The modulation of the excitation laser can be harmonic or have an alternative form. For instance, using a square-wave excitation instead of a harmonic excitation will increase the OPL amplitude by a factor of 4 / π⁠, or about 1.27. This factor can be determined from a square-wave Fourier series. Despite delivering a higher OPL amplitude, the signal for square-wave modulation resembles that of a harmonic excitation if the OPL amplitude is determined via LIAs. This resemblance is explained by the fact that LIAs typically use a sinusoidal reference signal.

The OP-configuration (collinear laser beams) has the advantage over the OX-configuration (crossed laser beams) in that the interaction path length can be scaled directly by increasing the laser beam’s axial overlap (cf. Sec. III A). The OX-configuration [cf. Fig. 7(d)], the probe laser beam, the excitation laser beam, and the advection are all orthogonal to each other. The analysis of this setup is considerably more complex than the previous configurations. The complexity makes optimization challenging, and one has to rely on numerical methods. Nevertheless, the dynamics of advection-induced signal changes are largely similar to those seen in the OP-configuration.

In the collinear beam configuration, δ T p a t h ( r p , t ) scales with the overlap length of the laser beams. In the crossed-beam configuration, however, the interaction path length is determined by the radii of both laser beams and the radial broadening of the temperature profile. The radial thermal broadening depends on further parameters (cf. Sec. III B). Therefore, the collinear beam configuration can provide a significantly higher signal than the crossed-beam configuration.

We obtain the time-independent amplitude of the periodic OPL change by substituting Eqs. (9) and (6) with (3) into Eq. (2). Following the substitution, the resulting expression still requires solving for the temporal convolution of the power source and the impulse response term. Considering a periodically modulated excitation laser beam, we can express the absorbed power as ( α ⋅ P ) ( t ) = α P a v ( 1 + exp ( i ω t ) )⁠, where P a v is the average laser power. Given a periodic power source, we can obtain the amplitude of an oscillating signal by applying a Laplace transform.1

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Figure 5 shows the functions f cross (solid blue curve) and f coll (dashed black curve) with respect to ω τ r ′ / 2⁠. It is evident for ω τ r ′ / 2 ≫ 1⁠, f cross and f coll are identical. However, it is essential to note that the prefactor in the crossed-beam configuration K 1 depends on the radii of the two laser beams.

In this section, we present an analytical description of the OPL amplitude, taking into account a finite probe laser beam radius. The analysis centers on describing the OPL amplitude, i.e., the signal, for the crossed and collinear laser beam configurations, assuming that the gas remains stationary. Among the most fundamental PTS parameters that determine the signal of the photothermal instrument are the modulation frequency f and the radius of the excitation laser beam a⁠. We specifically focus on the relationship between the angular modulation frequency ω = 2 π f and the characteristic radial thermal diffusion time, denoted as τ r = a 2 / ( 4 D T ) (cf. Table II). The dominance of radial thermal can be determined from their relationship, represented by ω τ r / 2⁠. There is a crucial difference between the regime characterized by ω τ r / 2 ≫ 1 and that characterized by ω τ r / 2 ≪ 1⁠: For ω τ r / 2 ≫ 1⁠, radial thermal diffusion has an insignificant impact on the measurable temperature distribution because the heated volume during one modulation period is primarily determined by the radius of the excitation laser beam. Conversely, in the regime where ω τ r / 2 ≪ 1⁠, radial thermal diffusion broadens the heated volume considerably beyond the radius of the excitation laser beam during one modulation period. In this regime, radial thermal diffusion does affect the signal considerably.

In the orthogonal-flow case [Eq. (7b) and Fig. 2(b)], the origins of the x and z axes coincide with the excitation laser beam axis. The distance, in the x-direction, between the excitation laser beam axis and a position x > 0 is denoted as D⁠. It is notable from Eq. (7b) that δ T ( x , y , z , t ) is invariant in the y-direction. The corresponding flow-related signal analysis with further variable allocations is found in Sec. III C.

In the PP-configuration, the excitation and the interaction path lengths are considered equal, i.e., L e = L p⁠. The interaction between the laser beams and the gas occurs between x = 0 and x = L e⁠. The illustration in Fig. 7(b), however, does not depict a specific realization of how the excitation and interaction limits could be technically implemented. Such limits might be realized, for example,38 by kinks in the flow guiding geometry and the use of laser windows. When v x → 0 (in both configurations) and x ≫ 0 (in the crossed-beam configuration), both Eq. (7) simplify to Eq. (6).

See the supplementary material for (1) the derivation of Eqs. (11) and (12) presented in Sec. III B and for (2) the derivation of Eq. (A1) presented in the  Appendix. The derivation of Eqs. (11) and (12) is obtained by applying the Laplace transform to the time-dependent part of the respective temperature impulse response term. The Laplace transform is similarly used in the derivation of Eqs. (16) and (20) for the parallel-flow configurations in Sec. III C.

Sketch of the interaction path length for the crossed-beam and the collinear beam configuration. (a) In the crossed beam configuration, the integration boundaries extend from − ∞ to ∞⁠, although the main contribution stems from the volume the laser beams cross. (b) In the collinear beam configuration, the axial overlap length determines the interaction length. The integration boundaries range from 0 to L p⁠. Radial thermal diffusion expands the volume influenced by the excitation laser beam and can significantly affect the signal (cf. Sec. III B).

Figure 12(b) presents b p e a k in units of a (refer to the left-hand side axis) and the corresponding gain in Δ T p a t h when b = b p e a k (right-hand side axis) as a function of ω τ r / 2⁠. This gain, labeled as g ( b p e a k )⁠, represents the increase in Δ T p a t h for b = b p e a k compared to Δ T p a t h for b → ∞⁠. The radial thermal diffusion causes b p e a k to increase toward reduced ω τ r / 2 in both configurations (unmarked, black curves). The marked, green curves represent g ( b p e a k )⁠. For a crossed-beam configuration, g ( b p e a k ) saturates at roughly 12% as ω τ r / 2 becomes significantly less than one. For the collinear beam configuration, g ( b p e a k ) reduces with decreasing ω τ r / 2 and peaks when the cut-off criterion b p e a k = 2 a is reached.

Advection can substantially influence the signal (Sec. III C). For a parallel-flow crossed-beam configuration (PX-configuration), advection can significantly alter the signal, depending on the ratio between modulation frequency, flow velocity, and excitation length. A high axial Péclet number is assumed, for which the advection time dominates over the axial thermal diffusion time. The influence of advection on the signal depends on the extent to which radial thermal diffusion is prevalent (cf. Fig. 9): When ω τ r / 2 ≫ 1⁠, advection can potentially double the signal if the ratio τ A / τ M o d is 0.5 + m⁠. However, the signal can be quenched if this ratio is precisely m⁠. When ω τ r / 2 ≫ 1⁠, advection has less effect on the signal, and the maximum signal increase is limited to approximately 1.4 at τ A / τ M o d = 0.34⁠. Suppose the advection time (⁠ τ A⁠) greatly exceeds the modulation period (⁠ τ M o d⁠). In this case, the advection’s influence becomes critical: Instead of the modulation period, advection primarily determines the excitation duration of a target. For ω τ A = 1⁠, the signal is about 95% of its limit value at ω → 0⁠.

Advection brings two further arrangements to the description of the photothermal signal. There is considerable research on how flow affects the probe laser beam’s OPL change, and OPL amplitude, i.e., the signal. Studies including the theoretical descriptions that are particularly relevant in the context of PTI can be found in Refs. 38,50,52–55,66,68–70. In this section, we analyze the signal-influencing capabilities of advection beyond the previously described aspects.

The resulting four configurations are illustrated in Fig. 7, denoted as PX-, PP-, OP-, and OX-configuration, respectively. We introduce a flow-guiding PT cell with a cylindrical geometry (i.e., a duct), whose axis is oriented in the advection direction and with its center at ( y , z ) = ( 0 , 0 )⁠. For the PX-, OP-, and OX-configuration [cf. Figs. 7(a), 7(c), and 7(d)], we assume that the PT cell duct augments considerably beyond the x-position of the probe laser beam. This assumption allows the influence of the downstream PT cell opening to be neglected.

(a) Distance D o p t between excitation and probe laser beam axes as a function of the ratio τ r / τ A for τ A / τ M o d → 0⁠, where the highest signal is obtained at the duct center. The y axis is scaled in units of a⁠. D o p t increases with higher v and a⁠, respectively. The top x axis gives a comparison using representative parametric variables. (b) Temperature amplitude at the duct center for an exemplary v of 0 and 0.5 m/s, respectively, and D of 0 and D o p t⁠, respectively. Further, a = 0.5 mm and D T is similar to ambient air. The comparison of the two advection-influenced cases (solid green and dashed blue curves) shows that Δ T c e n t e r is higher for D = D o p t than for D = 0 in the low-frequency regime (gain by beam separation). An advection-induced signal enhancement may be achieved in the frequency domain before signal saturation dominates at lower frequencies (gain by advection). The thin gray curve corresponds to the situation for gas at rest with D = D o p t⁠.

Definitions of characteristic times used in this work. Characteristic times related to thermal diffusion are marked with lowercase indices, while those not related to thermal diffusion are marked with uppercase indices.

The temperature amplitude Δ T p a t h is obtained from the time-dependent, oscillating δ T p a t h ( t )⁠. For b → ∞⁠, Eq. (A1) becomes Eq. (6). We restrict our analysis to cases where the duct radius is at least twice as large as the 1/e 2-radius of the excitation laser beam, i.e., b ≥ 2 a⁠. This restriction ensures that the excitation laser radiation does not significantly interact with the duct wall, while the duct still receives (almost) the entire radiation of the excitation laser beam.

Again, the maximum amplitude becomes independent of the excitation laser beam size as its radius is reduced. The amplitude reaches 1 / 2 ≈ 70 % of its maximum value for a = D T L e v⁠.

The functions f cross and f coll approach infinity for an unlimited excitation duration (i.e., for ω → 0 +⁠) because the energy input is infinite in this case, and heat removal mechanisms such as advection (cf. Sec. III C) or system boundaries held at a constant temperature (cf.  Appendix) are not considered in this section.

This description uses the characteristic times τ r and τ A⁠, respectively, and introduces the thermal diffusion in the excitation laser’s axial direction, referred to as axial thermal diffusion time τ a x = L e 2 / ( 4 D T )⁠. The superscript in δ T p a t h P X refers to the PX-configuration. The expression in the parentheses after the temporal convolution sign comprises two dependencies: The first term describes the dependency in the radial direction. The second term, i.e., the complementary error function, describes the dependency in the axial direction. Furthermore, the argument of the complementary error function itself comprises two specific components: The term in brackets, ( t − τ A )⁠, describes the transport of the impulse response in the flow direction. The fraction preceding this bracketed term describes the smoothing of the axial thermal response over time. At t = 0⁠, or when τ a x / τ A tends to infinity, erfc can be represented by a Heaviside step function, denoted as Θ⁠.

Ulrich Radeschnig: Conceptualization (lead); Formal analysis (equal); Investigation (equal); Methodology (lead); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Alexander Bergmann: Funding acquisition (equal); Investigation (supporting); Project administration (equal); Resources (equal); Supervision (supporting); Writing – review & editing (equal). Benjamin Lang: Conceptualization (supporting); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (supporting); Project administration (equal); Resources (equal); Supervision (lead); Writing – review & editing (equal).

Consider gas with a target species (i.e., a molecule or particle of interest) moving steadily and homogeneously at a velocity v through a guiding geometry. Inside this geometry, the target is irradiated by a modulated laser beam along a specific excitation path L e [cf. Figs. 7(a) and 7(b)]. The time taken for a target molecule or particle to traverse along L e may be equivalent to the duration of the excitation period. This situation is met when half the modulation cycle duration, 1 / 2 f⁠, equals L e / v⁠. In the following, the modulation duration and the advection time are further expressed in the form τ M o d = 1 / f and τ A = L e / v⁠, respectively. Supposing the irradiation duration is much longer than the time taken for a target to flow through the excitation path, i.e., τ A ≪ τ M o d / 2⁠. In this case, the prevailing factor for the maximum achievable temperature difference is the time a target resides on the excitation path rather than the modulation duration itself. In contrast, when τ A ≫ τ M o d / 2⁠, a target stays on the excitation path for a duration significantly exceeding the irradiation time.

As expressed in Eq. (7), the flow analysis focuses on two fundamental orientations based on the relative propagation directions of the excitation laser beam and the gas flow (i.e., parallel- and orthogonal-flow cases). We add the two flow configurations into the (already discussed) crossed-beam and collinear beam configurations. Thereby, we make the simplification by assuming a probe laser beam that is much smaller than the excitation laser beam and assign a p → 0⁠. We use this simplification for the sake of mathematical manageability and conciseness while recognizing that a more general approach not limited to a p ≪ a might provide further nuance but is beyond the scope of this study.

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This work presents a PTI instrument design and parameter analysis, providing a deeper understanding of the interactions affecting the OPL amplitude, i.e., the basis of the instrument’s signal. Our analysis focuses on several components influencing the OPL in the interferometer’s measurement arm. We describe the determining factors for the OPL amplitude in the two primary excitation-to-interferometer laser beam configurations, including the excitation and interferometer beam parameters. We further analyze the influences of gas flowing through the excitation volume in a PT cell and provide insight into limitations when downsizing PTI instruments. The study centers on using characteristic times, which enables the OPL amplitude to be represented as a function of their ratios. Our results shed light on how to maximize PTI signals and avoid signal-reducing effects.

In practice, however, advection requires a sufficiently large flow velocity or excitation length relative to the modulation frequency in order to affect the signal considerably. From an engineering perspective, one can relate this requirement to either the flow velocity or the excitation length. Referring to the latter, the excitation length L e should be substantially longer than the axial diffusion length over an excitation duration, given by L d i f f = 2 D T τ M o d⁠. The requirement L e > L d i f f can be expressed in terms of the characteristic times as 1 < τ M o d / ( 2 τ a x )⁠. When tuning a PTI sensor under these conditions, care should be taken to avoid ratios that could significantly reduce the signal.

This equation shows that toward ω τ r ′ / 2 ≪ 1⁠, the crossed-beam signal becomes proportional to 1 / ω and independent of the laser beam radii. Figure 6 shows the ratio of the crossed-beam OPL amplitude for finite τ r ′ to the highest achievable amplitude at τ r ′ → 0 as a function of the (angular) modulation frequency. The lines show the fractions from 75 to 95% of the maximum OPL amplitude.

Ulrich Radeschnig, Alexander Bergmann, Benjamin Lang; Optimization of the optical path length amplitude for interferometric photothermal gas and aerosol sensing considering advection: A theoretical study. J. Appl. Phys. 7 March 2024; 135 (9): 094501. https://doi.org/10.1063/5.0184357

In photothermal interferometry, the signal is derived from the phase shift δ ϕ of an interferometer’s laser beam relative to a reference. This phase shift occurs as the laser beam propagates through a sample volume whose RI properties are altered by photothermal heating.1,60 The interferometer laser beam’s phase shift accumulates along its propagation direction1,60 and depends on its radial intensity distribution.29,32,40 In the following, the interferometer laser beam is referred to as probe laser beam.

When ω τ r / 2 ≫ 1⁠, it can be expected that Δ T p a t h for b = 2 a is equal to that for b → ∞⁠. This expectation can be explained, as described in Sec. III B, because radial thermal diffusion during one modulation period does not significantly expand the heated region beyond the radius of the excitation laser. Thus, if ω τ r / 2 ≫ 1⁠, and given a duct whose radius is sufficiently sized to capture the entire excitation laser beam radiation, the wall influence on the signal is negligible. By contrast, in the regime ω τ r / 2 ≪ 1⁠, the influence of the duct wall requires additional considerations when optimizing a PTI sensor.

When ω τ r ′ / 2 ≪ 1⁠, radial thermal diffusion is dominant. In this regime, the dependence between signal, modulation frequency and laser beam radii is different from the regime where radial thermal diffusion is not dominant. The behavior of function f coll ( ω τ r ′ / 2 ) mirrors that of the absolute value of the natural logarithm, ln ⁡ ( ω τ r ′ / 2 )⁠, characterized by a similar dependency on changes in the argument. Consequently, within this regime, the signal for the collinear beam configuration is proportional to both ln ⁡ ( 1 / f ) and ln ⁡ ( 1 / ( a 2 + a p 2 ) )⁠.

The main advantage of the crossed-beam configuration is that it generally allows for a more simplistic design. It can lead to an improved signal-to-noise ratio by requiring fewer beam-splitting elements. Fewer beam-splitting elements can mitigate effects like imperfect filtering, unwanted reflections, dispersion, and mechanical alignment challenges. Several studies have shown the great potential of the crossed-beam configuration. The measurement of RI change is, for example, performed with an FPI34–39,41,42 or inside a diode-pumped solid-state laser resonator (DPSSL).45,47

Summarizing the key findings of this analysis, the collinear beam configuration can achieve a significantly higher OPL amplitude, i.e., signal, due to the scalable interaction path length (cf. Sec. III A). However, several external factors can influence the selection of a basic configuration for practical applications.

Photothermal spectroscopy, and more specifically photothermal interferometry (PTI), is a highly sensitive technique for measuring gas and aerosol concentrations. Numerous implementations of different PTI configurations have demonstrated the versatility of the technique. This theoretical study presents a comprehensive analysis and an optimization of the PTI optical path length (OPL) amplitude using characteristic times. We investigate how the OPL amplitude depends on the dimensions and orientations of the interferometer laser beam and the continuous-wave excitation laser beam. This analysis quantifies the impact of advection on the OPL amplitude based on the relative orientation of the two laser beams and the gas flow direction. It is analytically shown that the possibilities for photothermal OPL amplitude optimization are limited when thermal diffusion is dominant. Theoretically, advection has the potential to double or cancel the OPL amplitude, depending on the specific configurations. In summary, we provide an in-depth understanding of the design and parameter considerations required when tailoring and optimizing a PTI sensor for different fields of applications.

The demonstration of efficient flow-induced thermal transport may lead to the question of how the signal might be affected if the laser beams do not intersect. As stated, Eq. (7b) describes the temperature change for the orthogonal-flow configurations. However, both orthogonal-flow configurations (i.e., OP- and OX-configurations) do not provide a mathematically distinct argument separation of the radial and the axial components. The lack of a distinct argument separation contrasts the parallel-flow configurations in Eq. (7a), where δ T is separated into the radial component (i.e., exponential term) and the flow component (i.e., complementary error function term). An additional challenge in the description of the OP-configuration arises from the orthogonality of both the probe and excitation laser beams to the flow direction. As a consequence, an inhomogeneous radial flow profile within the duct cannot be disregarded. Therefore, we limit our analysis to the temperature distribution at the center of the duct, i.e., at ( y , z ) = ( 0 , 0 )⁠.

Examining the influences of both laser beam radii for the collinear and cross-beam setups (cf. Sec. III B), we describe how the signal increases as ω τ r ′ decreases (cf. Fig. 6). The laser beam with the larger radius has a more significant influence on τ r ′ and is, therefore, the more significant contributor to the signal. This implies that if the probe laser beam is much narrower than the excitation laser beam, its radial size will have minimal effect on the signal and vice versa. In the regime ω τ r ′ / 2 ≫ 1⁠, the signal is proportional to 1 / f for both configurations and proportional to 1 / a 2 + a p 2 for the crossed-beam and to 1 / ( a 2 + a p 2 ) for the collinear beam configuration. In the regime ω τ r ′ / 2 ≪ 1⁠, however, radial thermal diffusion has a critical effect on the signal. Here, radial thermal diffusion dominates over the excitation laser beam’s radius in determining the heated region during one modulation period. The capability to increase in the signal by lowering ω τ r ′ / 2 is reduced because thermal broadening leads to a diminished temperature difference over a modulation cycle: When ω τ r ′ / 2 ≪ 1⁠, the signal is proportional to 1 / f for the crossed-beam, and to ln ⁡ ( 1 / f ) and ln ⁡ ( 1 / ( a 2 + a p 2 ) ) for the collinear beam configuration, respectively. In the cross-beam configuration, the signal reaches saturation when the laser beams are narrowed below specific values, depending on the modulation frequency. We provide a quantitative estimate (cf. Fig. 6) of the point where reducing the laser radii will no longer result in a considerably beneficial signal increase.

Suppose the flow velocity is set to zero. In that case, this result is equivalent to Eq. (11) because the advection time then approaches infinity, and the terms in parentheses are reduced to a complementary error function with argument i ω τ r / 2⁠.

Figure 8 shows the advection-dependent signal behavior over ω τ r / 2⁠: The solid curves represent the normalized path-integrated temperature amplitude for several ratios of 2 τ A / τ r⁠. It is desirable to identify a cut-off frequency below which the signal remains above a certain percentage of the threshold given by Eq. (18). If we define a cut-off angular modulation frequency ω c = 2 π f c = 1 / τ A⁠, the signal is approximately 95 % of its limit value, irrespective of τ r at this frequency. When ω is chosen with the definition of ω c⁠, then ω c τ r / 2 = τ r / 2 τ A⁠. In this case, the path-integrated temperature amplitude in Eq. (18) depends only on the ratio 2 τ A / τ r⁠. The inverse of this ratio can be used to define a radial Péclet number P e r = τ r / 2 τ A⁠.

This analysis includes suggestions for PTI and related PTS applications, such as for recent gas and aerosol sensing,34–37,39,41,42,45–47,71 including using a narrow probe and excitation laser beam radius, respectively, and a reduced modulation frequency with a square-wave excitation modulation. However, the signal enhancement from laser beam narrowing is limited in the crossed-beam configuration. The values for which this limitation becomes significant depend on further parameters, such as the excitation modulation frequency. Nevertheless, it should also be kept in mind that narrowing the laser beams can increase the susceptibility to mechanical vibrations since the total volume of the overlap can vary more quickly when the laser beams are thinner. A lower excitation modulation frequency will produce a higher signal. However, the relation between a lowered modulation frequency and a higher signal is not constant and depends on the regime given by ω τ r ′ / 2⁠. Additionally, the choice of modulation frequency is often a compromise between signal and (ambient) noise. When designing and engineering fast response sensors, the effects of advection on the signal are essential to consider: The gas flow through the measurement volume can double the signal, but it can also limit or severely reduce it. Other parameters, such as the excitation modulation frequency and the excitation length, must be tailored appropriately to the gas flow, or vice versa. When focusing on miniaturized PTS sensors with small system volumes, including hollow-core fiber PTI systems, a potential impact on the signal due to heat loss at the system boundaries should be considered.

Illustration of the spatial and temporal temperature change for the (a) parallel and (b) orthogonal flow-to-excitation laser beam orientations from Eqs. (7a) and (7b), respectively. (a) The excitation laser beam propagates in the same direction as the flowing gas. Excitation of the gas is considered for x ≥ 0 (starting at the y–z-plane). The distance between x = 0 and x > 0 is denoted as excitation length L e⁠. (b) The excitation laser beam propagates orthogonally to the flowing gas. The distance between x = 0 and x > 0 is denoted as D⁠.

Particularly useful in the description of ΔOPL are characteristic times. Representing the underlying mechanisms in terms of relations between parameters and characteristic times can allow for a more general description. The characteristic times used in our analysis are listed in Table II.

Sketch of the interaction path length for the crossed-beam and the collinear beam configuration. (a) In the crossed beam configuration, the integration boundaries extend from − ∞ to ∞⁠, although the main contribution stems from the volume the laser beams cross. (b) In the collinear beam configuration, the axial overlap length determines the interaction length. The integration boundaries range from 0 to L p⁠. Radial thermal diffusion expands the volume influenced by the excitation laser beam and can significantly affect the signal (cf. Sec. III B).

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Our analysis considers a modulated, continuous-wave excitation laser beam as the heat source. The resulting oscillating part of the OPL change, given by Eq. (2), determines the PTI signal. These generated, oscillating OPL changes might also be termed OPL modulation. The signal of interest in this analysis is the amplitude of the oscillating part of δOPL(t).

PTI signal of the PX-configuration as normalized path-integrated temperature amplitude Δ T p a t h P X for P e a x ≫ 1⁠. Δ T p a t h P X is plotted over ω τ r / 2 for various ratios of the characteristic advection time τ A and the radial thermal diffusion time τ r (blue, orange, green, and cyan curves) and for gas at rest (black curve). Advection limits the signal if τ A is significantly lower than the modulation frequency (horizontally running lines). A 95% frequency cut-off relative to the value at ω → 0 occurs at f = 1 / ( 2 π τ A ) (dotted cyan curve). Notably, the possibility of a nearly complete signal cancelation should be considered for 2 τ A / τ r ≪ 1 (green curve).

The solid black curve corresponds to the situation with gas at rest. The dotted cyan curve indicates the 95% normalized path-integrated temperature amplitude over ω c τ r / 2 (i.e., over P e r⁠). A periodic modulation of the signal with ω τ r / 2 can be observed, which is absent for gas at rest. A near-complete signal cancelation can be seen for the curve with 2 τ A / τ r = 0.001 in the range of ω τ r / 2 ≫ 1 (cf. green curve).

Summary of the analyzed parameters and their influence on the PTI signal, specified as being the amplitude of the OPL change. The map shows, per segment, the proportionality between parameter and signal (indicated by the symbol ∝⁠) and significant relationships. The extent to which most parameters affect the signal depends on the configuration and the regime determined by ω τ r ′ / 2 or ω τ r / 2⁠. Adding two flow-to-excitation laser beam directions to the crossed-beam and collinear beam scenarios results in four configurations. The parallel-flow configurations allow a more extended analysis than the orthogonal-flow configurations. The limits of the advection-induced multiplication factors (values in the square brackets) are obtained under the assumption τ A ≪ τ a x⁠. An infinitely narrow probe laser beam is assumed in the analysis of the advection and the PT cell’s duct radius.

Photothermal spectroscopy (PTS) is a highly accurate sensing technique capable of measuring a wide variety of gas and aerosol species. Its fundamental principle relies on quantifying the heating of a sample by the absorption of electromagnetic radiation. Typically, modulated laser radiation is used to excite the analyte, resulting in periodic thermal fluctuations in the irradiated volume. The thermal fluctuations are proportional to the concentration of the substance.1

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Numerous publications have addressed the dependencies between parameters and signal-related aspects such as temperature distribution, phase shift, and OPL change on a theoretical basis for multiple PTS systems.1,17,38,50–59 However, several aspects of the relationships between system design parameters and the OPL amplitude have remained unaddressed. Particularly for fast-response instruments, which must exhibit a significant gas exchange rate, the description of the dependence of OPL amplitude and gas flow has remained incomplete.

Parameter optimization for the orthogonal-flow cases (OP- and OX-configuration) is particularly challenging due to the nature of their mathematical representations. However, advection significantly limits the signal when v / 2 f exceeds the effective excitation path length. Increasing the signal at comparatively low modulation frequencies or high flow velocities is possible by positioning the excitation laser upstream of the probe laser beam (cf. Fig. 11). The separation between the laser beams becomes particularly relevant if there is no intention to introduce the excitation laser beam into the PTI cell at the same position as the probe laser beam. Experimental results confirm that direct laser beam overlap is not a prerequisite for achieving a stable and effective signal in this configuration.42

Considering the fundamental directional possibilities of a moving gas, four distinct setups derive from the crossed-beam and collinear beam configurations: (a) PX-configuration: a parallel-flow crossed-beam configuration, (b) PP-configuration: a parallel-flow collinear beam configuration, (c) OP-configuration: orthogonal-flow collinear beam configuration, and (d) OX-configuration: orthogonal-flow crossed-beam configuration. For the advection analysis, the length and radius of the PT cell duct (illustrated as a circular duct) are presumed to be large enough not to affect the signal. Furthermore, the probe laser beam (green laser beam) is considered a line. The excitation length L e is the distance a gas is irradiated prior (PX, OP, OX-configuration) or along (PP-configuration) the probe laser beam axis. The interaction path length L p is, in these illustrations, the length of the probe laser beam. In the PP-configuration, L e = L p⁠.

(a) Distance D o p t between excitation and probe laser beam axes as a function of the ratio τ r / τ A for τ A / τ M o d → 0⁠, where the highest signal is obtained at the duct center. The y axis is scaled in units of a⁠. D o p t increases with higher v and a⁠, respectively. The top x axis gives a comparison using representative parametric variables. (b) Temperature amplitude at the duct center for an exemplary v of 0 and 0.5 m/s, respectively, and D of 0 and D o p t⁠, respectively. Further, a = 0.5 mm and D T is similar to ambient air. The comparison of the two advection-influenced cases (solid green and dashed blue curves) shows that Δ T c e n t e r is higher for D = D o p t than for D = 0 in the low-frequency regime (gain by beam separation). An advection-induced signal enhancement may be achieved in the frequency domain before signal saturation dominates at lower frequencies (gain by advection). The thin gray curve corresponds to the situation for gas at rest with D = D o p t⁠.

This work provides an in-depth analysis of the main parameters influencing the PTI signal, specified as amplitude of the optical path length, for various architectural configurations. In scenarios excluding advection, our study focuses on the spatial arrangements and dimensions of the excitation and probe laser beams. A differentiation is made between a collinear and a crossed-beam configuration. We express specific signal dependencies as ratios by employing characteristic times. Using ratios allows categorizing the sensor’s operational regimes into distinct domains, each subject to unique optimization criteria. Our findings extend existing descriptions of the PTI signal from sources such as Refs. 1,17,32,60, and 63 in several aspects. In our analysis of gas flow through the PT cell, we characterize its effect for four configurations. We assess and quantify the extent to which advection can increase, limit, or decrease the signal. The results contribute to flow-related descriptions found in the literature, such as Refs. 38,50,53,54, and 60.

Signal gain function for the PX-configuration as a function of the ratio between the characteristic advection time τ A and the modulation duration τ M o d for P e a x ≫ 1⁠. For ω τ r / 2 → ∞⁠, advection can lead to a doubling of the signal when τ A / τ M o d = 1 / 2 + m⁠, where m ∈ N 0⁠. Conversely, complete signal annihilation can occur when τ A / τ M o d = m⁠. When ω τ r / 2 approaches values equal or lower one, the peak positions start to shift toward lower τ A / τ M o d (dashed black and dashed–dotted red curve). Further, the influence of advection on the signal diminishes with decreasing ω τ r / 2⁠. For ω τ r / 2 → 0⁠, the maximal value of g A is 1.37.

Summary of the analyzed parameters and their influence on the PTI signal, specified as being the amplitude of the OPL change. The map shows, per segment, the proportionality between parameter and signal (indicated by the symbol ∝⁠) and significant relationships. The extent to which most parameters affect the signal depends on the configuration and the regime determined by ω τ r ′ / 2 or ω τ r / 2⁠. Adding two flow-to-excitation laser beam directions to the crossed-beam and collinear beam scenarios results in four configurations. The parallel-flow configurations allow a more extended analysis than the orthogonal-flow configurations. The limits of the advection-induced multiplication factors (values in the square brackets) are obtained under the assumption τ A ≪ τ a x⁠. An infinitely narrow probe laser beam is assumed in the analysis of the advection and the PT cell’s duct radius.

Signal gain function for the PP-configuration plotted against the ratio of the characteristic advection time τ A to the modulation duration τ M o d for P e a x ≫ 1⁠. Toward ω τ r / 2 → ∞ (solid green curve), the flow-induced signal gain peaks at approximately 1.26 for τ A / τ M o d at approximately 0.65. Toward ω τ r / 2 → 0 (dashed–dotted red curve), the signal ceases to be influenced by advection unless τ A / τ M o d approaches zero.

The relationship between a lowered modulation frequency and an increased signal has been repeatedly confirmed both theoretically and experimentally.1,17,36,38,60,63,64 In the (more straightforward) case for which ω τ r / 2 ≫ 1⁠, the relation between signal and modulation frequency can be understood as follows: A decrease in modulation frequency increases the duration of the modulation cycles. The increased duration results in more prolonged heating and cooling periods. These prolonged periods, in turn, generate higher measurable temperature differences (i.e., δ T⁠), contributing to an increase in the signal. The functional relationship between a reduced excitation laser beam radius and an enhanced temperature change is, in the broader PTS context, outlined in Refs. 1,52,53,63,65, and 66. The explanation for the reciprocal relationship between the squared radius of the excitation laser beam and the signal at a particular position covered by the excitation laser beam is that by reducing the radius, irradiance increases.

Values of the signal-determining terms f cross (crossed-beam configuration, solid blue curve) and f coll (collinear beam configuration, dashed black curve) as a function of ω τ r ′ / 2⁠. For ω τ r ′ / 2 ≫ 1⁠, f cross and f coll are identical and both configurations yield equal OPL amplitudes if K 1 = K 2⁠. Both curves increase toward reduced ω τ r ′ / 2 but exhibit a noticeable curvature toward the horizontal in the regime where radial thermal diffusion becomes significant (i.e., ω τ r ′ / 2 < 1⁠). Both curves approach infinity for ω τ r ′ / 2 → 0 +⁠. The top x axis offers a comparison using representative parametric variables.

Signal gain function for the PX-configuration as a function of the ratio between the characteristic advection time τ A and the modulation duration τ M o d for P e a x ≫ 1⁠. For ω τ r / 2 → ∞⁠, advection can lead to a doubling of the signal when τ A / τ M o d = 1 / 2 + m⁠, where m ∈ N 0⁠. Conversely, complete signal annihilation can occur when τ A / τ M o d = m⁠. When ω τ r / 2 approaches values equal or lower one, the peak positions start to shift toward lower τ A / τ M o d (dashed black and dashed–dotted red curve). Further, the influence of advection on the signal diminishes with decreasing ω τ r / 2⁠. For ω τ r / 2 → 0⁠, the maximal value of g A is 1.37.

As in the previous section, the parameters n 0 and T 0 are summarized as constant c⁠. With c as the proportionality constant, the OPL change is directly proportional to the temperature change. Since a p → 0⁠, the following analysis focuses on the description of the temperature, which we consider preferable to the OPL change for tangibility.

The functions f cross and f coll increase with decreasing ω τ r ′ / 2⁠. For ω τ r ′ / 2 ≫ 1⁠, the OPL amplitude is proportional to (1) 1 / f (both configurations) and to (2) 1 / a 2 + a p 2 (crossed-beam configuration) and 1 / ( a 2 + a p 2 ) (collinear beam configuration). However, radial thermal diffusion begins to adversely affect the signal when approaching ω τ r ′ / 2 < 1⁠. The adverse effect of radial thermal diffusion is caused by the expansion of the heated volume beyond the initial radius of the excitation laser within the time of one modulation cycle. Reducing ω τ r ′ / 2 below one shifts the primary interaction volume from the volume directly irradiated by the excitation laser beam to the volume affected by thermal diffusion. This shift is responsible for the curvature of the curves toward the horizontal in Fig. 5.

Values of τ r ′ for which the crossed-beam OPL amplitude ranges between 75 and 95% of its maximum at τ r ′ → 0⁠, plotted as a function of the modulation frequency. The signal starts to saturate toward smaller laser beam radii if the modulation frequency remains constant. The y axis on the right-hand side shows the combined laser beam size for a thermal diffusivity similar to ambient air.

The signal increase at b p e a k can be attributed to a beneficial utilization of heat dissipation at the duct wall: During the heating period, the heat sink is sufficiently far away from the heated region to significantly reduce the temperature rise on the integration path (i.e., the probe laser beam). However, during the subsequent cooling period, this previously generated heat spreads further toward the heat sink, where it dissipates. During a complete modulation cycle, the heat dissipation, if properly timed, can lead to an increased overall temperature difference (i.e., Δ T p a t h⁠). Yet, when b < b p e a k⁠, the heat sink begins to counteract heat generation even during the heating period. In this case, Δ T p a t h notably diminishes with reducing duct radii. This diminution causes the drop in the curves for b / b p e a k < 1 in Fig. 12(a).

Hollow-core fiber PTI systems provide very high sensitivities,24–26 using interferometer types like Mach–Zehnder,27,28 Sagnac,29,30 or Fabry–Pérot.31–33 However, the comparatively small flow-guiding geometry of hollow-core fibers increases response time and can induce modal interferences that raise susceptibility toward mechanical disturbances.3,24 By contrast, recent PTI systems incorporating an FPI into a PT cell (i.e., a gas cell) have presented being a highly sensitive and selective gas34–41 and aerosol42 sensing method with potential for miniaturization. A fiber-coupled PTI-FPI system with a rigid Fabry–Pérot etalon inside a PT cell has demonstrated remarkable insensitivity against mechanical vibrations.36 Presented PTI-FPI systems have relied on either commercially available36–38,42 or self-built34,35,39,41,43 Fabry–Pérot etalons. These properties make fiber-coupled PTI systems using a PT cell particularly suitable when aiming toward a highly sensitive and selective gas or aerosol sensor with good mechanical robustness, short response times, and reduced size. Other configurations related to PTI gas sensing, such as intracavity PTS, have also demonstrated high sensitivity with reduced system size.44–47 Here, intracavity refers to excitation and heat absorption inside the interferometer laser resonator.48,49

Depending on the specific modulation frequency targeted, there is a limit to the size of the laser beams, below which a considerable benefit in signal optimization can no longer be expected. Consider measuring a gas mixture with a thermal diffusivity similar to that of ambient air (i.e., D T = 2.2 × 10 − 5 m 2 a − 1⁠).67 When, for example, having the modulation frequency set to 100 Hz and using laser beams with a 2 + a p 2 below 100  μm, the maximal attainable OPL amplitude is already above 85% compared to that for infinitesimally small laser radii (cf. gray circle in Fig. 6).

Illustration of the spatial and temporal temperature change for the (a) parallel and (b) orthogonal flow-to-excitation laser beam orientations from Eqs. (7a) and (7b), respectively. (a) The excitation laser beam propagates in the same direction as the flowing gas. Excitation of the gas is considered for x ≥ 0 (starting at the y–z-plane). The distance between x = 0 and x > 0 is denoted as excitation length L e⁠. (b) The excitation laser beam propagates orthogonally to the flowing gas. The distance between x = 0 and x > 0 is denoted as D⁠.

Advection can enhance the PTI signal compared to the amplitude with gas at rest. We introduce an advection gain function, denoted as g A⁠. The advection gain function represents the advection-induced change in the signal compared to the signal with the gas at rest. If the modulation frequency is held constant, it is reasonable to plot g A as a function of the ratio between the advection and the modulation cycle duration, i.e., τ A / τ M o d⁠. Figure 9 illustrates the signal gain with respect to τ A / τ M o d for ω τ r / 2 that ranges from zero to infinity: For ω τ r / 2 ≫ 1⁠, advection can result in a signal amplification toward a factor of two, when τ A / τ M o d = 1 / 2 + m⁠, with m ∈ N 0⁠. In contrast, the signal can be quenched when τ A / τ M o d = m⁠. For ω τ r / 2 ≪ 1⁠, the advection gain function is considerably less distinct and approaches one for τ A / τ M o d ≫ 1⁠. The gain function approaches a peak of 1.37 at τ A / τ M o d approximately 0.34 (cf. dashed–dotted red curve).

The advection-induced effects, at least in the center of the duct, are comparable to that of the PX-configuration. In all three configurations (i.e., PX, OP, OX), gas is heated upstream of a probe laser beam whose axis is orientated orthogonally to the flow direction. However, given the “blurry” excitation path, achieving an advection-induced signal gain beyond that of the PX-configuration can be considered not plausible for both orthogonal-flow configurations. Similar to the previous configurations, an advection-induced signal limitation predominates for τ A / τ M o d ≪ 1⁠, evident by the horizontal part of the curves in Fig. 11(b).

An outline of how the analyzed parameters affect the signal, i.e., the OPL amplitude, for a given configuration is presented in Fig. 3. The figure summarizes the main findings and provides a compact overview.

With advection, an optimal distance between the laser beam axes exists that maximizes the signal. This distance is denoted as D o p t and can be calculated numerically, depending on the particular flow velocity and excitation laser beam radius [cf. sketch in Fig. 11(a)]. Figure 11(a) presents D o p t at the duct axis in units of a (y axis) to the ratio τ r / τ A (x axis) for τ A / τ M o d → 0⁠. It can be seen that D o p t increases with increasing gas flow velocity. For a chosen gas flow velocity, D o p t increases with a growing radius of the excitation laser beam. If the distance between the laser beam axes is below D o p t⁠, the maximum temperature difference decreases because of a shortened effective excitation path. Conversely, suppose the distance exceeds D o p t⁠: In this case, the radial thermal diffusion, whose signal-lowering effect grows with increasing beam axes separation, starts to dominate, reducing the maximum achievable temperature difference.

The heating within a modulation period is, among others, determined by the delivered optical power from the excitation source and the absorption coefficient of the target. In most cases, the resulting temperature change is too small to be measured by conventional means. Measuring the density modulation-induced change of the refractive index (RI) is a method to provide the high sensitivity necessary. Among the first photothermal (PT) methods for measuring the change in the RI caused by the absorption of a periodically modulated excitation laser beam were thermal lensing and deflection spectroscopy,1–5 which are considered to offer a broad range of applications in analytical chemistry, ranging from gas sensing to microfluidics.6–13

In the PX-, OP-, and OX-configuration, respectively, the probe laser beam is orientated orthogonally to the direction of the advection. Such an orientation could be advantageous for certain applications. More specifically, it could ease the integration of a gas-open Fabry–Pérot cavity into the PT cell. Unlike single-pass interferometers, Fabry–Pérot interferometers can provide an increased signal owing to their finesse.18 The PP-configuration is commonly associated with the use of single pass interferometers,14,17,20,22,23 which lack this advantage.

In conclusion, the parameters with the most substantial contribution to the signal are the orientation of the laser beam axes relative to each other, the modulation frequency, and the radii of the laser beams. Depending on the configuration and parameters used, advection can increase, limit, or reduce the signal. Separating the laser beams may be a favorable option if an orthogonal-flow configuration is used. The influence of the PT cell wall can be considered insignificant when heat diffusion does not significantly expand the irradiated volume radially over a modulation period.

Illustration of the expression given in Eq. (2) to obtain the volume-integrated OPL change for a collimated probe laser beam: When propagating through a volume with a spatial and temporal RI distribution, the volume-integrated OPL change depends on the laser beam’s interaction path length (propagation direction) and intensity distribution (radial direction). Equation (2) expresses the volume integration separated as (1) a line integration along the interaction path ( d l ) followed by (2) an area integration over the intensity-weighted cross section of the laser beam ( d A )⁠. For a probe laser beam that has a Gaussian intensity distribution, the central region has the most significant contribution to the OPL change.

A wider probe laser beam increases ω τ r ′ / 2 and thus reduces the influence of the radial thermal diffusion. This influence reduction is because a broader probe laser beam covers a larger interaction volume. However, increasing the probe laser beam radius inherently lowers the signal. Yet, this lowering can be considered insignificant if the excitation laser beam radius, along with the corresponding radial thermal diffusion, expands substantially beyond the radius of the probe laser beam.

As with the previous configuration, advection can have an increasing or decreasing effect on the signal in the PP-configuration. Similarly, the advection’s impact on the signal depends on the ratio of modulation frequency, flow velocity, and excitation length. Again, advection becomes a limiting factor for the signal when the excitation duration of the gas is primarily determined by the time a target spends on the excitation path rather than by the modulation period.

In the parallel-flow case [Eq. (7a) and Fig. 2(a)], the origins of the y- and z-axes coincide with the excitation laser beam axis. The origin of the x axis is at the position where the initial irradiation of the gas is considered [indicated by the plane at x = 0 in Fig. 2(a)]. The distance gas flows from x = 0 to x > 0 is referred to as the excitation length L e⁠. An example of a PTI application with a specified excitation length can be found in Ref. 38.

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(a) Path-integrated temperature amplitude Δ T p a t h as a function of the duct radius for the crossed-beam (top) and collinear beam (bottom) configurations, plotted for exemplary values of f and a⁠. The axes are scaled in terms of b p e a k (x axis) and Δ T p a t h ( b p e a k ) (y axis). Each curve peaks at b p e a k⁠, where heat removal by the duct wall causes the temperature difference to increase. For b < b p e a k⁠, the heat sink starts to significantly reduce Δ T p a t h⁠. For b ≫ b p e a k⁠, the duct wall does not affect the signal. We exclude cases for b < 2 a⁠, considering direct radiation on the duct wall insignificant. (b) Estimation of b p e a k in units of a (left y axis, black curves) and the associated gain in Δ T p a t h when compared to the condition with b → ∞ (right y axis, green curves). When ω τ r / 2 ≫ 1⁠, the duct wall can be considered to have a negligible effect on the signal if its radius is kept large enough to capture the radiation of the excitation laser beam completely.

Illustration of the expression given in Eq. (2) to obtain the volume-integrated OPL change for a collimated probe laser beam: When propagating through a volume with a spatial and temporal RI distribution, the volume-integrated OPL change depends on the laser beam’s interaction path length (propagation direction) and intensity distribution (radial direction). Equation (2) expresses the volume integration separated as (1) a line integration along the interaction path ( d l ) followed by (2) an area integration over the intensity-weighted cross section of the laser beam ( d A )⁠. For a probe laser beam that has a Gaussian intensity distribution, the central region has the most significant contribution to the OPL change.

This study assumes instantaneous excitation relaxation times and excludes mass diffusion and natural convection. In general, mass diffusion effects can be assumed to be negligible if the total excitation relaxation time is much shorter than the mass diffusion time.1 Natural convection can be assumed to be negligible if the natural convection velocity is much smaller than the thermal diffusion velocity.17 Variations in the radius of the probe laser due to density variations are not considered. Our analysis further assumes signal extraction via a lock-in amplifier (LIA).

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(a) Path-integrated temperature amplitude Δ T p a t h as a function of the duct radius for the crossed-beam (top) and collinear beam (bottom) configurations, plotted for exemplary values of f and a⁠. The axes are scaled in terms of b p e a k (x axis) and Δ T p a t h ( b p e a k ) (y axis). Each curve peaks at b p e a k⁠, where heat removal by the duct wall causes the temperature difference to increase. For b < b p e a k⁠, the heat sink starts to significantly reduce Δ T p a t h⁠. For b ≫ b p e a k⁠, the duct wall does not affect the signal. We exclude cases for b < 2 a⁠, considering direct radiation on the duct wall insignificant. (b) Estimation of b p e a k in units of a (left y axis, black curves) and the associated gain in Δ T p a t h when compared to the condition with b → ∞ (right y axis, green curves). When ω τ r / 2 ≫ 1⁠, the duct wall can be considered to have a negligible effect on the signal if its radius is kept large enough to capture the radiation of the excitation laser beam completely.

Figure 2 illustrates the context of Eqs. (7a) and (7b). Both flow cases have the advection directed in the x-direction with the velocity v x⁠.

The PX-configuration benefits from an extended excitation path, and the PP- as well as the OP-configurations exploit a more directly scalable interaction path length. The OX-configuration lacks both of these advantageous features. However, the orthogonal orientations in the OX-configuration could allow for a simpler design, providing advantageous flexibility when integrating lasers into a PTI geometry.

When reducing the size of the PTI sensor, it may be desirable to determine the radius of the circular duct of the PT cell where the proximity of the wall begins to affect the signal. In the following, the duct wall is considered an infinite heat sink. This consideration is valid for sufficiently small temperature changes, which can be assumed in this photothermal context.1,17 Heat sinks can reduce and limit the rise in temperature during the heating period by the dissipation of heat at the duct wall. Therefore, they are generally considered to have a negative effect on the signal. However, our analysis shows that if the duct radius is chosen appropriately, the heat dissipation at the duct wall can also increase the temperature amplitude relative to configurations with an infinitely large radius.

This analysis centers on several main parameters influencing the OPL change and amplitude for different PTI designs. These parameters are briefly outlined in Table I. In the following, we use the Greek capital letter Δ to represent the amplitude. Specifically, the amplitude of the oscillating part of δ OPL ( t ) is denoted as ΔOPL, and the amplitude of the oscillating part of δ T ( t ) is denoted as Δ T⁠.

Although the two curves in Fig. 5 bend toward the horizontal as ω τ r ′ / 2 decreases, both curves diverge toward infinity when ω τ r ′ / 2 → 0 +⁠. This can be seen from the representation of f cross and f coll in Eqs. (11b) and (12), respectively. However, from the alternative representation of f cross given by Eq. (11a), it can be seen that the OPL amplitude in the crossed-beam configuration is limited even for ω τ r ′ / 2 → 0 + if the modulation frequency is non-zero. Since both functions f cross ( ω τ r ′ / 2 → 0 + ) and f coll ( ω τ r ′ / 2 → 0 + ) approach infinity, the only point of reference found in Fig. 5 is at ω τ r ′ / 2 = 1⁠. These values are approximately 0.84 and 0.71, respectively. The essential advantage of the dimensionless representation in Fig. 5 is that it is universally valid for all systems, regardless of the specific value of either τ r ′ or ω⁠, respectively.

The  Appendix addresses the influence of the PT cell’s duct radius on the signal. Assuming a duct wall that acts as an infinite heat sink, its effect is negligible when radial thermal diffusion is not dominant, i.e., when ω τ r / 2 ≫ 1⁠. However, this still requires that the duct radius be large enough to receive the total power of the excitation laser beam. A signal enhancement is feasible for ω τ r / 2 ≪ 1 when heat dissipation at the duct wall is properly utilized. We present an estimate for the duct radius as a function of ω τ r / 2 and a⁠, where signal enhancement can be expected (cf. Fig. 12). This estimate further marks the lower limit for the cell radius below which the cell walls cause a significant signal reduction.

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The orientation of the probe and excitation laser beam axes with respect to each other is a fundamental element in designing a PTI instrument. We group the laser beam orientation into two primary configurations: The crossed-beam configuration, in which the two laser beam axes intersect perpendicularly, and the collinear configuration, in which the laser beams are coaxial. Figure 4 provides an illustration of these configurations.

Equations (11) and (12) use a radial thermal diffusion time τ r ′ that also takes into account the radius of the probe laser beam. We denote this characteristic time as the probe laser beam-weighted, or more briefly, the effective radial thermal diffusion time. The effective radial thermal diffusion time is defined as τ r ′ = ( a 2 + a p 2 ) / ( 4 D T )⁠. Further, Γ is the upper incomplete gamma function, and the prefactors from Eq. (2) and (3) are contained in the variable c = ( n 0 − 1 ) / T 0⁠. The expression given by Eq. (11b) provides a better comparison between the crossed-beam and collinear beam configurations. The absolute value of the terms that are a function of ω τ r ′ / 2 are denoted as f cross and f coll⁠, respectively. The configuration-specific prefactors are grouped as K 1 and K 2⁠, respectively.

A key finding from these equations is that the influence of the probe laser beam radius (⁠ a p⁠) on the signal is identical to that of the excitation laser beam radius (⁠ a⁠). Furthermore, the equations show that the broader laser beam dominates the influence. This size-dependent influence implies that the radius of the probe laser beam can be considered negligible when a p ≪ a and vice versa.

The insensitivity of the PP-configuration to changes or variations in the flow velocity can be attributed to the similar orientation of flow direction, excitation, and probe laser beam axes. Given their similar orientation and the fact that ambient air is constantly flowing into the duct, the probe laser beam propagates through both heated (far from the duct opening) and yet-unheated (near the duct opening) regions. The insensitivity and the shift in the ratio τ A / τ M o d are a consequence of the axial temperature averaging over the probe laser beam. Compared to the PX configuration, the constant temperature near the inlet may explain the increased excitation length (or reduced flow velocity or modulation frequency) required for τ A / τ M o d to yield the highest g A⁠.

Values of τ r ′ for which the crossed-beam OPL amplitude ranges between 75 and 95% of its maximum at τ r ′ → 0⁠, plotted as a function of the modulation frequency. The signal starts to saturate toward smaller laser beam radii if the modulation frequency remains constant. The y axis on the right-hand side shows the combined laser beam size for a thermal diffusivity similar to ambient air.

Signal gain function for the PP-configuration plotted against the ratio of the characteristic advection time τ A to the modulation duration τ M o d for P e a x ≫ 1⁠. Toward ω τ r / 2 → ∞ (solid green curve), the flow-induced signal gain peaks at approximately 1.26 for τ A / τ M o d at approximately 0.65. Toward ω τ r / 2 → 0 (dashed–dotted red curve), the signal ceases to be influenced by advection unless τ A / τ M o d approaches zero.

In practice, however, technical constraints often require a setup in which the beam axes are not aligned completely coaxially, resulting in a reduced interaction volume. Some authors claim the alignment to be optimally overlapping41,44 or nearly coaxial.14,18,22 Others state the specific value of the angle between the laser beams, for example, 8 mrad,49 1–2 degrees,20 and 5 degrees.62 A stable alignment of the two laser beams is crucial to prevent the introduction of noise that outweighs the benefits of increased interaction volume.20,23

In the crossed-beam configuration, the interaction path cuts through the cross section of the duct [cf. upper sketch in Fig. 12(a)]. The interaction path length is L p = 2 b⁠. The path-integrated temperature change δ T p a t h ( t ) is calculated by integrating δ T ( r , t ) over r from − b to b⁠. In contrast, for the collinear beam configuration, the temperature change is only relevant at the center of the duct, i.e., at r = 0 [cf. lower sketch in Fig. 12(a)]. The interaction path is oriented along the duct axis. Therefore, as described in Sec. III A, L p is given by the length of the duct, thus δ T p a t h ( t ) = L p δ T ( r = 0 , t )⁠.

Figure 10 illustrates the advection gain function for the PP-configuration. A notable observation is that the PP-configuration shows less distinct advection-induced signal oscillation compared to the PX-configuration: For ω τ r / 2 ≫ 1⁠, g A reaches a maximum of approximately 1.26. In addition, this maximum occurs for τ A / τ M o d at an approximate value of 0.65. By comparison, this is a ratio shift of roughly 30% relative to the first peak position in the PX-configuration. This shift implies that the maximum signal gain, compared to the PX configuration, occurs either with (1) an expanded excitation length, (2) a reduced flow velocity, or (3) a decreased modulation frequency. For ω τ r / 2 ≪ 1⁠, the impact of advection on the signal is minimal, except when τ A / τ M o d approaches zero.

Figure 11(b) presents the temperature amplitude at the duct center Δ T c e n t e r O plotted against the modulation frequency for exemplary flow velocities v and distances D⁠. The superscript in Δ T c e n t e r O indicates that the description is valid for both orthogonal-flow configurations, i.e., the OP- and OX-configuration. The center temperature amplitude is obtained from the oscillating temperature change calculated from Eq. (7b). The calculations use an excitation laser beam radius of 0.5 mm and a flow velocity of 0 and 0.5 m/s, respectively, and compare to a gas stationary case. In the low modulation frequency range, it is observed that the signal increases when the excitation laser beam is placed upstream of the probe laser beam, reaching a peak when D = D o p t⁠. Figure 11(b) further reveals that the temperature amplitude influenced by advection can exceed the signal without advection within a specific frequency range. As observed in the previous configurations, this gain occurs in the regime preceding the onset of advection-induced signal limitations at lower frequencies. Using parameters where thermal diffusion is not considered dominant, this range can be estimated from the ratio between f⁠, v⁠, and the length of the effective excitation path. For D > a⁠, the effective excitation path length is roughly equivalent to the previous definition of the (total) excitation length and approximately the diameter of the excitation laser beam. The effective excitation path length, in this case, is 2 a⁠. Conversely, when D < a⁠, the effective excitation path length can be approximated as the sum of a and D⁠. In this case, the effective excitation path length is a + D⁠. However, in both orthogonal-flow cases (i.e., OP- and OX-configuration), the effective excitation path length is only an approximation because the irradiance is not constant along the excitation path. The irradiance gradually increases (and decreases) along the excitation path because of the excitation laser beam’s radial Gaussian intensity distribution.

The orthogonal-flow configurations [cf. Figs. 7(c) and 7(d)] can be advantageous in situations where short response times are required. Orthogonal-flow configurations can provide short response times because the averaging over the concentration (excitation laser beam) and temperature (probe laser beam) is performed almost exclusively in the radial direction of the duct.

Considering the fundamental directional possibilities of a moving gas, four distinct setups derive from the crossed-beam and collinear beam configurations: (a) PX-configuration: a parallel-flow crossed-beam configuration, (b) PP-configuration: a parallel-flow collinear beam configuration, (c) OP-configuration: orthogonal-flow collinear beam configuration, and (d) OX-configuration: orthogonal-flow crossed-beam configuration. For the advection analysis, the length and radius of the PT cell duct (illustrated as a circular duct) are presumed to be large enough not to affect the signal. Furthermore, the probe laser beam (green laser beam) is considered a line. The excitation length L e is the distance a gas is irradiated prior (PX, OP, OX-configuration) or along (PP-configuration) the probe laser beam axis. The interaction path length L p is, in these illustrations, the length of the probe laser beam. In the PP-configuration, L e = L p⁠.

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Since the early 1980s, researchers have intensified the focus on interferometric methods for measuring the photothermally induced RI change. The RI change is derived from the phase shift of the interferometer laser beam. A phase shift is caused by a change in the sensing volume’s optical path length (OPL). Early applications for gas sensing used Mach–Zehnner14–17 or Fabry–Pérot18 interferometers. Those systems achieved detection limits down to the single-digit parts-per-billion level. However, the setups were described as being highly sensitive to mechanical vibrations.16–18 Nowadays commonly termed photothermal interferometry (PTI),1 these systems were not considered to operate outside a laboratory environment until fiber-based components allowed for a more stable and less bulky setup.3 Fiber-based PTI systems promise a robust and sensitive application with increased flexibility in design. Further PTI systems have employed other types of interferometers, including the folded and the modified Jamin design.19–23 PTI systems that use hollow-core fibers and those using a Fabry–Pérot interferometer (FPI) are regarded as the most promising contemporary techniques for precise and selective trace gas measurements.3

When assuming τ r → 0⁠, Eq. (21) allows one to determine the maximum signal increase feasible when separating the excitation from the probe laser beam: The maximum increase in signal for D > 0 compared to D = 0 is (limited to) a twofold gain at the duct center. However, in the O-configurations, the excitation laser beam is orientated orthogonally to the flow direction. Therefore, it is essential to note that, depending on the geometry, an inhomogeneous flow velocity distribution and the duct wall are additional factors to be considered.

Values of the signal-determining terms f cross (crossed-beam configuration, solid blue curve) and f coll (collinear beam configuration, dashed black curve) as a function of ω τ r ′ / 2⁠. For ω τ r ′ / 2 ≫ 1⁠, f cross and f coll are identical and both configurations yield equal OPL amplitudes if K 1 = K 2⁠. Both curves increase toward reduced ω τ r ′ / 2 but exhibit a noticeable curvature toward the horizontal in the regime where radial thermal diffusion becomes significant (i.e., ω τ r ′ / 2 < 1⁠). Both curves approach infinity for ω τ r ′ / 2 → 0 +⁠. The top x axis offers a comparison using representative parametric variables.

In subsequent analyses, we consider the radius of the excitation laser beam and the radial thermal diffusion during a modulation cycle to be significantly smaller than the radius of the flow-guiding geometry. This consideration allows us to assume a uniformly distributed flow velocity in the radial direction over the region of interest. With this assumption, we can avoid the complexity often associated with a fully developed flow profile and disregard the cell wall’s influence without compromising the validity of the analysis.

PTI signal of the PX-configuration as normalized path-integrated temperature amplitude Δ T p a t h P X for P e a x ≫ 1⁠. Δ T p a t h P X is plotted over ω τ r / 2 for various ratios of the characteristic advection time τ A and the radial thermal diffusion time τ r (blue, orange, green, and cyan curves) and for gas at rest (black curve). Advection limits the signal if τ A is significantly lower than the modulation frequency (horizontally running lines). A 95% frequency cut-off relative to the value at ω → 0 occurs at f = 1 / ( 2 π τ A ) (dotted cyan curve). Notably, the possibility of a nearly complete signal cancelation should be considered for 2 τ A / τ r ≪ 1 (green curve).

In the parallel-flow collinear beam configuration (or PP-configuration), advection’s influence is less distinct compared to the PX-configuration (cf. Fig. 10). When ω τ r / 2 ≫ 1⁠, the highest signal gain is found to be approximately 1.3. However, advection scarcely affects the signal when ω τ r / 2 ≪ 1⁠. The robustness of the PP-configuration to advection may explain why flow is often of little importance in studies with this configuration.

Our previous work38 revealed the effects of advection on the photothermal signal compared to a static gas scenario for the PX-configuration. The study showed that periodic changes with ω τ r / 2 occur in the signal for varying ratios of the excitation modulation frequency to the gas flow rate through a PT cell. Here, we present a more in-depth insight into how and under which conditions advection impacts the signal.