1 Introduction

Ultrafine soot particles are a by-product of the incomplete combustion of biomass and fossil fuels. Emissions of soot particles from various combustion systems and biomass burning have been identified harmful to human health [1] and the second largest contributor to climate forcing only after CO2 [2, 3]. Because of its elemental carbon content, which has a strong ability of light absorption in the ultraviolet, visible, and infrared spectra, the soot emitted into the atmosphere is loosely referred to as black carbon (BC) particles in the atmospheric science community. Unlike CO2, BC particles have a short atmospheric lifetime on the order of several days to several weeks [4]. Therefore, reduction of BC emissions allows a rapid climate response and helps slow down global warming on a fairly short timescale [4, 5]. Our current quantitative understanding of the contribution of BC particles to radiative forcing is subject to large uncertainties, which stems from a lack of not only detailed knowledge of BC particle radiative properties, but also accurate information of the quantity of BC emissions from various sources. The latter is largely due to a lack of reliable and real-time measurement methods for BC mass concentration.

The transportation sector contributes about 19% of the global BC emissions and is considered a large BC emission source [6]. The environmental and health concerns of BC emissions have led to more stringent soot particle emission regulations in the automobile industry, which in turn motivate researchers to develop accurate, reliable, and real-time measurement methods to characterize BC emissions from engine exhaust in terms of mass concentration, size distribution, and compositions (organic carbon vs. elemental carbon). In the measurements of engine exhaust emissions, the raw exhaust is in general heavily diluted with filtered air. Consequently, the diluted exhaust temperature is close to that of the ambient. The BC particles in the atmosphere and soot in the diluted engine exhaust may be referred to as ambient-temperature soot.

Laser-induced incandescence (LII) has been used to develop powerful spatially and temporally resolved in situ laser diagnostics of soot volume fraction in combustion and atmospheric applications, including but not limited to laminar and turbulent flames, automobile and aero-engine exhausts, and the ambient environment; see [7, 8] and the references cited therein. The principle of LII techniques is to rapidly heat soot particles using a high-power pulsed laser of about 20-ns duration to temperatures close to carbon sublimation point (~4000 K) [7, 8]. In essence, LII techniques take advantage of the strong light absorption ability of the elemental carbon content of soot in the visible and infrared spectrum and its high sublimation temperature. Detection and analysis of the resultant broadband thermal radiation, i.e., the incandescence signals, reveal the soot concentration and possibly the primary soot particle size if time-resolved LII signal is detected [7, 9]. Because LII is sensitive to the BC content of the aerosols to be probed, this technique offers a better selectivity for BC measurements in aerosols containing significant content of organic carbon [8], such as in engine exhausts and atmospheric aerosols. LII employing a pulsed laser is also referred as pulsed LII to differentiate it from LII using a continuous-wave laser [8]. The present study concerns the pulsed LII.

Multimode Q-switched Nd:YAG lasers at their fundamental wavelength (1064 nm) have been commonly used as the excitation source of LII measurements. Such laser systems suffer from a fairly poor spatial intensity uniformity and shot-to-shot variation in the beam intensity distribution. The latter is caused by fluctuations in the laser medium, in the pump source, or the stochastic nature of the pulse generation process [10]. Both the temporal laser pulse profile and the spatial distribution of the laser beam energy at the measurement location affect the analysis of the detected LII signal. A top hat or uniform distribution of fluence across the laser beam is preferred because all the soot particles in the measurement volume are subject to the same fluence and are therefore heated to the same peak temperature in LII experiments conducted at the atmospheric pressure. However, it is very difficult to achieve a perfectly uniform spatial distribution of laser fluence even with the use of laser beam shaping optics [7, 8] or a beam homogenizer [10]. Pfadler et al. [10] demonstrated that the application of a microlens array as beam homogenizer can achieve a nearly uniform intensity distribution with less than 5% intensity variation in their planar LII setup. To minimize the spatiotemporal inhomogeneities of the laser beam, it is necessary to spatially filter the beam by placing an aperture at the beam center and relay the image of the aperture to the detection region [11]. Because of the importance of spatial laser beam intensity distribution in LII measurements and modeling, it has been recommended to report the spatial beam profile measured with a beam profiler [7].

The auto-compensating LII (AC-LII) [8, 12], also known as two-color LII in the literature [13], offers several advantages over the conventional ways to implement LII [7, 8] and has become the preferred LII technique in most applications. Calibration of the detection system of AC-LII is performed by placing a radiation source of known spectral radiance at the measurement location and recording the incandescence signals by the photomultiplier tubes [12]. In AC-LII, the absolute LII intensities at two or more wavelengths in the visible are detected. An effective soot temperature can then be derived based on the principle of two-color pyrometry. To this end, it is critical to know the soot absorption function E(m) at the two detection wavelengths. Although only the E(m) ratio is required for determining the soot temperature, the absolute value of E(m) at least at one of the detection wavelengths is needed for determining the soot volume fraction [12]. There is a relatively large uncertainty in the values of E(m) and how it varies with the wavelength in the visible spectrum, as shown in [14] and the references cited therein, which affects the estimated soot temperature by AC-LII and subsequently on the inferred soot volume fraction [15].

When the laser fluence used in LII measurements is sufficiently high to heat soot particles to above 4000 K, there is considerable soot mass loss through sublimation and LII is no longer non-intrusive. For accurate soot concentration measurements at atmospheric pressure, it is preferred to conduct AC-LII measurements in the low-fluence regime, where the peak soot temperatures remain below about 3800 K to avoid significant soot mass loss [12]. It should be noted that this recommendation is different from that made for the conventional LII, where the laser fluence should be in the plateau region (high fluence) to minimize the influences of laser energy attenuation or laser power fluctuations [7]. The fundamental reason for the different required laser fluence in AC-LII and the conventional LII lies in the facts that the soot particle temperature in the measurement volume is inferred from the principle of two-color pyrometry in AC-LII, which automatically accounts for laser energy attenuation or laser power fluctuations, while the conventional LII assumes that the LII signal detected at a single spectral band is proportional to the local soot volume fraction. The latter implies that all the soot particles in the measurement volume reach more or less the same temperature (the soot sublimation temperature). Another important difference between the two LII methods is related to calibration [7, 12]. AC-LII has become the preferred approach to conduct LII experiments, especially for fundamental research, since it is able to provide time-resolved soot temperature, which is not only highly useful from experimental perspective, but also valuable information for the validation of LII models.

It has been reported by Smallwood [16] that the measured soot volume fractions using AC-LII in the exhaust of a diesel engine and in the exhaust of an inverted diffusion flame burner increased dramatically, by a factor of about 5–8, with increasing the mean laser fluence from about 0.4–1.9 mJ/mm2 under the excitation of 1064-nm laser pulse. These experimental observations were obtained under the assumption of temperature-independent soot absorption function E(m). More recently, in their AC-LII measurements using a pulsed Nd:YAG laser at 1064 nm in atmospheric pressure laminar diffusion flames and in the exhaust of quenched diffusion flames, Migliorini et al. [17] also showed that the measured soot volume fraction increased significantly with increasing the laser fluence in the low- to intermediate-fluence regime, if the soot absorption function E(m) is assumed temperature independent. In the laminar flame case, the measured soot volume fraction increases about 40% when the mean laser fluence is increased from 1 to 2.2 mJ/mm2. However, in the exhaust of quenched flame case, i.e., near ambient-temperature soot, the measured soot volume fraction increases by about a factor of 2.5 when the mean laser fluence is increased from 1.6 to 5 mJ/mm2. The strong increase in the measured soot volume fraction with increasing laser fluence, especially for ambient-temperature soot, such as in the exhaust of an engine, a quenched flame, or black carbon in the atmosphere, currently cannot be explained by available LII theories or models. This phenomenon is referred as soot volume fraction anomaly hereafter. The significant underestimate of soot volume fraction at low laser fluences is likely directly associated with the overestimate of soot temperature determined by the ratio of LII signals at two detection wavelengths, albeit the cause of this temperature overestimate is currently not understood. A possible cause for the soot volume fraction anomaly in AC-LII measurements, suggested by Smallwood [16], is the change in E(m) due to several factors that are absent in flames, such as the increased degree of heating from a much lower initial temperature, the cooling process from flame temperatures to a much lower temperature, and the presence of coatings by condensed volatiles or water. Migliorini et al. [17] also explained the increased soot volume fraction with increasing laser fluence by assuming that the ratio of the soot absorption functions at the two detection wavelengths varies with temperature. It should be pointed out that the conjecture of E(m) variation with temperature is somewhat supported by the recent experimental findings that the rapid laser heating indeed induces changes in the soot absorption function E(m) on timescales relevant to LII measurements [18]; however, the observed variation in E(m) by laser heating is not strong enough to explain the significant increase in the inferred soot volume fraction with laser fluence documented in [16, 17].

When soot particle temperatures are non-uniform in the measurement volume, which can be caused by a non-uniform fluence distribution, the effective soot temperature determined by two-color pyrometry in AC-LII is strongly biased toward the peak temperatures [19]. This is simply because the hotter particles associated with the hot spots in the laser beam emit much higher incandescence signals due to the very strong temperature dependence of the blackbody radiation intensity. Although primary particle polydispersity can also lead to non-uniform soot temperature, such effect is significant only at elevated pressures or at delayed detection of LII signals after the laser pulse. Therefore, the laser fluence non-uniformity is likely one of the causes for the overestimation of soot temperature and hence underestimation of soot volume fraction in AC-LII measurements at low fluences. However, the effect of laser fluence non-uniformity on soot volume fraction anomaly has so far not been investigated. Although Migliorini et al. [17] identified the possibility of the contribution of spatial non-uniformity and the shot-to-shot variations of laser fluence to the soot volume fraction anomaly, they dismissed it without providing any evidence. In this study, three sets of AC-LII measurements of soot volume fraction in the exhaust of a MiniCAST soot generator [20] operated at conditions of high elemental carbon content in soot were conducted and the detailed histograms of laser fluence distributions in the measurement volume were compiled. Numerical modeling was also carried out by employing a primary particle-based LII model and the experimental laser fluence histograms to investigate the effect of laser fluence spatial non-uniformity on soot volume fraction anomaly.

2 Experimental setup

In the present study, AC-LII measurements of soot volume fraction over a wide range of laser fluence were conducted using two different setups, which have different lasers, different signal detection spectral bands, and different delays relative to the laser pulse for the analysis of soot volume fraction and temperature, to investigate how lasers and detection strategies affect the extent of soot volume fraction anomaly.

2.1 The MiniCAST soot generator

The Miniature Combustion Aerosol Standard (MiniCAST), from Jing Ltd., is a soot generator commonly used for research and instrument calibration [21–24]. In the present work, a MiniCAST Model 5201 Type C has been used which generates soot using a laminar propane/air coflow diffusion flame quenched by a nitrogen jet [20]. The flow rates of propane and oxidation air are 60 mL/min (at 1 bar and 293 K) and 1.55 L/min (at 1 bar and 293 K), respectively. The quench-nitrogen jet has a flow rate of 7 L/min (at 4 bar). The particle stream is diluted with 20 L/min (at 1 bar) of air. Under these conditions, the elemental-to-total carbon ratio (EC/TC) of the particles is 0.98, measured using a thermal–optical EC-OC Aerosol Analyzer from Sunset Laboratory, following a thermal–optical transmittance protocol (ASTM D6877-13e1, 2013). The soot generator provides a stable source of particles with minimal hour-to-hour and day-to-day fluctuations.

2.2 Three-channel LII setup

Figure 1 shows a schematic of the optical layout of the three-channel LII (3C-LII) setup. It is worth pointing out that this 3C-LII system was developed to investigate other potential causes of the soot volume fraction anomaly, such as the effect of detection wavelengths, that otherwise cannot be achieved with the 2C-LII setup described below. Soot particles are heated using a Q-switched Nd:YAG laser (Big Sky Ultra from Quantel, Serial number 99012502, manufactured in January 1999) at its fundamental wavelength, 1064 nm, with a frequency of 20 Hz and a pulse duration of about 6 ns (FWHM, full width at half of the maximum). The laser is operated at full power, and the beam energy is controlled by means of a half-wave plate combined with a thin-film polarizer. The final polarization direction is tuned to be vertical using a second half-wave plate. The latter was used only to perform 1064-nm scattering experiments but is not necessary for the present AC-LII measurement. Two pairs of metal slits are used to aperture the middle portion of the laser beam, which is imaged at the outlet of the MiniCAST using a pair of optically conjugate lenses, generating a beam with a nearly uniform energy distribution across a 2.2 mm × 2.2 mm profile at the focal plane. The laser beam power, ranging from 0.02 to 0.6 W, is monitored using a power meter. The size, intensity profile, and fluence of the laser beam, varying from 0.21 to 6.19 mJ/mm2, are measured using a beam analyzer. The beam analyzer is mounted at the LII probe volume location, i.e., at the focal plane of the LII signal collection lens assembly (LIISCLA) centered at the MiniCAST exhaust.

Fig. 1
figure 1

Optical layout of the three-channel LII setup, including an Nd:YAG laser, half-wave plates (λ/2), a thin-film polarizer (TFP), a beam dump (BD), an aperture (A), lenses (L), mirrors (M), a power meter unit (PMU), LII signal collection lens assembly (LIISCLA), a demultiplexer box (DM), a long-pass filter (LPF), a beam splitter (BS), band-pass filters (BP), and photomultipliers (PM)

Full size image

Two achromatic lenses, operating at infinite conjugate, collect the LII radiation at an angle of 90 degrees to the laser axis and focus it into an optical fiber. The LIISCLA can generate a 1-mm-diameter beam at its focal point. Therefore, the LII probe volume is about 1.73 mm3. The optic viewing axis is aligned to be coincident with the center of the sheet for LII measurements. The fiber carries the LII radiation to a demultiplexer box where it is collimated using a lens. The collimated LII radiation strikes a long-pass filter mounted at an incidence angle of 15°. The LII radiation to be collected by the blue channel is reflected onto band-pass filter 1 (BP1) and focused onto photomultiplier 1 (PM1, H10304-00-NF from Hamamatsu). Using the spectral intensity of the source, the transmission characteristics of the optics, and the response curve of the photomultiplier module, the center wavelength and FWHM of the light that reaches PM1 can be calculated using the equivalent filter approximation [12] as 444.7 and 61.3 nm, respectively. The same equivalent filter approximation holds over a large range of soot and calibration source temperatures. However, filter corrections need to be calculated from numerical integrations of the spectral terms contributing to the PM response curve under conditions of highly irregular spectral features of the signal detection system.

The LII radiation passing through the long-pass filter strikes a dichroic beam splitter mounted at an incidence angle of 15 degrees. The LII radiation to be collected by the red channel is reflected on BP2 and is focused onto PM2 (H10304-20-NF from Hamamatsu). The center wavelength and FWHM of the light that reaches PM2 are calculated to be 692.1 and 46.7 nm, respectively. The LII radiation passing through the beam splitter strikes BP3 and is focused onto PM3 (H10304-20-NF from Hamamatsu). The center wavelength of the light incident on PM3 is 796.5 nm, and its FWHM is 61.8 nm.

The control voltages applied to the PMs are in the range of 0–1.000 VDC, where a 1.000 VDC control voltage sets a PM bias voltage of 1000 VDC. In the present work, the PMs are operated in non-gated mode. PM output voltages are recorded using an oscilloscope. The oscilloscope is programmed to average 400–1000 time-resolved signals. The LII signals at the three wavelengths are measured simultaneously.

The spectral sensitivity of the PMs, needed for interpreting the measured LII signals, is determined by calibrating them with an extended source of known radiance [12], which is a large-area radiance calibration source (SR-3A from Sphere-Optics) in the present work. This source contains a filament lamp of known brightness temperature, and thus known radiance, and an integrating sphere whose output provides a calibrated, large area, uniform, diffusely radiating source with a near-normal luminance that can be varied over a wide range of radiance.

Although the 3C-LII setup permits detection of LII signals at three wavelengths, only the first and third channel signals detected by PM1 and PM3 were used in the present work. This is because there is a fairly good agreement between the results obtained from all three-channel combinations. Therefore, LII signals of any pair from the 3C-LII setup can be used in the present study without altering the conclusions.

2.3 Two-channel LII setup

Two sets of two-channel LII (2C-LII) measurements have been conducted in this study, one without and one with the use of a DOE. The goal of this preliminary effort is to investigate the effectiveness of a DOE in improving the laser fluence uniformity. The experimental setup for the 2C-LII systems is very similar to the 3C-LII system; therefore, only the different features of the 2C-LII setups are described below. However, it is important to point out that the 2C-LII setup used a different Q-switched Nd:YAG laser (Big Sky Ultra from Quantel, Serial number 00100601, manufactured in October 2000) from that of the 3C-LII setup described above.

2.3.1 2C-LII without DOE

Figure 2a shows a schematic of the optical layout of the conventional 2C-LII setup. The relayed image of the apertured laser beam has a nearly uniform energy distribution across a 3.2 mm tall by 2.0-mm-wide profile, as measured with a beam analyzer (LaserCam HR from Coherent) placed at the beam’s focal point. The maximum laser beam power, measured at the LII probe volume, is 0.38 W which corresponds to a maximum fluence of 3 mJ/mm2. A two-channel demultiplexer box, with the LII signal collection lens assembly directly coupled to the box, collects the LII signal at 33.7 degrees, which is the angle between the laser beam and the viewing optics axis shown in Fig. 2. The diameter of the image of the fiber tip at the measurement location is about 2 mm, leading to a LII probe volume about 11.32 mm3. The collimated radiation entering the demultiplexer box strikes LPF1, which reflects the blue-colored radiation onto BP1. The blue radiation is then focused onto PM1. The calculated center wavelength and band-pass FWHM for this channel are 419.9 and 14.5 nm, respectively. The radiation transmitted through LPF1 strikes LPF2, which reflects the red LII radiation onto BP2. The radiation is then focused onto PM2. The calculated center wavelength and FWHM of this channel are 780.0 and 19.7 nm, respectively. Radiation transmitted by LPF2 is terminated at the end wall of the demultiplexer box.

Fig. 2
figure 2

Schematics of the 2C-LII experimental setup without a DOE, (a), and with a DOE, (b), which include an Nd:YAG laser, half-wave plates (λ/2), a thin-film polarizer (TFP), a beam dump (BD), a ceramic aperture (A), lenses (L), mirrors (M), a power meter unit (PMU), LII signal collection lens assembly (LIISCLA), and a demultiplexer box (DM) with the following optical components: long-pass filters (LPF), band-pass filters (BP), and photomultipliers (PM)

Full size image

2.3.2 2C-LII with DOE

Figure 2b displays a schematic of the 2C-LII setup with a DOE. This setup uses a two-step technique to generate a more uniform laser beam profile. In the first step, a DOE (from Silios Technologies), which has the damage threshold about 80 mJ/mm2, and a plano-convex lens are installed to generate a 6.0 mm × 6.0 mm square image plane of the laser beam with a smooth intensity distribution. In the second step, an aperture, with a 2.5 mm × 2.5 mm central slit, is used to aperture the middle portion of the DOE-homogenized laser beam. The apertured laser beam is then relay-imaged at the outlet of the MiniCAST. The maximum laser beam power, measured at the probe volume, is 0.12 W which corresponds to a maximum fluence of 0.96 mJ/mm2. The DOE used in this setup is designed for a larger laser beam; therefore, the ceramic slit aperture (2.5 mm × 2.5 mm) blocks a significant fraction of the 1064-nm radiation and reduces the fluence at the LII probe volume location to values that cannot heat soot particles up to the optimal temperature near 4000 K. The LII signal collection lens assembly, demultiplexer box, PM controls, and calibration procedure are identical to the description for the conventional two-channel LII configuration described above.

3 LII model

To simulate the effects of non-uniform laser fluence distribution in the measurement volume on the measured soot temperature and volume fraction, the temporal evolution of soot temperature and primary particle diameter is calculated using an LII model. To focus the attention on the potential role of the non-uniform laser fluence to soot volume fraction anomaly, soot is assumed to be composed of isolated primary particles, thereby neglecting the effect of particle aggregation in this study.

The LII model used in this study accounts for only the commonly accepted heat and mass transfer processes, namely laser energy absorption, soot particle internal energy change, heat conduction, and thermal sublimation. Thermal radiation from soot particles contributes negligibly to particle heat loss during LII at atmospheric pressure. The LII model consists of the energy and mass balance equations for a single primary particle written as

$$\frac{1}{6}\pi d_{\text{p}}^{3} \rho_{\text{s}} c_{\text{s}} \frac{{{\text{d}}T}}{{{\text{d}}t}} = C_{\text{abs}} Fq(t) - q_{\text{c}} + \frac{{\Delta H_{\text{v}} }}{{M_{\text{v}} }}\frac{{{\text{d}}M}}{{{\text{d}}t}}$$
(1)
$$\frac{{{\text{d}}M}}{{{\text{d}}t}} = \frac{1}{2}\rho_{\text{s}} \pi d_{\text{p}}^{2} \frac{{{\text{dd}}_{p} }}{{{\text{d}}t}} = - \pi d_{\text{p}}^{2} \beta_{0} p_{\text{v}} \sqrt {\frac{{M_{\text{v}} }}{{2\pi R_{\text{u}} T}}}$$
(2)

where d p is the primary particle diameter, ρs and c s are the density and specific heat of soot, t is time, T is soot temperature, C abs is the absorption cross section of the primary particle and is evaluated using the Rayleigh regime expression, F is the laser fluence, q(t) is the laser power temporal profile corresponding to 1 mJ/mm2 laser fluence, q c is the heat loss rate due to conduction. The last term in Eq. (1) represents the sublimation heat loss rate with M being the mass of primary particle, M v the mean molecular weight of the sublimated species, and ΔH v the heat of sublimation. In Eq. (2), p v is the sublimation pressure, R u is the universal gas constant (8.3145 J/mol K), and β 0 is the effective sublimation coefficient (taken to be 0.2 in this study). Soot conduction heat loss, q c, was calculated using the Fuch’s approach [25, 26]

$$q_{\text{c}} = \alpha \pi \left(\frac{{d_{\text{p}} }}{2}\right)^{2} \frac{{p_{\text{g}} }}{2}\sqrt {\frac{{8k_{\text{B}} T_{\delta } }}{{\pi m_{\text{g}} }}} \frac{{\gamma^{*} + 1}}{{\gamma^{*} - 1}}\,\left( {\frac{T}{{T_{\delta } }} - 1} \right)$$
(3)

where α is the thermal accommodation coefficient, p g is the ambient pressure, k B is the Boltzmann constant, m g is the ambient gas molecule mass, T δ and γ* are the limiting sphere temperature and the averaged specific heat ratio, respectively [25]. In this study, the thermal accommodation α is assigned a value of 0.37 following the study of Snelling et al. [27] and the thermal properties of surrounding gas are assumed to be those of air.

The size parameter of primary particles (πd p/λ) at the blue channel (419.9 nm) of the 2C-LII setup is about 0.24, so the Rayleigh regime approximation can be used and the absorption cross section of primary particle is expressed as

$$C_{\text{abs}} = \frac{{\pi^{2} d_{\text{p}}^{3} E(m)}}{\lambda }$$
(4)

The value of soot absorption function E(m) is assumed to be 0.4 in this study [26]. Most of the thermal properties of soot described in several previous studies [27, 28], such as density, specific heat, thermal conductivity of the surrounding gas, mean molecular weight of the sublimated species, sublimation pressure, and sublimation enthalpy, were also used in this study.

The pulsed laser characteristics used in the modeling are specified according to the experimental conditions. In particular, the laser pulse has a duration of about 6 ns (FWHM) and its temporal profile has been given in [27]. The evolution of soot temperature and primary particle diameter from the onset of the laser pulse is obtained by solving the coupled equations expressed in Eqs. (1) and (2) under known initial conditions (local gas temperature T g, pressure p g, and the initial primary particle diameter d p) and a prescribed laser fluence F. The details of the solution technique and time steps used in the calculations were described in [29].

To properly account for the non-uniform laser fluence distribution across the laser beam, a histogram of the laser fluence distribution over the entire range of fluences has to be generated on a single-shot basis. This procedure is believed the correct way to incorporate the effect of non-uniform laser fluence distribution. In contrast, it is inappropriate to use spatially averaged and perhaps multi-pulse-averaged laser fluence profiles, as was done in a previous study [27], for such purpose. This is because the single-pulse histograms vary somewhat from shot to shot especially in the absence of the DOE. The histogram approach to characterize the spatial laser fluence non-uniformity has also been used by Bladh et al. [30].

To simulate the experimentally observed soot temperature inferred from AC-LII, the effective soot temperature derived from the LII signals at the two detection wavelengths is calculated as follows. The total LII signal at detection wavelength λ, contributed from all parts of the laser beam cross section viewed by the detection optics, can then be expressed as

$$S_{\lambda } = \varTheta \frac{{2\pi c^{2} h}}{{\lambda^{5} }}\sum\limits_{i = 1}^{N} {\left[ {\exp \left(\frac{hc}{{\lambda k_{\text{B}} T_{i} (d_{{{\text{p,}}i}} ,F_{i} )}}\right) - 1} \right]^{ - 1} \frac{{\pi^{2} d_{{{\text{p,}}i}}^{3} E(m)}}{\lambda }} p(F_{i} )$$
(5)

where Θ is a detection system constant, which accounts for the detection wavelength-independent parameters of the detection system such as the laser beam width, single collection solid angle, the filter bandwidth, and PM gain [12], and is independent of detection wavelength, p(F) is the frequency of laser fluence F, and N is the number of fluence bins used in the construction of the fluence histogram or frequency. The effective soot temperature T e, which would represent the temperature of soot particles as deduced from the ratio of LII signals, can be defined as follows:

$$\frac{{S_{{\lambda_{1} }} }}{{S_{{\lambda_{2} }} }} = \frac{{E(m_{1} )}}{{\lambda_{1}^{6} }}\frac{{\lambda_{2}^{6} }}{{E(m_{2} )}}\frac{{\exp (hc/k_{\text{B}} \lambda_{2} T_{\text{e}} ) - 1}}{{\exp (hc/k_{\text{B}} \lambda_{1} T_{\text{e}} ) - 1}}$$
(6)

After making the Wien approximation, i.e., \(\exp (hc/k_{\text{B}} \lambda T) > > 1\), which causes negligible error in LII applications, substitution of Eq. (5) into Eq. (6) leads to

$$T_{e} = \frac{hc}{{k_{\text{B}} }}\left( {\frac{1}{{\lambda_{2} }} - \frac{1}{{\lambda_{1} }}} \right)/\ln \left( {\frac{{S_{{\lambda_{1} }} }}{{S_{{\lambda_{2} }} }}\frac{{\lambda_{1}^{6} }}{{\lambda_{2}^{6} }}\frac{{E(m_{2} )}}{{E(m_{1} )}}} \right)$$
(7)

It should be noticed that the soot temperature in the integrals of Eq. (5) is dependent on time t, the laser fluence F, and the detection wavelengths λ 1 and λ 2. The choice of two detection wavelengths in the visible has a minor influence on the effective soot temperature (less than about 5 K) and can be neglected. In this study, the effective soot temperatures are based on the ratio of modeled LII signals at the two detection wavelengths of the 2C-LII experiments, i.e., 420 and 780 nm. A series of solutions for a range of laser fluence is first obtained by solving the energy and mass conservation equations discussed earlier. These solutions serve as a database for the evaluation of the total LII signals to obtain the effective temperature T e from Eqs. (5)–(7), along with the assumption that E(m) values at the two detection wavelengths are equal.

Once the effective soot temperature is available, the derived relative soot volume fraction can be obtained as

$$f_{\text{v}} = \frac{{{\text{LII}}\text{ }{\text{signal}}}}{{{\text{LII}}\text{ }{\text{signal based}}\text{ }{\text{on}}\text{ }d_{{{\text{p}}0}} \text{ }and\text{ }T_{\text{e}} }} = \frac{{S_{{\lambda_{i} }} (T,d_{\text{p}} )}}{{S_{{\lambda_{i} }} (T_{\text{e}} ,d_{\text{p0}} )}}$$
(8)

where d p0 is the initial primary particle diameter and the wavelength can be λ 1 or λ 2. It should be noted that the derived soot volume fraction reaches a maximum value of unity in the low laser fluence regime (there is negligible soot sublimation) and when all the soot particles across the laser beam have the same temperature. These conditions are in general met only at very low laser fluences for a non-uniform laser beam, which have no practical significance since the resultant LII signals are too low to be detected with reasonable signal-to-noise ratios. In general, the derived soot volume fraction is less than unity.

4 Results and discussion

To investigate the role of laser fluence non-uniformity across the laser beam in the soot volume fraction anomaly in ambient-temperature soot measurements, numerical calculations were conducted for d p0 = 32 nm and T g = 350 K based on our unpublished transmission electron microscope (TEM) analysis of emitted soot and thermocouple measurements. The size of primary soot particles was obtained from the high-magnification (45,000 times) TEM images of the emitted particles. The arithmetic mean of d p is 32.8 nm with a standard deviation of ±6.2 nm, for 157 primary particles. Calculations conducted for a somewhat different value of d p0 around 32 nm showed that the numerical results presented later are not sensitive to the value of d p0. The soot temperature and volume fraction were obtained at 12 ns after the peak of the laser in the 2C-LII experiments. It should be pointed out that the choice of evaluating the soot temperature and volume fraction at 12 ns after the laser peak is somewhat arbitrary. However, it is mainly based on the consideration that it is preferred to derive soot volume fraction at or shortly after the peak of the laser to minimize the influence of differential cooling (different sized particles cool at different rates) on the measurement. Nevertheless, this choice does not significantly affect the experimental results and has no influence on the findings of this study. Correspondingly, the effective soot temperature and the derived relative soot volume fraction were also evaluated at 12 ns after the peak of the laser pulse from the LII modeling. However, results of the 3C-LII experiments were obtained in a different way: The soot temperature was reported as the peak value and the soot volume fraction was reported as the value when its temporal variation is minimal.

4.1 Laser beam fluence distributions

To illustrate the spatial and temporal distributions of beam intensity of the laser used in the two-channel LII setup, Figs. 3 and 4 display four typical single-shot laser beam intensity distributions and the corresponding 50-shot averaged one without and with DOE, respectively. It is clear that there are fairly large spatial non-uniformities within each image and large variations in the laser beam intensity from shot to shot, as shown in Fig. 3a–d, without DOE. The presence of hot spots in single-shot distributions leads to higher soot temperatures, which is reflected in the collected single-shot LII signals, though the multi-shot averaged laser beam intensity distribution is considerably smoother and more uniform, Fig. 3e. Therefore, the use of the spatially averaged laser beam intensity distribution in LII modeling should be avoided. Instead, the histogram approach, which represents the non-uniform laser beam intensity distribution in terms of the frequency of a given laser fluence value, is more adequate to better represent the laser fluence non-uniformity.

Fig. 3
figure 3

Four typical single-shot laser beam intensity distributions at the LII measurement location, (a–d), and the corresponding 50-shot averaged one, (e), in the 2C-LII setup without DOE

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Fig. 4
figure 4

Four typical single-shot laser beam intensity distributions at the LII measurement location, (a–d), and the corresponding 50-shot averaged one, (e), in the 2C-LII setup with the use of DOE

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A comparison between the profiles depicted in Figs. 3 and 4 shows that the use of DOE apparently improves the laser beam intensity uniformity. The use of DOE has two effects on the laser energy profiles. First, the sizes of non-uniform pockets are substantially reduced. Second, the shot-to-shot variation of laser energy distribution seems significantly reduced, since the hot spots appear in similar locations. However, the difference between the 50-shot averaged fluence distribution, Fig. 4e, and the single-shot distributions, Fig. 4a–d, implies that there are still some shot-to-shot variations. In addition, there are still many hot spots of much smaller size in the single-shot laser beam intensity distributions at the LII measurement volume with the use of DOE, as shown in Fig. 4a–d, albeit the overall uniformity seems improved. The spatially averaged laser beam intensity distribution, depicted in Fig. 4e, again leads to a more uniform laser beam intensity distribution than the single-shot ones.

The histograms of the two lasers used in the two AC-LII setups are shown in Fig. 5. Hereafter, Laser 1 and Laser 2 represent the lasers used in two-channel and three-channel LII experiments, respectively. It is useful to emphasize that the laser beam fluence histograms of the two LII setups are the only concern from this point on. These histograms were obtained by analyzing and averaging 50 single-shot laser fluence distributions in the area viewed by detection optics across the laser beam as follows. A Coherent LaserCam HR Head (Serial Number 0390A06) captured the full image of laser beam at a distance of 415 mm from the laser head corresponding to the sample region. Fifty individual images were recorded at a spatial resolution of 6.5 µm with a 10 bit A to D. A region of the image corresponding to the sample area was selected, and individual histograms were calculated for each of the 50 images. The intensity data from 50 individual images were combined to derive a single histogram representing the 50 individual histograms. A histogram was also derived for the average image. With the DOE in place, it was noted that the histogram derived from the individual pulses was almost identical to that derived from the average pulse. Without the DOE, the histogram from the 50 individual images was distinguishable from that of the average image. The DOE appears to impose an image intensity pattern that is independent of the pulse to pulse variations in the image intensity distribution.

Fig. 5
figure 5

Histograms of laser fluence in the region viewed by the detection optics across the laser beam

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It is evident from Fig. 5 that Laser 1, without or with DOE, has better fluence uniformity than Laser 2, characterized by the smaller variances of fluence around the mean (0.1578 and 0.1510 for Laser 1 without and with DOE, respectively, and 0.2055 for Laser 2), and that the use of DOE slightly improves the laser fluence uniformity of Laser 1. To illustrate the differences observed in the histograms obtained by 50 single-shot averaging and a single-shot, the histogram of a single-shot of Laser 2 is also plotted in Fig. 5. It is seen that the averaged histogram, for 50 single-shot distributions, does not deviate significantly from that of a single-shot histogram. To demonstrate the effect of a highly non-uniform laser fluence distribution on soot temperature and the derived soot volume fraction, an assumed double-δ histogram, also shown in Fig. 5, is considered in the numerical calculations. This assumed double-δ histogram has two peaks: one is at F = 0.18 mJ/mm2 with a frequency of 0.175 and the other one is at F = 0.3 mJ/mm2 with a frequency of 0.075, leading to an average fluence slightly larger than that of Laser 2 (0.216 vs. 0.212 mJ/mm2). It is worth pointing out that the choice of this double-δ histogram is somewhat arbitrary, but based on the following two considerations: First, the higher fluence peak representing identical hot spots has a lower frequency than that of the lower fluence peak. Second, the two fluence peaks are arranged on both sides of the mean fluence of Laser 2 so that the resultant mean value is close to that of Laser 2. The purpose of considering the double-δ histogram is to demonstrate how such a highly non-uniform laser fluence distribution affects the soot temperature and the AC-LII derived soot volume fraction over a fairly wide range of mean laser fluence. Finally, a perfectly uniform or top hat laser fluence, which corresponds to a δ-histogram peaked at the mean fluence (not shown in Fig. 5), is also considered in numerical calculations.

4.2 Effective soot temperature

The AC-LII measured and numerically modeled effective soot temperatures are compared in Fig. 6 over a wide range of mean laser fluences up to about 4 mJ/mm2. In the experiments, the mean laser fluences were calculated based on the laser beam equivalent-width approach discussed in [12] and a constant E(m) of 0.4 was assumed. The numerical results are plotted against the mean laser fluences calculated from the histograms shown in Fig. 5. It has been verified that the mean laser fluences calculated from the two approaches differ only slightly (less than about 3%).

Fig. 6
figure 6

Comparison of the measured and modeled soot effective temperatures. The experimental data of Laser 2 in the 3C-LII experiments are the peak soot temperatures. All other experimental and numerical results are for soot temperatures at 12 ns after the peak of the laser

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It can be seen from Fig. 6 that the two runs of 2C-LII experiments using Laser 1 without DOE show that the repeatability is quite good. The effective soot temperatures in the 2C-LII experiments using Laser 1, without or with DOE, are very similar up to the mean laser fluence of about 1 mJ/mm2 (higher fluences cannot be achieved when DOE is used in the present setup), suggesting that the effect of DOE is very small or even negligible. The reason for the small impact of DOE on the AC-LII measurements perhaps lies in the fact that the use of DOE only slightly improves the fluence non-uniformity as characterized by the small decrease in the variance of the fluence histogram mentioned earlier, albeit the use of DOE appears to redistribute the non-uniformities into a smaller structure shown in Fig. 4. As expected, the use of DOE leads to slightly lower modeled effective soot temperatures than those without the use of DOE by up to about 100 K at the mean fluences around 1.3–1.4 mJ/mm2. It is noticed that the modeled effective soot temperatures of 2C-LII experiments (using Laser 1) without DOE are almost the same as those of the modeled 3C-LII experiments at 12 ns after the peak of the laser, i.e., the red and green curves in Fig. 6 overlap.

The experimental effective soot temperatures in the 3C-LII experiments (using Laser 2) are evidently higher than those in the 2C-LII experiments (using Laser 1) at relatively low fluences below about 2 mJ/mm2. The main reason for much higher soot temperatures in the 3C-LII experiments is that these temperatures are the peak values, not those at 12 ns after the peak of the laser. It is interesting to observe that the modeled effective soot temperatures for Laser 1 without DOE are nearly the same as the modeled ones for Laser 2, though the latter has a larger variance of the fluence histogram than the former (0.2055 vs. 0.1578). This result may be explained by the similarity of their fluence histograms around and above their respective mean values and the fact that the ‘hot-spots’ in the laser beam associated with the higher fluences above the mean plays a dominant role than the fluences below the mean. This implies that the somewhat large frequencies of Laser 2 between fluences of 0.1 and 0.2 mJ/mm2 shown in Fig. 5 contribute only slightly to LII signals, but contribute significantly to the increase in the variance of its fluence histogram. The modeled effective soot temperatures based on the experimental laser fluence histograms shown in Fig. 6 display very weak dependence on either the laser or the use of DOE. The latter is in agreement with the experimental soot temperatures in the 2C-LII experiments using Laser 1 with and without DOE.

The modeled temperatures of soot particles heated by a top hat laser are lower than any other non-uniform laser beams considered, which is indeed expected due to the absence of hot spots. The calculated effective temperatures of soot particles heated by the assumed laser fluence distribution of double-δ histogram are significantly higher than those of other fluence distributions at mean fluences below about 1.5 mJ/mm2, implying that a strongly non-uniform laser beam can result in significantly higher soot temperatures than those corresponding to a top hat laser fluence distribution. At high laser fluences above 2 mJ/mm2, the differences in the modeled effective soot temperatures almost disappear and they are in fairly good agreement with the measurements.

4.3 Soot volume fraction

The measured soot volume fractions, which were obtained at 12 ns after the peak of the laser pulse in the 2C-LII experiments using Laser 1 and at the moment of minimum temporal variations of soot volume fraction in the 3C-LII experiments using Laser 2, over a wide range of mean laser fluences are shown in Fig. 7. It is evident from Fig. 7 that the measured soot volume fractions from all three AC-LII experiments display a large increase with increasing the mean laser fluence at fluences below about 2 mJ/mm2 by a factor of about 5–8, i.e., the phenomenon of soot volume fraction anomaly. The soot volume fractions measured by two sets of 2C-LII experiments (Laser 1 without DOE) first increase fairly quickly with increasing the mean laser fluence to a peak value. At higher mean laser fluences after the peak, the measured soot volume fractions in these experiments start to decrease due to sublimation. The two runs of 2C-LII measurements (Laser 1 without DOE) display very good repeatability at fluences below about 1.5 mJ/mm2, but differ in the measured soot volume fractions by about 20% for higher fluences. The soot volume fractions measured using the 3C-LII experiments (Laser 2, no DOE) also display a similar level of soot volume fraction anomaly and overall good agreement with those from the 2C-LII experiments, albeit they were obtained at different moments after the laser peak. In agreement with the soot temperatures shown in Fig. 6, the use of DOE in the 2C-LII measurements (Laser 1, with DOE) has almost no impact on the measured soot volume fractions, i.e., it has negligible influence on soot volume fraction anomaly, at least for mean laser fluences up to 1 mJ/mm2.

Fig. 7
figure 7

Comparison of the measured soot volume fractions from the three sets of LII experiments at different mean laser fluences

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The modeled relative soot volume fractions for different laser fluence histograms at 12 ns after the peak of the laser pulse are compared in Fig. 8. When the laser fluence distribution is non-uniform, the modeled relative soot volume fraction indeed first increases with increasing the mean laser fluence to reach a peak and then starts to decrease at higher mean fluences. Therefore, the modeled soot volume fraction displays an overall qualitative agreement with the AC-LII measurements. However, the increase in the modeled relative soot volume fractions using the experimental laser fluence histograms with increasing the mean laser fluence is less than a factor of 2, which is much weaker than that of the measured soot volume fractions shown in Fig. 7 (about a factor of 5 or more increase).

Fig. 8
figure 8

Comparison of the modeled relative soot volume fraction for different laser fluence histograms at different mean laser fluences

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The modeled relative soot volume fractions using the experimental laser fluence histograms (Lasers 1 and 2 without DOE) display very similar results, as expected from their nearly identical effective soot temperatures shown in Fig. 6. The modeled relative soot volume fractions obtained using Laser 1 with DOE (2C-LII measurements) are significantly higher than those without DOE at fluences below about 2 mJ/mm2, and the difference gradually vanishes at higher fluences. The use of DOE with Laser 1 results in a higher modeled relative soot volume fraction than that obtained without DOE (0.932 vs. 0.900). This is expected from the modeled effective soot temperatures shown in Fig. 6, where the effective soot temperatures of Laser 1 with DOE are slightly lower because of the slightly improved fluence uniformity with the use of DOE. The results shown in Figs. 6 and 8 suggest that a small difference in soot temperature can lead to a fairly large difference in the derived soot volume fraction.

It is interesting to point out that the modeled relative soot volume fraction using the experimental fluence histograms only peaks up to 0.93, i.e., it does not recover the expected value of unity before sublimation takes place. These results imply that the AC-LII measured soot volume fraction is likely to be about 7% lower than the true value due to the laser fluence non-uniformity alone in the measurement volume. The modeled relative soot volume fraction for a top hat laser beam remains unity in the low-fluence regime, as expected, and then starts to drop in the high-fluence regime, where sublimation takes place. The differences in the modeled relative soot volume fraction between the top hat laser fluence and those of non-uniform fluences are solely caused by the non-uniform laser fluence distributions. Therefore, the laser fluence non-uniformity is indeed contributing to the soot volume fraction anomaly, though it is insufficient to explain the degree of soot volume fraction anomaly observed in the AC-LII experiments. The assumed laser fluence distribution of double-δ histogram results in the lowest modeled relative soot volume fraction at mean fluences below about 1.5 mJ/mm2, which is consistent with the highest modeled effective soot temperatures for this assumed distribution shown in Fig. 6. At high laser fluences above 2 mJ/mm2, the double-δ histogram results in the lowest modeled relative soot volume fractions, due to enhanced soot sublimation. Even using such a strongly non-uniform fluence distribution, the modeled relative soot volume fraction only increases by about a factor of 3, i.e., the degree of soot volume fraction anomaly caused by such a non-uniform laser beam is still not as large as that observed in AC-LII experiments, suggesting that additional mechanisms, other than the laser fluence non-uniformity, play a role in the soot volume fraction anomaly.

5 Conclusions

Auto-compensating laser-induced incandescence (AC-LII) experiments and numerical calculations were conducted using 1064-nm pulsed Nd:YAG lasers and a standard LII model in this study to investigate the effects of laser fluence non-uniformity on the measured volume fraction of near ambient-temperature soot in the exhaust of a MiniCAST soot generator over a wide range of mean laser fluences. Two AC-LII systems were implemented in this study: one was a two-channel LII setup and the other was a three-channel LII setup. Measurements using the two-channel LII setup were carried out without and with a diffractive optical element (DOE). To investigate the effects of different LII setups characterized by different laser sources, different LII signal detection delays, and different detection wavelengths and DOE on the soot volume fraction anomaly, three sets of AC-LII experiments were performed: two using the two-channel LII setup without and with DOE with a 12-ns delay after the laser peak, one using the three-channel LII setup without DOE and without delay. The following conclusions can be drawn based on the present experimental and numerical results:

  1. 1.

    The measured soot volume fractions in all three AC-LII experiments increase significantly with increasing the mean laser fluence over the range of 0.5–2 mJ/mm2 before sublimation occurs by about a factor of 5–7 under the present conditions, regardless of the laser source and the LII signal detection delay and wavelengths in the visible spectral range.

  2. 2.

    The use of DOE as a laser beam homogenizer only slightly improves the laser fluence distribution and has little influence on the measured soot volume fractions and temperatures, as well as the degree of soot volume fraction anomaly in the present experimental setups.

  3. 3.

    The non-uniform laser fluence distribution varies from shot to shot, and the laser fluence distribution in the measurement volume should be represented by the averaged fluence histogram rather than by the spatially averaged distribution.

  4. 4.

    The LII model results indicate that a non-uniform laser fluence distribution results in a higher effective soot temperature than a top hat distribution at a given mean laser fluence in the low-fluence regime. This bias of the effective soot temperature toward the higher soot temperatures associated with the higher local fluences in the measurement volume results in a lower derived soot volume fraction in AC-LII. The influence of such a bias in the effective soot temperature on the derived soot volume fraction becomes more pronounced with decreasing the mean laser fluence.

  5. 5.

    The numerical results indicate that the non-uniform laser fluence is responsible for the appearance of soot volume fraction anomaly due to the strong temperature dependence of LII signals and the non-uniform soot temperatures in the measurement volume associated with the non-uniform laser fluence. However, the non-uniform laser fluence only results in an increase in the modeled relative soot volume fractions by a factor of less than 2 over the 0.5–2 mJ/mm2 fluence range. Therefore, the non-uniform laser fluence alone cannot explain the much stronger increase in the measured soot volume fraction with increasing mean laser fluence in the low-fluence regime.

  6. 6.

    The numerical results suggest that AC-LII measurements using a slightly non-uniform laser fluence commonly deployed in LII research and application cannot fully recover the true soot volume fraction and the inferred soot volume fraction is in general about 10% lower.

Finally, it is worth pointing out that the increasingly lower measured soot volume fraction with decreasing the mean laser fluence using AC-LII appears to be associated with increasingly higher inferred soot temperature from the ratio of LII signals detected at two different wavelengths than that predicted from the theoretical model, though the cause of such deviation is not understood. Further research is required to explore other potential causes of the soot volume fraction anomaly in AC-LII measurements of ambient-temperature soot.