1 Introduction

Surface temperature fields are important data for various heat and mass transfer engineering applications so that measurements of these scalars need to be addressed from a metrology science point of view. This includes the design and implementation of sensors that can easily and correctly sense inputs. Moreover, associated complex calibration procedures to obtain outputs require specific precautions to ensure acceptable accuracy in measurements.

There are many types of temperature measurement techniques that are commonly used: thermocouple, infra-red thermometry, thermal paint, temperature-sensitive paint, etc. Although thermocouples provide accurate, sensitive and cheap temperature measurements over a wide temperature range, this technique is intrusive and a close contact between the two dissimilar metals of the thermo-electric junction and the surface is still challenging. Moreover, they are also subject to corrosion, affecting their calibration and accuracy [1, 2]. Infrared thermography is a contactless and non-destructive imaging technique. Its premise consists in detecting the infrared radiation emitted by an object with an IR camera. Although it can be implemented passively or actively, it can necessitate a particular attention with signal processing and inversion algorithms. Furthermore, high-temperature measurements are still possible but they require a detailed knowledge of the surface emissivity, and can suffer from non-negligible interferences. Thus, a detailed understanding of the underlying physics between the measurement and the environment is required to correctly interpret data and to successfully convert the radiation measurements into surface temperature scalars. Temperature-indicating paint can be also envisaged to perform surface temperature measurements up to \( 1500{\text{ }}^{^\circ } {\text{C}} \). Temperature-indicating paint is functional paint which measures the temperature distribution by judging the color changes of the paint film. This effect arises from the thermochromic properties of the paint composition, comprising mineral compounds such as carbonates, phosphates or aluminates. However, the color change with respect to the temperature is irreversible and only maximum temperature distributions can be assessed [3, 4]. Temperature-sensitive paint is another useful non-contact technique which employs thin luminescent polymer-based coatings. Thermal quenching mechanisms enable to observe a reduction in the luminescence as temperature increases. Unfortunately, the chemical composition of temperature-sensitive paints restricts its application to temperature measurements below \( 200{\text{ }}^{\circ } {\text{C}} \) [5, 6].

Thermographic phosphor thermometry is a semi-invasive optical technique which has been successfully applied to a large variety of applications to measure surface temperature [7, 8]. This technique utilizes the temperature dependent luminescent properties of phosphor materials that consist of a host material/matrix doped with rare-earth ions (lanthanides) or transition metals. Coated on the surface of interest, the phosphorescence signal is detected after an energy excitation (e.g. laser source) [9]. The phosphor emission spectrum being well defined, it is possible to distinguish it from broadband background emission using optical bandpass filters. The luminescent emission is also independent of blackbody radiations from the surface and of the surface emissivity.

Surface temperature can be measured using various detection strategies. The decay-time approach can be classified as a time-resolved strategy [9, 10]. A pulsed excitation light is used to excite the phosphor material. The subsequent optical emission induced from the relaxation of electrons from the excited state to the ground state via radiative and non-radiative transition pathways is then observed. For most phosphors, the intensity signal is likely to decay exponentially with a time constant, or lifetime, which is temperature dependent. A direct bijective relation linking temperature and lifetime can thus be established thanks to a calibration procedure. Depending on the phosphor characteristic emission timescales, relatively important acquisition rates are required. Photomultiplier tube (PMT) detectors are well suited for point-wise measurements while CMOS cameras are good sensors for two-dimensional measurements. However, in order to achieve a sufficient framing rate, the number of useful pixels of the active-pixel sensor must be significantly reduced, limiting the region of interest as well as the spatial resolution [11,12,13]. To alleviate these hardware limitations, alternatives have been proposed. Omrane et al. [14] used a framing system with eight CCD detectors, each of them being triggered separately to reconstruct the decaying signal. However, the important cost-in-use of this system constitutes a significant drawback to its routine use. Cai et al. [15] proposed an excitation system made of a long-pulse LED. By carefully controlling the excitation waveform, it was possible to use a low frame rate CCD camera. Nevertheless, the method actually suffers from a low repetition rate (1.3 Hz), and may be sensitive to interferences from the environment (thermal radiation, flame emission, etc).

The second technique, named spectral intensity ratio, is classified as a time-integrated strategy [9, 10]. It is based on the temperature-dependent spectral distribution of the phosphorescence signal. Two spectral bands with different temperature sensitivities are collected with an ICCD camera, leading to construct a temperature-dependent ratio R. Following a calibration step to relate R with the temperature, 2D temperature maps can be obtained by recording two phosphorescence images with two CCD cameras equipped with optical bandpass filters [13, 16], or seldom with one image doubling device [17, 18]. Notwithstanding that the main difficulty of the intensity ratio technique consists in carefully matching pixel-by-pixel of both regions of interest [9], only one single-shot acquisition is required to obtain an instantaneous temperature field. This approach is suitable when the phosphorescence lifetime is too long compared to the physical process to observe: fast-moving objects (piston in internal combustion engines, turbomachine blades), or fast thermal events (cooling of electronic chips). Another application is the gas or liquid thermometry, where phosphor particles are seeded into the carrier fluid. Quasi-instantaneous measurement is crucial to restrict the particle displacement during one acquisition.

The temperature uncertainty of both approaches indicates that the decay-time method is preferred to the intensity ratio method, mostly due to a greater relative sensitivity at elevated temperatures [13]. By inspecting the various parameters that controls the temperature uncertainty with the intensity ratio technique, the spectral properties of the optical filters are of primary importance. Indeed, an increase of temperature produces a cumulative and nonlinear intensity and spectral shift variations of the emission spectrum. A simple use of optical bandpass filters centered on the dominant emission peaks observed at one temperature may not be the best solution in terms of uncertainty at another temperature. Preliminary sensitivity tests to select optical filters with regards to an application are routinely used [19,20,21,22,23]. This can be either conducted with virtual filters applied to an experimental emission spectra database or by directly testing several alternatives. These methods can be limited for applications with large temperature variations, such as near-wall combustion. Indeed, spectra variations on a large temperature range are significant to discredit optical filters selected at given temperatures or on a restricted temperature range, thus producing undesired temperature uncertainties.

The current study aims to evaluate the improvement of the temperature uncertainty with the spectral intensity ratio method by taking advantage of a numerical multi-objective optimization that selects the best optical filters. This investigation is solely performed on the Mg4FGeO6:Mn4+ (MFG) phosphor which is a good candidate for temperature measurements in a wide range. Following some general considerations on the technique, an algebric estimate of the temperature uncertainty is introduced as well as the general framework of the multi-objective optimization procedure (Sect. 2). The experimental setup as well as the data post-processing and data reduction are detailed in Sect. 3. Section 4 details the spectroscopic properties of the MFG phosphor with temperature and details the new detection strategies while using several optimization strategies. Finally, the performances of the multi-objective optimization procedure are experimentally examined by data processing phosphorescence signals recorded with an ICCD camera equipped with an image doubler in order to quantify the enhancement of the temperature uncertainty.

2 Theoretical considerations

2.1 Phosphorescence dependencies

Phosphor thermometry measurements are performed by illuminating the wall-coated phosphor particles with a pulsed UV laser beam. The phosphor particles re-emit a temperature dependent signal through spontaneous emission. Assuming a weak (i.e. non-saturating) laser excitation, the time-integrated spectral phosphorescence signal, I, can be modelled as

$$\begin{aligned} I(T, \lambda ) = \eta _{opt} \frac{\varOmega }{4\pi } \frac{E}{hc/\lambda } n_p \sigma _{cs}(T, \lambda ) \varPhi (T, \lambda ), \end{aligned}$$
(1)

where \(\eta _{opt}\) and \(\varOmega / 4\pi \) are the optical efficiency and the fractional solid angle of the signal collection, E is the laser fluence (J/cm\(^2\)), \(hc/\lambda \) is the energy per laser photon, \(n_p\) is the number of phosphor particles in the collection volume, \(\sigma _{cs}\) is the phosphor absorption cross-section at the laser wavelength and \(\varPhi \) is the phosphorescence quantum efficiency [21]. In common experiments, the collection optics, laser wavelength, and collection volume are assumed to be constant. Then, the computation of a ratio R of phosphorescence signals recorded simultaneously in two spectral ranges (\(\varDelta \lambda _1\) and \(\varDelta \lambda _2\)) can be simplified into a simple relationship between R and the wall temperature T:

$$\begin{aligned} R(T, \varDelta \lambda _1, \varDelta \lambda _2) = \frac{\int _{\varDelta \lambda _1} I(T, \lambda )\mathrm {d}\lambda }{\int _{\varDelta \lambda _2} I(T, \lambda )\mathrm {d}\lambda }. \end{aligned}$$
(2)

Experimentally, the spectral domains are determined by optical filters where \(\varDelta \lambda _i\) (\(i=\{1, 2\}\)) is the spectral domain of the optical filter i regrouping the central wavelength, \(\lambda _{c,i}\), and its full-width at half-maximum linewidth, \(\mathrm {FWHM}_i\). In the following, properties of a specific optical filter will be expressed as \(\varDelta \lambda _i\ = \left[ \lambda _{c,i}, \mathrm {FWHM}_i\right] \) (in nm). A set of optical filters used to record both phosphorescence signals is thus expressed as \(\mathrm {FS} = \left( \varDelta \lambda _1, \varDelta \lambda _2\right) \). The ranking of the two filters is defined such as all calibration curves increase with respect to temperature.

2.2 Temperature uncertainty

Assuming that temperature is solely dependent on the phosphorescence ratio R, and using a Taylor expansion-based error propagation, the relative temperature uncertainty, \(\delta T / T\), can be analytically expressed as [20,21,22,23,24,25,26,27]

$$\begin{aligned}&\frac{\delta T}{T} = \underbrace{\frac{1}{T \times S_r\left( T, \varDelta \lambda _1, \varDelta \lambda _2\right) }}_{\varPsi } \nonumber \\&\quad \times \underbrace{\sqrt{ \frac{1}{\mathrm {SNR}^2_{\varDelta \lambda _1}(T)} + \frac{1}{\mathrm {SNR}^2_{\varDelta \lambda _2}(T)}}}_{\varGamma }. \end{aligned}$$
(3)

\(S_r\) represents the relative temperature sensitivity of R (in \(\%/K\)), and \(SNR_{\varDelta \lambda _i}\) is the signal-to-noise ratio of the phosphorescence signal in a given spectral domain \(\varDelta \lambda _i\). As shown in Eq. 3, the temperature uncertainty is dependent on a sensitivity factor \(\varPsi \) and a SNR factor \(\varGamma \), both being dependent on the spectral features of FS. Various detection strategies allowing an enhancement of the temperature uncertainty can therefore be tested and compared.

2.3 Numerical optimization strategy

Fig. 1
figure 1

Schematic of the optimization procedure leading to a set of filters, \(\left( \mathrm {FS}\right) _{opt}\) that improves the temperature uncertainty (Eq. 3). a Database of phosphorescence spectra recorded with a spectrograph. b Multi-objective optimization procedure that numerically integrates the phosphorescence spectra with various FS. c Optimum filter set \(\left( \mathrm {FS}\right) _{opt}\) returned by the procedure. The optimization process varies the properties of the spectral bands \(\varDelta \lambda _1\) and \(\varDelta \lambda _2\)

Full size image

A constrained optimization model is developed to find the best filter set \(\left( \mathrm {FS}\right) _{opt}\) that optimizes the relative temperature uncertainty \(\delta T / T\) with respect to some evaluation criteria. Figure 1 displays the architecture of the optimization model, consisting of three main blocks: input conditions, model matching and results.

The input conditions consist of a series of experimental phosphorescence spectra that are recorded with a spectrograph/CDD detection system at various temperatures (Fig. 1a).

In the model matching, the phosphorescence spectra are numerically integrated with two numerical top-hat bandpass filters (transmission equal to 1 in the transmitting band, 0 elsewhere), corresponding to the two spectral domains \(\varDelta \lambda _1\) and \(\varDelta \lambda _2\) (Fig. 1b). This numerical integration enables to determine a calibration curve R(T) and an estimator of the relative temperature uncertainty using Eq. 3. The optimization process consists in varying the specifications of the two spectral domains (central wavelength \(\lambda _{c,i}\) and transmitting bandwidth \(\mathrm {FWHM}_i\)) in order to reduce the temperature uncertainty estimator. \(\lambda _c\) is varied from 630 to 680 nm, and FWHM is varied from 4 to 40 nm, in accordance with the emission properties of the MFG phosphor. Three criteria are implemented to constraint the numerical optimization:

  1. (i)

    non bijective calibration curves are discarded;

  2. (ii)

    ratio values must be bounded between 0.14 and 7. This criterion is intended to avoid too different signal levels on each spectral band since the current experimental setup is composed of an image doubler and a ICCD single camera (e.g. acquisition parameters);

  3. (iii)

    signal dynamic must not exceed 1:14 (ratio between the maximum and the minimum signal levels on one channel for a specific temperature range). This criterion enables to record signals in the linear response of the detector.

The two last criteria are dependent on the performances of the available experimental setup, and could be less restrictive with sensors presenting extended linear responses. The limit values of the ratio as well as those of the signal dynamics are selected from the performances of the experimental setup described in Sect. 3.3.

Depending on the experimental conditions and the optimization goals, several objectives (named objective functions in the following) can be simultaneously targeted and a multi-objective function [28,29,30] can be introduced. Eventually, the optimization process returns the top-rated filter set, called optimal filter set \(\left( \mathrm {FS}\right) _{opt}\).

In the following section, the experimental procedure developed to record the phosphorescence signals as well as the model employed to compute the relative temperature uncertainty estimator \(\delta T / T\) are described.

3 Experimental procedure

3.1 Setup

A 5 % in mass of manganese-activated magnesium fluorogermanate, Mg4FGeO6:Mn4+ (MFG) (Osram SV067), is mixed with a water-based binder (Zyp Coatings HPC) [7, 31,32,33], and coated on a stainless-steel sample. To ensure a spatial homogeneity of the paint, five layers are applied with an air brush. The phosphor coating thickness is then measured with an electromagnetic thickness gauge (Sauter TE 1250-0.1FN) [19, 34] and remains lower than 10 µm. An estimation of the temperature shift based on a simple resistance model indicates an additional systematic temperature error to be around 1 K [35, 36].

Excitation of thermographic phosphors is provided at 266 nm by the fourth harmonic of a Nd:YAG laser (Quantel Qsmart 450) operating at a repetition rate of 10 Hz. The laser fluence is adjusted with an optical attenuator until a maximum value of 1.0 mJ/cm2. The laser beam diameter is first reduced in size by an iris diaphragm to 5 mm and shaped with a top-hat homogenizer diffuser (Holo/Or HM-273-W-Y-A) to ensure a 28 × 28 mm2 squared homogeneous laser energy area. This homogeneous distribution is ensured by recording acetone fluorescence images in an optical cell filled with a constant gaseous acetone/air mixing. A single dichroic mirror reflects the excitation beam towards the coated sample and transmits the emitted phosphorescence signal. The stainless-steel sample is fixed horizontally into a furnace (Nabertherm RHTH 80-300) and its temperature is controlled and monitored with five type-K thermocouples mounted vertically and horizontally on its backside, the distance between the phosphor coating and the thermocouple contact being less than 2 mm. The furnace is heated up to 850 K until a thermal steady-state is reached. Acquisition of the phosphorescence spectra is conducted during cooling and heating periods with a maximum rate is 0.1 K.s-1, giving a maximum temperature difference of 3 K during one acquisition sequence.

3.2 Phosphorescence spectra

Phosphorescence spectra are collected through an optical fiber (NA = 0.22) located behind the dichroic mirror and then directed into a symmetric crossed Czerny-Turner miniaturized spectrograph equipped with a 50 µm entrance slit and an HC-1 grating, with a spectral resolution of 0.89 nm (OceanOptics Maya2000 Pro). The sensor is a 2D back-thinned linear CCD-array (Hamamatsu S10420) with 2048x64 active pixels. All the measurements are made using an identical viewing geometry and integration time. For each temperature, a background spectrum is recorded after the acquisition of the phosphorescence spectra. The experimental spectra are then background subtracted.

3.3 Phosphorescence images

In order to validate the performances of the multi-objective optimization procedure (Sect. 2.3) an ICCD camera (Princeton Instruments PIMAX-II) equipped with an image doubler (LaVision 1108765) is used instead of the spectrograph/CCD detection system. Two colored bandpass filters are mounted on the image doubler to collect phosphorescence signals. Elastic scattering from the laser beam is suppressed by a high-pass optical filter (Semrock, cut-off wavelength of 280 nm). An additional bandpass filter (Asahi spectra, [660/140] nm) is also used to suppress the near-infrared contribution. The gain of the intensifier is adjusted to not exceed 30 000 counts, and thus prevents from a non-linear response of the camera in its upper range, mainly arising from the microchannel plate saturation [37]. The signal dynamic is not higher than 1:14. 100 frames are acquired for each temperature level with an exposure time of 300 µs, delayed by 10 µs after the excitation pulse. The record of the phosphorescence images is performed inside a furnace as detailed in Sect. 3.1. Phosphorescence images are background subtracted and corrected with a spatial optical transfer function, before being normalized by the maximum count level value of each frame. After a spatial matching between the two phosphorescence images, a pixel-to-pixel ratio is computed for each temperature. A spatio-temporal average is used to construct a calibration curve, and the temperature uncertainty is lastly calculated using Eq. 3.

3.4 Data reduction

3.4.1 Ratio and relative sensitivity estimation

Adopting a set of optical filters, Eq. 2 is used with the shot-averaged phosphorescence signals integrated on \(\varDelta \lambda _1\) and \(\varDelta \lambda _2\) to calculate the calibration curves of R(T). A third-order polynomial law is used to fit the data with a resulting determination coefficient larger than 0.99. Once this operation is carried out, \(S_r\) is computed from the fitted curve of R(T).

3.4.2 Signal-to-noise computations

As shown in Eq. 3, the main contributions to the uncertainty of the ratio R arise from random noise coming from the CCD camera. Some of them are set by the sensor design itself, but also how the sensor is integrated into the electronics of the CCD camera. The significant sources of noise, i.e. readout (\(\sigma _{readout}\)), dark noise (\(\sigma _{dark}\)) and shot noise (\(\sigma _{signal}\)), can be introduced into the SNR expression as [38]:

$$\begin{aligned} \mathrm {SNR} = \frac{\eta _{QE}N_{p}}{\sqrt{\sigma ^2_{dark} + \sigma ^2_{readout} + \sigma ^2_{signal}}}, \end{aligned}$$
(4)

\(\eta _{QE}\) is the photocathode quantum efficiency of the CCD, and \(N_p\) is the number of incident photons collected by the CCD sensor. In our spectroscopic experiments, the appropriate strategy for optimizing the phosphorescence spectra consists in maximizing the number of photons detected per unit time. For that, 10 consecutive phosphorescence spectra are recorded, each being acquired with an exposure time of 100 ms. An averaged phosphorescence spectrum is then computed. With those specifications and in accordance with the specifications of the CDD manufacturer, the maximum peak intensity of the phosphorescence spectra recorded at various temperatures is approximately 15 000 counts, with an associated pixel readout noise being in the order of 5–6 electrons and about 1 electron for the dark noise Record of the experimental spectra is then performed in a photon-noise limited regime and the photon shot noise can be simply reduced as

$$\begin{aligned} \sigma _c = \sqrt{\overline{N_c}}. \end{aligned}$$
(5)

\(\overline{N_c}\) is the time-averaged count value issued from the averaged phosphorescence spectra on one pixel, and \(\sigma _c\) is the standard deviation calculated by assuming a Poisson distribution for the photon arrival rate on the detector.

Using the shot-noise-limited assumption, the SNR for each spectral filtering detection is derived from the integration of the phosphorescence signal on a given spectral band \(\varDelta \lambda \) by using the following expression

$$\begin{aligned} \mathrm {SNR}_{\varDelta \lambda _i}(T) = \frac{\int _{\varDelta \lambda _i} \overline{N_c}(\lambda , T) \mathrm {d}\lambda }{\sqrt{\int _{\varDelta \lambda _i} \overline{N_c}(\lambda , T) \mathrm {d}\lambda }}. \end{aligned}$$
(6)

Note that the phosphorescence spectra are all normalized by the peak value of a spectrum recorded at 300 K (see Fig. 2a).

For imaging experiments, the signal-to-noise ratio is computed assuming a shot-noise limited regime for an intensified CCD sensor. In that case, the SNR is given by

$$\begin{aligned} \mathrm {SNR}_{\varDelta \lambda _i}(T) = \sqrt{\frac{N_{c, \varDelta \lambda _i}(T)}{K}}, \end{aligned}$$
(7)

where K is a supplementary noise factor dependent of the intensifier used and typically equal to 2 [38]. \(N_{c, \varDelta \lambda _i}\) is the count level integrated on the spectral band \(\varDelta \lambda _i\).

4 Results and discussion

4.1 Temperature dependence of phosphorescence spectra

Fig. 2
figure 2

Phosphorescence spectra of Mg4FGeO6:Mn4+ (MFG) at various temperatures. Spectra are normalized (a) with the maximum of phosphorescence at 298 K, (b) by their own maximum value

Full size image

Figure 2a, b show experimental phosphorescence spectra of MFG recorded at various temperatures. Two emission peaks, centered around 630 nm and 660 nm are observed, corresponding to the transitions of this oxide phosphor when it is activated by Mn4+ [39]. Figure 2a displays various phosphorescence spectra normalized by the intensity peak at 298 K. When temperature increases from 298 to 710 K, the 660 nm emission peak is reduced by a factor of ten. The 630 nm peak is constant for temperatures between 298 K and 480 K and then reduced by a factor five between 480 K and 710 K. This behaviour mainly arises from important lattice relaxations. these one lead to non-radiative relaxation processes which naturally decrease the intensity of phosphorescence signals [39].

Figure 2b shows the same phosphorescence spectra, each of them being normalized by their own intensity peak. Though the relative intensity of the 660 nm emission band is five times larger than the one at 630 nm at room temperature, it dwindles to 1.5 at 710 K. It is worth noting that the emission spectra are broadened at higher temperatures. Moreover, a red-shift of the full spectrum is observed: the 660 nm emission peak shifts by \(1.2 \times 10^{-2}\, \mathrm {nm}.\mathrm {K}^{-1}\), while the 630 nm peak is shifted by \(1.45\times 10^{-2}\,\mathrm {nm}.\mathrm {K}^{-1}\). From these results, the nonlinear change of the phosphorescence spectral profiles with temperature suggests that a variation of the temperature uncertainty is attributed to the combined change of \(S_r\) and SNR. A rigorous selection of the integrated spectral bands, \(\varDelta \lambda _1\) and \(\varDelta \lambda _2\), cannot then be undertaken by a simple analysis of the emission spectra. A suitable selection of the two spectral bands \(\varDelta \lambda _1\) and \(\varDelta \lambda _2\) is then conducted by applying a numerical multi-objective optimization procedure (Sect. 2.3) on an large set of MFG phosphorescence spectra.

4.2 Optimization of the temperature uncertainty

4.2.1 Optimization parameters

The multi-objective optimization procedure is performed with the aim of selecting the most suitable set of optical filters to perform 2D surface temperature measurements for near-wall combustion applications, i.e. in the typical temperature range \(\varDelta T=\left[ 300 -750\,\mathrm {K}\right] \). As illustrated by Fuhrmann et. al [13], the temperature measurements with the intensity ratio method with MFG are known to significantly lose in accuracy at high temperature. Instead of selecting optical bandpass filters at a given temperature in this temperature range, the optimization procedure is performed on the entire temperature range in order to obtain a similar temperature uncertainty on \(\varDelta T\) [27]. To apply the multi-objective optimization procedure, two objective functions are simultaneously considered to determine the best spectral domains of the optical filters: (i) the average relative temperature uncertainty \(\left\langle \delta T / T \right\rangle \) on \(\varDelta T\) and (ii) the standard deviation of the relative temperature uncertainty \(\sigma \left( \delta T / T \right) \), on \(\varDelta T\). The first objective function aims to have a filter set that enables the lowest temperature uncertainty on a temperature range). The goal of the second objective function is to minimize the variations of temperature uncertainty values on the whole temperature range. This criterion has been added for near-wall combustion applications, where large temperature gradients must be accurately measured. It is important to notice that the choice of the objective functions is user-dependent, and should be defined in accordance to a specific application. A discussion on the definition of the multi-objective function is presented in Sect. 4.3.2.

In order to conduct the optimization procedure, the spectral properties of a reference set of filters are first taken from the literature [13, 19]. These ones are centred on the two main emission peaks, \(\left( \mathrm {FS}\right) _{ref}= (\left[ 630, 11\right] , \left[ 658, 20\right] )\). The different objective functions for optimizing the target and for determining the optimal set of filters are then normalized by the quantity determined with \(\left( \mathrm {FS}\right) _{ref}\) so that gain or loss in performances can be easily evaluated. Note that the optimization procedure has been found to be independent on the choice of the normalization parameter. Hereafter, a quantity X that is normalized by the quantity determined with \(\left( \mathrm {FS}\right) _{ref}\) will be noted as \(\left( X\right) ^* = X/X_{ref}\). Finally, a multi-objective function \(Q_1\) considering simultaneously the effects of the two objective functions previously introduced is defined as

$$\begin{aligned} Q_1 = \frac{\left( \left( \left\langle \delta T / T \right\rangle \right) ^* + \left( \sigma \left( \delta T / T \right) \right) ^*\right) }{2}. \end{aligned}$$
(8)

The minimization of \(Q_1\) then provides the best spectral characteristics of the detection optical filters.

4.2.2 Optimal filter set

Fig. 3
figure 3

Comparison between the reference filter set \(\left( \mathrm {FS}\right) _{ref}=\left( \left[ 630, 11\right] , \left[ 658, 20\right] \right) \) (blue) and the optimal filter set \(\left( \mathrm {FS}\right) _{opt}=\left( \left[ 658, 18\right] , \left[ 678, 18\right] \right) \) (red) that minimizes the multi-objective function \(Q_1\) (Eq. 8). a Phosphorescence spectra at various temperatures with the transmittance bands of the two spectral domains \(\varDelta \lambda _1\) (dashed line) and \(\varDelta \lambda _2\) (solid line), b  experimental calibration curves (left axis) and associated related relative sensitivities (right axis), c relative sensitivity term \(\varPsi =\left( TS_r\right) ^{-1}\) (open markers, left axis) and SNR term \(\varGamma =\sqrt{\sum _i\mathrm {SNR}_{\varDelta \lambda _i}^{-2}}\) (filled markers, right axis) (Eq. 3), and d temperature error \(\delta T / T\) (right axis) and normalized temperature error \(\left( \delta T/T\right) ^*\) (left axis)

Full size image

Figure 3a shows a set of phosphorescence spectra on which are plotted the spectral properties of the reference filter set \(\left( \mathrm {FS}\right) _{ref}\) (blue curves) and the optimal filter set \(\left( \mathrm {FS}\right) _{opt}\) (red curves). The filters of \(\left( \mathrm {FS}\right) _{ref}\) are centred on the two phosphorescence peaks, while the filters of \(\left( \mathrm {FS}\right) _{opt}\) are shifted towards the red part of the spectra. The first filter of \(\left( \mathrm {FS}\right) _{opt}\) is centred on the 660 nm peak, and almost matches the second filter of the reference filter set, while the second filter is fully shifted to the extremity of the red wing of the phosphorescence spectra (Sect. 4.1). Interestingly, the two filters of \(\left( \mathrm {FS}\right) _{opt}\) have about the same spectral width, i.e. \(\mathrm {FWHM}_{1,2,opt}=18\,\mathrm {nm}\).

Figure 3b presents the calibration curves for the two set of filters. The optimal filter set leads to a configuration presenting a large variation of ratio values within \(\varDelta T = \left[ 300 - 750\,\mathrm {K}\right] \), with a ratio value of 0.15 at 300 K, and 0.66 at 750 K. The signal dynamic of both channels is 1:11 and 1:4. On the contrary, the ratio variation of the reference filter set is limited to a factor 2 for the same \(\varDelta T\), from 0.19 at 300 K to 0.37 at 700 K, while the signal dynamic for both channels is 1:6 and 1:10. This result suggests that a detection system allowing an extended signal dynamic range may improve the temperature uncertainty. Furthermore, the optimal calibration curve has the advantage to converge to one [27], corresponding to similar signal intensities on both channels. This enables to adjust either the excitation laser energy and/or the signal collection settings to improve the signal-to-noise ratios recorded on both channels at elevated temperature, and thus to enhance the temperature uncertainty in this critical temperature range. In addition, except at 300 K, the relative sensitivity of the optimal filter set is always larger than the one of the reference filter set. Finally, even if the signal dynamic of the two channels is relatively high for the reference filter set, the variation of the ratio is limited, leading to weak sensitivity values (\(S_r=0.05\,\%/\mathrm {K}\) at 600 K). Regarding the optimal filter set, the signal dynamic of the most sensitive channel is not significantly larger than for the reference filter set ((1:11) against (1:10) for the reference filter set), only one signal varies with the temperature while the other one remains constant (signal dynamic of 1:4), leading to larger ratio variations, and thus higher relative sensitivities. With the same signal dynamic, the optimal filter set enhances the relative sensitivity by a factor of 5 (\(S_r=0.26\,\%/\mathrm {K}\) at 600 K).

Figure 3c depicts the two terms of the relative temperature uncertainty estimator for both filter sets: open symbols indicate the relative sensitivity term \(\varPsi =\left( TS_r\right) ^{-1}\), while filled symbols are for the SNR term \(\varGamma = \sqrt{\sum _i\mathrm {SNR}_{\varDelta \lambda _i}^{-2}}\). Both terms should be minimized to improve the relative temperature uncertainty \(\delta T / T\). The SNR term is nearly constant up to 650 K for both FS, while it dramatically increases at higher temperature, mainly due to the advent of strong non-radiative relaxation processes [8, 9, 13, 19]. Besides, \(\varGamma \) is roughly similar for \(\left( \mathrm {FS}\right) _{opt}\) and \(\left( \mathrm {FS}\right) _{ref}\), and is even slightly worse for temperature lower than 500 K with \(\left( \mathrm {FS}\right) _{opt}\). The relative sensitivity term of \(\left( \mathrm {FS}\right) _{ref}\) strictly increases up to 3.5 at 650 K, before decreasing down to 2.5 at 750 K. The trend of \(\varPsi \) completely changes for \(\left( \mathrm {FS}\right) _{opt}\), this one being approximately constant between 300 K and 750 K at 0.6 with a maximum value of 0.75 at 300 K. The optimal filter set reduces the relative sensitivity term \(\varPsi \) by a factor up to 5 at 650 K (compared to the reference filter set), as also illustrated in Fig. 3b.

Figure 3d presents the relative temperature uncertainty estimator \(\delta T / T\) for both filter sets. The normalized relative temperature uncertainty \((\delta T / T)_{opt}^* = (\delta T / T)_{opt} / (\delta T / T)_{ref}\) is also depicted (green markers). The temperature uncertainty increases monotonously between 300 K and 750 K for \(\left( \mathrm {FS}\right) _{ref}\), while it remains nearly constant for \(\left( \mathrm {FS}\right) _{opt}\). Except for temperature lower than 325 K, the optimal filter set presents an enhanced temperature uncertainty, up to a factor 5 at high temperature. The temperature uncertainty gain is thus mainly due to the improvement of the relative sensitivity term, whereas the effect of the signal-to-noise ratio term is rather marginal.

4.2.3 Shape of \(Q_1\) spaces at optimum

Fig. 4
figure 4

Multi-objective variable \(Q_1\) for \(\left( \mathrm {FS}\right) _{opt}=\left( \left[ 630, 11\right] , \left[ 658, 20\right] \right) \) in the a \(\varDelta \lambda _1\) subspace, valid for \(\varDelta \lambda _2=\left[ 658, 20\right] \), and in the b \(\varDelta \lambda _2\) subspace, valid for \(\varDelta \lambda _1=\left[ 630, 11\right] \). The two optimal filters are depicted by a green star. The blue cross corresponds to a different local minimum in the \(\varDelta \lambda _2\) space. Blank areas correspond to discarded filter sets (Sect. 2.3)

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In order to evaluate the robustness of the multi-objective optimization procedure, various subspaces of \(Q_1\) are analysed. Since four decision variables are considered (\(\lambda _{c,1}, \mathrm {FWHM}_1, \lambda _{c,2}, \mathrm {FWHM}_2\), see Sect. 2.3), the optimization of \(Q_1\) is performed in a four dimension space. For clarity, Fig. 4 presents the multi-objective function \(Q_1\) in the vicinity of the optimal filter set \(\left( \mathrm {FS}\right) _{opt}\), where two maps of \(Q_1\) are extracted in accordance with the best specifications.

Considering the \(\varDelta \lambda _1\) subspace (Fig. 4a), a region with low \(Q_1\) values is noticeable for narrow filters (\(\mathrm {FWHM}_1=10 - 20 \,\mathrm {nm}\)) centred around \(\lambda _{c,1} = 656\,\mathrm {nm}\), and slightly shifted (\(\approx 4\,\mathrm {nm}\)) to the left of the 660 nm emission band (see Fig. 2).

Considering the \(\varDelta \lambda _2\) subspace (Fig. 4b), two regions showing low \(Q_1\) values stand out, one around \(\lambda _{c,2} = 640\,\mathrm {nm}\) (left blue patch), and a second one more preponderant around \(\lambda _{c,2} = 675\,\mathrm {nm}\) (right blue patch). The first zone is representative of the inter-peak emission region (see Fig. 2) while the second zone is located on the red wing of the spectra. The presence of top-rated filters for longer wavelengths is attributed to the red-shift of the spectra with temperature enabling then a significant increase of the relative sensitivity while maintaining good SNR at high temperature.

For instance, by considering the optimal FS in the inter-peak region for \(\varDelta \lambda _2\) (blue cross) while keeping \(\varDelta \lambda _1\) constant (\(\left( \mathrm {FS}\right) \,=\,\left( \left[ 658,18\right] , \left[ 640,14\right] \right) \)), \(Q_1\) is 0.29 while it is 0.16 for the original \(\left( \mathrm {FS}\right) _{opt}\). Besides, the gain in percentage in the averaged relative sensitivity factor \(\left( \left\langle \varPsi \right\rangle \right) ^*\) is 27.1 % for this filter set, whereas it is 45.6 % for \(\left( \mathrm {FS}\right) _{opt}\) (green star). Thus, the new detection strategy clearly presents better performances in terms of temperature uncertainty.

4.3 Impact of the optimization parameters

In this section, the sensitivity of various optimization parameters is investigated. The effect of the temperature range on which the multi-objective optimization procedure is performed is discussed first. The choice of the multi-objective function \(Q_1\) is then investigated, before testing alternative optical filters. In the following, the set of filters determined by the multi-objective optimization procedure are systematically compared to the reference filter set.

4.3.1 Impact of the temperature range

Fig. 5
figure 5

Results of the multi-objective optimization procedure performed for various temperature ranges.  a–d Optimal filter set specifications for different temperature ranges, \(\varDelta \lambda _1\) (green dashed line) and \(\varDelta \lambda _2\) (orange solid line). Phosphorescence spectra between 300 K and 720 K are also represented for information. e Normalized relative temperature uncertainty for each temperature range. Values lower than unity indicate an enhancement in comparison to the reference filter set \(\left( \mathrm {FS}\right) _{ref}\). f Specifications of the filter sets returned by the optimization procedure for different temperature ranges

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In order to study the effect of the temperature range on which the optimization procedure is conducted, the procedure is performed on three limited temperature ranges, similarly to other studies [19]. Figure 5 presents the optimization results for three temperature ranges (\(\left[ 300 - 400\,\mathrm {K}\right] , \left[ 300 - 600\,\mathrm {K}\right] \), and \(\left[ 600 - 700\,\mathrm {K}\right] \)) as well as the reference temperature range \(\left[ 300 - 750\,\mathrm {K}\right] \). These temperature ranges are selected to compare the performances of temperature uncertainty determined with the multi-objective optimization procedure with those reported in previous studies when using another set of filters [8, 9, 13].

Figures 5a–d presents the specifications of the spectral domains of the best filters superimposed on the phosphorescence spectra. The choice of the temperature range \(\varDelta T\) does not seem to be a preponderant factor for the selection of the best optimal filter sets. For \(\varDelta T\,=\,\left[ 300 - 600\,\mathrm {K}\right] \) or \(\left[ 600 - 700\,\mathrm {K}\right] \), the filter sets are \(\left( \left[ 654, 14\right] , \left[ 678, 28\right] \right) \) and \(\left( \left[ 660, 12\right] , \left[ 678,18\right] \right) \) respectively, these one being very close to \(\left( \mathrm {FS}\right) _{opt}\). Despite the observation of small variations on the spectral width, the filter spectral properties have the same features: one narrow filter centred around the 660 nm peak, and the other located on the red wing of the phosphorescence spectra, far from the properties of the reference filter set. It is also noticeable that the FWHM of the filter centered on the red wing of the phosphorescence spectrum is reduced when increasing the temperature: 34 nm for \(\varDelta T = \left[ 300 - 400\,\mathrm {K}\right] \) against 18 nm for \(\varDelta T = \left[ 600 - 700\,\mathrm {K}\right] \). Since the red-shifting combined with the spectrum broadening are preponderant at high temperature, the reduction of the filter width enables to enhance the strength of the relative sensitivity term, counterbalancing the diminution of the signal-to-noise ratios. Besides, the central wavelength of the filter located around the 660 nm peak is also function of the red-shift of the phosphorescence spectrum when increasing temperature: from \(\lambda _c=654\,\mathrm {nm}\) for \(\varDelta T = \left[ 300 - 400 \,\mathrm {K}\right] \), this central wavelength is moving to \(\lambda _c=660\,\mathrm {nm}\) for \(\varDelta T = \left[ 600 - 700\,\mathrm {K}\right] \).

Considering the normalized relative temperature uncertainty (Fig. 5e), all the filter sets show a significant enhancement compared to the reference filter set, whatever the temperature range investigated. The filter sets that are optimized at low temperature also present good performances for intermediate temperatures (500–600 K), while they are less convenient at high temperature (750 K). However, they still show an enhancement of the temperature uncertainty which can reach 40 % or more at 750 K compared to the reference filter set. Furthermore, the normalized relative temperature uncertainty curves for \(\varDelta T = \left[ 300 - 750\,\mathrm {K}\right] \) and \(\left[ 600 - 750\,\mathrm {K}\right] \) overlap, even at low temperature. This indicates that the behaviour of the MFG phosphor at high temperature has a non-negligible weight during the multi-objective optimization process.

Finally, it appears that the selection of the temperature range \(\varDelta T\) moderately affects the outputs of the multi-objective optimization procedure. the red-shift and broadening features of these phosphor particles are used by the optimization to seek detection spectral domains having important sensitivities. Moreover, the temperature uncertainty shown in Fig. 3d indicates that the multi-objective optimization procedure improves the temperature uncertainty in the high-temperature range.

4.3.2 Selection of the multi-objective function

Fig. 6
figure 6

Sensitivity of the multi-objective optimization procedure to the definition of the function Q. a Normalized relative temperature uncertainty for four multi-objective functions. The relative temperature uncertainty \(\delta T / T\) is normalized by that of the reference filter set \(\left( \mathrm {FS}\right) _{ref}\). b Filter set spectral properties returned by the multi-objective functions. c Values of three specific temperature uncertainty parameters for different functions Q. Colors indicates low and high values

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As presented in Sect. 4.2, the multi-objective optimization procedure is based on the minimization of a multi-objective function \(Q_1\) (Eq. 8). In our case, this function is defined in order to find the most suitable filter set that simultaneously minimizes the mean relative temperature uncertainty \(\left\langle \delta T/ T\right\rangle \) and its standard deviation \(\sigma \left( \delta T / T\right) \). Note that this definition is user-dependent so that alternative constraints could be defined according to an application. In this section, the definition of the function \(Q_1\) is modified in order to study its impact on the choice of the optimal filter set. Three different definitions of Q are then investigated: (i) mean relative temperature uncertainty (\(Q_2\)), (ii) standard deviation of the relative temperature uncertainty (\(Q_3\)), (iii) maximum of the relative temperature uncertainty (\(Q_4\)). As stated above, these parameters are normalized by the same variables obtained for the reference case so that gain or loss can be easily assessed.

Figure 6a shows the normalized relative temperature uncertainty \(\left( \delta T / T\right) ^*\) for the three above-mentioned multi-objective functions (\(Q_2\) to \(Q_4\)) as well as the original multi-objective function \(Q_1\) (Sec. 4.2). The spectral properties of the filter sets are mostly identical, except for \(Q_3\), where the central wavelengths slightly differ. Furthermore, a difference on the evolution of the relative temperature uncertainty with temperature is observed for \(Q_3\) and only an improvement in accuracy is observed for temperatures above 500 K. Table 6c gives the performances of the different uncertainty parameters for the four objective functions. As expected, each objective function gives the best performance regarding the uncertainty parameter used in the multi-objective optimization procedure. Thus, \(Q_2\) gives the best mean relative temperature uncertainty, \(Q_3\) reduces it standard deviation and \(Q_4\) offers good potential to minimize the maximum relative temperature uncertainty. Instead, \(Q_3\) delivers low performances, and should not be retained. Note also that \(Q_1\) and \(Q_2\) have similar performances, indicating that the mean relative temperature uncertainty criterion plays a major role as compared to its standard deviation. \(Q_1\) and \(Q_4\) also depicts the same tendencies, indicating that consideration of the maximum and/or the mean relative temperature uncertainties are redundant in the multi-objective optimization procedure.

In conclusion of this section, \(Q_1\) appears to be a good compromise to select the spectral properties of the optical filters. It offers a mean relative temperature uncertainty 40 % better than that of \(Q_3\), only 5 % worse than that of \(Q_2\). The multi-objective optimization process is also mainly driven by the mean relative temperature uncertainty and the spectral specifications of the selected filter sets are marginally correlated to the definition of the functions Q tested in the current study.

4.3.3 Alternative detection strategies

Fig. 7
figure 7

Results of the multi-objective optimization procedure for various combinations of optical filters. a Performances of each optimal filter set when compared to the reference filter set, in terms of the normalized relative sensitivity factor \(\varPsi ^*\) (green bars), normalized signal-to-noise ratio factor \(\varGamma ^*\) (purple bars) and relative temperature uncertainty (blue bars), see Eq. 3. These parameters have been temperature-averaged. b Normalized relative temperature uncertainty for relevant filter combinations. Values lower than unity mean an enhancement of the uncertainty with respect to the reference filter set \(\left( \mathrm {FS}\right) _{ref}\). c Specifications of the best filter sets for various detection strategies (the first value \(\lambda _c\) indicates the central wavelength of a bandpass filter or a notch filter whereas it stands as the cut-off wavelength for a shortpass filter or a longpass filter)

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In this section, the bandpass filters are replaced by notch filters, shortpass filters and longpass filters, in order to investigate the performances on the temperature uncertainty coming from the modification of the spectral profile of the optical filters. Sixteen filter combinations have been tested, and Fig. 7 only shows the most relevant cases. The optimization procedure is performed using the function \(Q_1\) defined in Eq. 8 on \(\varDelta T = \left[ 300 - 750\,\mathrm {K}\right] \).

Figure 7a presents the normalized mean relative sensitivity factor (green bars), normalized mean signal-to-noise ratio factor (purple bars) and the normalized mean relative temperature uncertainty (blue bars) for various filter combinations, Eq. 3. A value lower than unity indicates improved performances compared to the reference filter set (combination of two bandpass filters) \(\left( \mathrm {FS}\right) _{ref}\). Three filter combinations stand out: bandpass-bandpass, bandpass-notch and bandpass-longpass. It appears that the bandpass-notch and bandpass-longpass filters combinations are able to reach a normalized mean relative temperature uncertainty 5 % lower than the original bandpass-bandpass configuration, due to the significant enhancement of the signal-to-noise ratio factor, even if the relative sensitivity term is similar. The bandpass-shortpass, shortpass-longpass and longpass-longpass combinations demonstrate similar and even worse performances in temperature uncertainty (loss between 1 % and 5 %). On contrary, the shortpass-shortpass configuration is not a good combination, with weak relative sensitivity factor resulting in mean performances 40 % worse than the bandpass-bandpass configuration.

The normalized relative temperature uncertainty estimator is plotted versus temperature in Fig 7b for the same filter combinations. As expected the normalized relative temperature uncertainty is better for the bandpass-notch and bandpass-longpass configurations compared to the bandpass-bandpass configuration on the entire temperature range. The gain is particularly noticeable at high temperature (750 K), where \(\left( \delta T / T\right) ^*=0.0791\) for the bandpass-notch configuration, against \(\left( \delta T / T\right) ^*=0.1568\) for the bandpass-bandpass configuration. The shortpass-longpass combination gives similar performances below 400 K, slightly better in the 400 - 700 K and much lower at higher temperature. Finally, whereas the trend for the longpass-longpass configuration is similar to that of the bandpass-bandpass combination, the bandpass-shortpass combination gives a poorer relative temperature uncertainty for temperature lower than 700 K.

The specifications of the best filter sets for all the filter combinations are shown in Fig. 7c. Interestingly, the specifications of the bandpass-bandpass, bandpass-notch and bandpass-longpass filter sets are very close: one filter centered around the 660 nm peak, and one filter located on the red wing of the phosphorescence spectra. These spectral specifications are also observed when considering a shortpass-longpass filter configuration: one collecting wavelengths lower than 668 nm, and the other one detecting longer wavelengths above 668 nm.

Contrary to the above-mentioned filter combinations, the bandpass-shortpass filter configuration consists of a bandpass filter located on the 660 nm phosphorescence peak and a shortpass filter cutting off at \(\lambda _c = 650\,\mathrm {nm}\), thus allowing the detection of the 630 nm emission band of the emission spectrum. However, as depicted in Fig. 7a, b, this combination presents lower performances in terms of relative temperature uncertainty (16 % worse) even if the signal-to-noise ratio factor is improved.

A limitation of this parametric study originates from the single use of the phosphorescence spectrum during the optimization procedure. Considering technical flame environments with the production of light-emitting interferences, such as thermal radiations, soot emissions, or light emission (chemiluminescence, laser interferences...) could undoubtedly generate interfering sources of noise, leading to strong biases between the calibration and the experimental measurements. Moreover, a reduction of the dynamic range of phosphorescence signals could lead to measurement bias in terms of temperature uncertainty. When possible, additional filters should be added to remove these interferences but this will complicate the experiments and therefore moderate performances obtained with the bandpass-notch and bandpass-longpass filter combinations.

4.4 Camera validation

Fig. 8
figure 8

Validation of the best filter set \(\left( \mathrm {FS}\right) _{opt}\) (red lines and markers) compared to the reference filter set \(\left( \mathrm {FS}\right) _{ref}\) (blue lines and markers) conducted with an ICCD camera and commercial optical bandpass filters. a Experimental calibration curves for both sets of filters (markers) and corresponding relative sensitivity (dotted lines, right axis), from images acquired with the ICCD. b The normalized relative temperature uncertainty \(\left( \delta T / T\right) ^* = \left( \delta T / T\right) _{opt} / \left( \delta T / T\right) _{ref}\), computed for the spectrograph detection (dashed purple line, using Eq. 3 and Eq. 6) and images acquired with the ICCD camera (green solid line, using Eq. 3 and Eq. 7). c Relative sensitivity factor \(\varPsi = \left( TS_r\right) ^{-1}\) (see Eq. 3) (opened markers, left axis), and the signal-to-noise ratio factor \(\varGamma = \sqrt{\sum _i\mathrm {SNR}_{\varDelta \lambda _i}^{-2}}\)(closed markers, right axis) for the ICCD camera. d Relative temperature uncertainty \(\delta T/T\) computed with Eq. 3 and Eq. 7 (solid lines), and measured relative temperature uncertainty standard deviation \(\sigma (T)\) (closed markers), for the ICCD camera

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An experimental validation of the results determined with the multi-objective optimization procedure is performed by replacing the spectrograph/CCD detection system by an ICCD camera (see Sec. 3.3). Two experimental calibration curves are recorded, one using the optical filters corresponding to the reference filter set \(\left( \mathrm {FS}\right) _{ref}\) (commercial filters Semrock FF01-631/4 and FF01-660/13), and a second one corresponding to the optimal filter set \(\left( \mathrm {FS}\right) _{opt}\) detailed in Sect. 4.2 (Semrock FF01-660/13 and FF01-680/13). Note that these commercial filters were selected because of the similarities to those returned by the multi-objective numerical optimization procedure.

Figure 8a shows the two calibration curves. The evolutions of the ratio R with temperature show significant differences. The relative sensitivity \(S_r\) of each filter set (dotted lines, secondary axis) also confirms that the optimal filter set gives a sensitivity 3.2 times larger on the whole temperature range, when compared to the one observed for the reference case.

Figure 8b shows the normalized relative temperature uncertainty between the reference and the optimal filter sets by using Eq. 3. It compares the performances calculated with the multi-objective optimization procedure (dashed purple line) using the phosphorescence emission database, and those recorded with the ICCD camera (green solid line). The normalized relative temperature uncertainty of both acquisition systems (spectrometer and ICCD camera) overlaps until \(T=600\,\mathrm {K}\), while small deviations are observed at higher temperature, possibly attributed to the decrease of the signal-to-noise ratios. The good similarity while having two detection systems validates the methodology to use a database of phosphorescence spectra combined with virtual optical filters to enhance the temperature uncertainty for phosphor thermometry measurements.

The contribution of the relative temperature uncertainty (Eq. 3), namely the relative sensitivity factor \(\varPsi \) and the signal-to-noise factor \(\varGamma \), is shown in Fig. 8c for the ICCD camera. These results clearly confirm that, regardless of the filter set considered, the dominant source of improvement in the temperature uncertainty is the relative sensitivity factor [21]. It is also interesting to note that the signal-to-noise ratio factor does not change when using the reference filter set or the optimal filter set, the latter showing an exponential growth at high temperature. In contrast, the relative sensitivity factor of the optimal filter set is improved, reaching a factor 4.5 at 650 K.

The absolute relative temperature uncertainty \(\delta T / T\) determined with Eq. 3 for the ICCD camera is shown in Fig. 8d (solid lines) for the two filter sets. Firstly, the optimized filter set \(\left( \mathrm {FS}\right) _{opt}\) allows to reduce the mean relative temperature uncertainty between 300 and 750 K by a factor 3, down to 1.14 % compared to 3.38 % for \(\left( \mathrm {FS}\right) _{ref}\). For \(\left( \mathrm {FS}\right) _{opt}\), the relative temperature uncertainty is slightly decreasing for temperature lower than 750 K, whereas reaches a value of 2.8 % at 850 K. A different trend is observed for \(\left( \mathrm {FS}\right) _{ref}\), with an increase of the relative temperature uncertainty from 300 to 600 K, a plateau at 4.6 % between 600 and 700 K, followed by a larger increase at higher temperature (up to 12 % at 850 K).

Markers represented in Fig. 8d correspond to the measured relative standard deviation \(\sigma (T)/T\) determined from the experimental images. It is interesting to note that \(\sigma (T)/T\) and the relative temperature uncertainty estimator \(\delta T/T\) present similar trends. However, \(\sigma (T)/T\) appears to be 2.5 times larger than the relative temperature uncertainty estimator (whatever the FS considered). This result can be explained by the fact that the measured standard deviation gathers the noise induced by the detection system (shot-noise), but also additional sources of noise arising from the spatial characteristics of the ICCD sensor (inhomogeneity of the pixel-to-pixel sensitivity, matching of the signal of the two images, temperature inhomogeneity, etc.).

Finally, the optimal filter set \(\left( \mathrm {FS}\right) _{opt}\) demonstrates a mean standard deviation between 300 K and 750 K of \(\overline{\sigma _{opt}} = 2.9\,\%\), while it is 8.4 % for the reference case. This enhancement is even more pronounced at high-temperature, being of 24.5 % at 847 K for the reference case, while it is only 6.3 % for the optimal configuration. Although the measured temperature uncertainty with the reference filter set is better at 300 K (3.50 % for the reference case vs. 4.73 % for the optimal one), the optimal filter set enables to achieve better temperature uncertainties for T > 340 K, with a gain being of a \(\left( \mathrm {FS}\right) _{opt}\) 3.5 factor at 500 K, and even a 4.5 factor at 600 K.

5 Conclusion

The current study presents a multi-objective optimization methodology to improve the temperature accuracy of measurements performed with the spectral intensity ratio phosphor thermometry. Based on the use of the relative temperature uncertainty calculated in the the shot-noise limited regime as the estimator of the optimization procedure, a single multi-objective variable is defined by considering the mean relative temperature uncertainty and its standard deviation on a large range of temperature. This method is implemented on a set of experimental phosphorescence spectra of the Mg4FGeO6:Mn4+ luminescent particles. The best combination of filters consists of a filter centered on the 660 nm phosphorescence emission peak, the other one being shifted towards the red-wing emission spectrum. A detailed analysis of the results reveals that the temperature sensitivity of the ratio between the two phosphorescence intensities is the main factor for enhancing the temperature uncertainty when comparing to a reference case from the literature. In contrast, the improvement of the signal-to-noise ratio (SNR) recorded on each channel compared to the reference case plays a marginal role. This set of optical filters greatly differs from usual arrangements reported in the literature. A robustness study of the optimization procedure is performed. Although the results of this study are roughly the same while varying different input parameters, it appears that high-temperature spectral features of MFG phosphor, combined with the mean relative temperature uncertainty are considerably weighting the convergence of the optimization. A validation of this new set of optical filters is then performed when using an ICCD camera. Similar trends in improvements are achieved when compared to the numerical routine. The relative temperature uncertainty is 1.2 % between 300 and 750 K, being three times better than the reference set of filters reported in the literature. The measured relative temperature uncertainty of the optimized configuration is 2.9 % between 300 and 700 K, while it is 8.4 % for the reference case. This trend is even more significant at high-temperature, where the temperature uncertainty of the reference configuration is four times larger at 750 K when compared to the output of the optimization. The measured average relative sensitivity of the optimized configuration is 0.32 %/K, while it is 0.15 %/K for the reference setup. Depending on the temperature range where temperature measurements must be performed, the results issued from a parametric study using different objective functions clearly show the significance of the definition of the multi-objective function Q on the definition of the spectral parameters of the optical filters and notably on the resulting temperature uncertainty. In addition to the potential offered by the multi-objective optimization procedure, the investigation on the importance of light interferences in flames on the selection of optical filters should be considered in the future as an additional step to extend the performances of the spectral intensity ratio-based phosphor thermometry with the use of more sophisticated multi-objective optimization criteria.