, E_ref(f), with corresponding dynamic range values of DR = 5.8 × 10³, 5.7 × 10³, 2.4 × 10³, 3.0 × 10², 2.8 × 10¹, and 2.8 × 10⁰ (with DR being defined as the maximum of E_ref(f) divided by the noise floor). It is clear that DR decreases as A_n increases. This DR along with the sample thickness, d, will limit the maximum measurable (intensity) absorption coefficient, α_M, according to

$${\alpha }_M=\frac{2}{d}\ln \left[DRF(f)\frac{4n}{{(n+1)}^2}\right]$$

(7)

where the dynamic range function is DRF(f) (being E_ref(f) normalized with respect to its noise floor) and n is the refractive index of water over the THz spectrum. (This fundamental equation was initially reported by Jepsen and Fischer [34] and its derivation is given in the Appendix.) A piece-wise equation for the refractive index of water is fit to data reported by Wang et al. [38]. The piece-wise function for refractive index is n = −1.8f + 3.1, for f < 0.5 THz with respective slope and intercept uncertainties of 10% and 1%; and n = −0.1f + 2.3, for f > 0.5 THz with respective slope and intercept uncertainties of 7% and 1%. Figure 3a shows the maximum measurable absorption coefficient for the dynamic range values of DR = 5.8 × 10³, 5.7 × 10³, 2.4 × 10³, 3.0 × 10², 2.8 × 10¹, and 2.8 × 10⁰ and a constant sample thickness of d = 120 μm, plotted as red, yellow, green, blue, indigo, and violet solid lines, respectively. The absorption coefficient of water, α_water = 121f + 109 cm⁻¹, is quantified by fitting data from Wang et al. [38] to a linear fit of R² = 0.978 with both slope and intercept uncertainties being 4%, and is plotted as a solid black line. The intersection of the α_water curve with the maximum measurable absorption coefficient curves defines each of the maximum measurable frequencies of f_0–5, which correspond to the respective DR values of 5.8 × 10³, 5.7 × 10³, 2.4 × 10³, 3.0 × 10², 2.8 × 10¹, and 2.8 × 10⁰. These maximum measurable frequencies range from f₀ = 2.0 THz to f₅ = 0 and decrease with decreasing DR values. From these results, it is clear that a THz-TDS system with a limited dynamic range value will severely limit the application of a microfluidic-based THz-TDS system, and care must be taken in the design of such systems.

Figure 3b shows the maximum measurable absorption coefficient for a constant dynamic range value of DR = 2.4 × 10³ and sample thicknesses of d = 30, 60, 120, 240, and 480 μm, plotted as blue long-dashed, medium-dashed, solid, short-dashed, and dashed-dotted lines, respectively. The absorption coefficient of water is plotted as a solid black line. The intersection of the α_water curve with the maximum measurable absorption coefficient curves defines each of the maximum measurable frequencies of f_6–10, which correspond to respective sample thicknesses of d = 30, 60, 120, 240, and 480 μm. These maximum measurable frequencies range from f₆ = 2.3 THz to f₁₀ = 0 and decrease with increasing sample thickness. From these results, it is clear that a microfluidic platform with a large sample thickness will severely limit the application of a microfluidic-based THz-TDS system.

These results can be summed up with a set of measurements rastering through the foreseeable combinations of dynamic range values and sample thicknesses, as presented in Figure 4. Figure 4a,b plot maximum measureable frequency versus sample thickness and logarithm of dynamic range value, respectively. The equations for the curves in Figure 4a, from highest to lowest, are: f_max = −0.0055 d + 2.82, f_max = −0.0055 d + 2.81, f_max = −0.0055 d + 2.63, f_max = −0.0075 d + 2.36, f_max = −0.0101 d + 1.87, and f_max = −0.0094 d + 0.94. The equations for the curves in Figure 4b, from highest to lowest, are: f_max = 0.66 log(DR) + 0.51, f_max = 0.63 log(DR) + 0.19, f_max = 0.55 log(DR) − 0.09, f_max = 0.44 log(DR) − 0.47, and f_max = 0.45 log(DR) − 1.34. (All of these linear equations have a fit of R² ≥ 0.94.) These results can inform the design of future THz-TDS microfluidic platforms. For example, consider the THz-TDS system of Venkatesh et al. that is able to achieve a dynamic range value up to 10³ [39], with performance beyond this value being unattainable or unpractical; and the microfluidic platform of Ng et al. that is able to achieve sample thickness down to 180 μm [40]. Combining these into one THz-TDS microfluidic platform would yield a maximum measurable frequency of 1.0 THz. As such, this THz-TDS microfluidic platform would be suited for lysozyme measurements with absorption over 0.4–0.9 THz. However, this microfluidic-based THz-TDS system would be unsuited for lactose measurements with absorption at 1.4 THz. In this way, the data set of Figure 4 can be used for the design of future microfluidic-based THz-TDS systems.

4. Conclusions

This work explored design considerations for microfluidic-based terahertz time-domain spectroscopy (THz-TDS) systems wherein microfluidic platforms and THz-TDS systems are integrated. The critical parameters of sample thickness (of the microfluidic platform) and dynamic range (of the THz-TDS system) were considered as they greatly influence the maximum measurable frequency of the overall system. The work developed a semiclassical computational model to simulate the emission of THz radiation from a GaAs photoconductive THz emitter. This model incorporated white noise with increasing noise amplitude corresponding to decreasing dynamic range values. Through the results of this semiclassical computational model and defined sample thicknesses, the maximum measurable absorption coefficient was found for numerous microfluidic-based THz-TDS systems with varying dynamic range values and sample thicknesses. The corresponding maximum measurable frequencies for such microfluidic-based THz-TDS systems were found through the intersection of the maximum measurable absorption coefficient and a representative absorption coefficient for biofluids. These results will serve as a design tool for the development of future microfluidic-based THz-TDS systems.