An algorithm for the characterisation of multi-exponential decay curves

ARTICLE IN PRESS Optics and Lasers in Engineering 47 (2009) 75–79

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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng

An algorithm for the characterisation of multi-exponential decay curves J. Bru¨bach , J. Janicka, A. Dreizler ¨t Darmstadt, Petersenstraße 30, 64287 Darmstadt, Germany Fachgebiet Energie- und Kraftwerkstechnik, Technische Universita

a r t i c l e in fo

abstract

Article history: Received 20 August 2007 Received in revised form 14 July 2008 Accepted 30 July 2008 Available online 24 September 2008

An algorithm for the non-ambiguous reduction of multi-exponentially decaying luminescence signals to scalar lifetimes is described and characterised. Whereas this investigation stems from the research of thermographic phosphor thermometry, the technique can also be useful for the analysis of various other decay phenomena. Thermographic phosphors are activator-doped ceramics, whose phosphorescence decay characteristics depend on their temperature. Although the underlying energy transfer processes of the generally multi-exponential phosphorescence decay are not understood in detail, this property can be exploited for remote thermometry. Thereby, the temperature measurement is converted to the measurement of a phosphorescence lifetime evaluated by the approximation of a single-exponential term within a fitting window where mono-exponential decay characteristics prevail. In contrast to multi-exponential approaches, ambiguity due to the attribution of various exponential terms is eliminated provided that the fitting window is the same for the evaluation of all measurement data obtained at unknown temperatures and all temperature-referenced calibration data. In order to achieve this, an iterative algorithm was applied that selects a fitting window dependent on the decay waveform itself. This algorithm was characterised in detail using data detected for two different thermographic phosphors (Mg4 FGeO6 : Mn and Y2 O3 : Eu), at two different temperatures each. Compared to less elaborate routines applying a predefined fitting window depending on an ambiguous setting of the phosphorescence signal’s detection length, the evaluation-induced systematic error of temperature determination could be reduced by a factor of 103 at similar precision. & 2008 Elsevier Ltd. All rights reserved.

PACS: 07.20.Dt 32.50.þd 42.30.Lr Keywords: Multi-exponential decay Lifetime Decay time Thermographic phosphors Phosphor thermometry Fluorescence spectroscopy

1. Introduction Thermographic phosphors (TP) are activator-doped ceramics, whose phosphorescence decay characteristics depend on their temperature. Although the photo-physics of these materials is not understood in detail, this property can be exploited for remote thermometry [1–9,17]. This presumes calibration measurements under controlled conditions for various temperatures and data reduction by fitting an analytical waveform to the measured decay curve. As generally the decay characteristics are not monoexponential, this reduction takes considerable effort [10–14]. In this context, Omrane [13] as well as several earlier studies of this work encountered problems with regards to ambiguous attribution of the various exponential terms, because a variation of the lifetime components compensates for a variation of the amplitude components and vice versa. However, from a more practical point of view a mono-exponential approach could be applied, because phosphor thermometry claims to deliver a quantity that is a nonambiguous measure for the temperature rather than to explore the energy transfer structure or a detailed photo-physical model.

 Corresponding author. Tel.: +49 6151162502; fax: +49 6151166555.

E-mail address: [email protected] (J. Bru¨bach). 0143-8166/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2008.07.015

In case of a mono-exponential description, the fitting window has to be the same for all measurements independent from the different settings of the detection system and particularly independent from the inherently different settings of the overall observation window length. In this context, the present study characterises an accurate, precise and reliable method for a nonambiguous reduction of phosphorescence decay waveforms to one single lifetime by applying an iteratively defined fitting window that depends on the decay characteristics themselves. Accuracy and precision of this method are compared to that of a predefined fitting window depending on the observation window of the detection system.

2. Data acquisition Temporally resolved intensity decay curves were acquired for two different phosphors (Mg4 FGeO6 : Mn and Y2 O3 : Eu), at two different temperatures each. The phosphor was excited by a pulsed (10 Hz), Q-switched, frequency tripled (355 nm, Mg4 FGeO6 : Mn) or frequency quadrupled (266 nm, Y2 O3 : Eu) Nd:YAG laser. The subsequently emitted phosphorescence was imaged onto a photomultiplier after passing an interference filter ðFWHM ¼ 10 nmÞ. The filter was used to suppress spurious

ARTICLE IN PRESS ¨ bach et al. / Optics and Lasers in Engineering 47 (2009) 75–79 J. Bru

scattering of the laser and other disturbing light from the laboratory environment. The photomultiplier current was read out by a digital oscilloscope (Tektronix 5032B) at an input resistance of 512 O, whereas for the discretisation of the overall observation window a temporal resolution of 2000 samples per pulse was chosen. The oscilloscope provided a high resolution mode, which utilises more information than the appointed sample rate. It always exploits its maximum sample rate and reduces the data by temporal batch bulk averaging. At long time scales, this is the reason for a better signal-to-noise ratio and a dynamic range greater than the nominal 8 bit.

1.5

Signal Fit

Δt 1 t0

Intensity

76

0.5

3. Algorithm 0

As several tests applying bi- or multi-exponential fitting routines showed problems with regard to an ambiguous attribution of the various exponential terms, the single-shot waveforms were approximated by a mono-exponential decay   t (1) þ Ioffset IðtÞ ¼ I0 exp 

0

3.1. Predefined fitting window One way to ensure this, is to keep constant the settings of the detection system (observation length Dt, laser trigger point t 0 ; . . . cp. Fig. 1) and to predefine a fitting window within fixed start and end points related to t 0 and Dt. For example, the fitting window would start at (2)

and end at (3)

However, since the dynamic range of the decay characteristics of TP often exceeds more than three orders of magnitude, the length of the observation window Dt generally cannot be kept constant. The location and the length of the fitting window would therefore depend on the adjustable setting of the observation length of the detection system and would cause systematic errors of the evaluated lifetime and the temperature, respectively. 3.2. Iteratively adapted fitting window To avoid this inaccuracy, an algorithm was applied that is related to the lifetime t instead of Dt: therefore, the fitting window starts at (4)

and ends at t2 ¼ t0 þ c2  t

4

5 6 Time (ms)

7

8

9

10

Signal Fit

Intensity

10−1 Fit 2

10−2

10−3

10−4

t1 ¼ t0 þ c1  t

3

Fit 1

After the offset intensity Ioffset was eliminated by subtracting the mean value of the respective single-shot’s intensity prior to the laser pulse, a Levenberg–Marquardt algorithm [15,16] approximated the initial amplitude I0 as well as the phosphorescence lifetime t. Whereas t is a measure for the temperature of the phosphor material, I0 only depends on the signal strength and is thus not considered any further. Applying this mono-exponential fit to a multi-exponential signal, the evaluated lifetime t strongly depends on the beginning and the end of the fitting window (Fig. 1). Therefore, a precisely defined fitting window consequently has to be the same for the evaluation of all waveforms.

t 2 ¼ t 0 þ c2  Dt

2

100

t

t 1 ¼ t 0 þ c1  Dt

1

(5)

0

1

2

3

4

5 6 Time (ms)

7

8

9

10

Fig. 1. Normalised single-shot phosphorescence signal for Y2 O3 : Eu at 773 K as well as two mono-exponential fits for two different fitting windows in a linear (upper plot) and a logarithmic scale (lower plot). In the logarithmic scale, the absolute value of the slope of the fit corresponds to the reciprocal of the evaluated lifetime t.

As t is one of the unknown quantities, the fitting routine has to be applied iteratively. In this context, only the initial guess t0 was chosen to be related to the length of the observation window Dt

t0 ¼ c0  Dt

(6)

whereas the constant c0 was varied to investigate the sensitivity of this parameter. Since t0 is influenced by c0 and by Dt in the same way, this variation is corresponding to a variation of the observation length assuming that the length of the observation window after laser triggering is much longer than the considered fitting window (Dt  t 0 bt 0 þ c2 t). Therefore, the parameter Dt did not have to be varied separately. The initial guess for I0 was chosen to be half the intensity of the maximum of the overall signal. For the iterative determination of t, the evaluated lifetime of the current iteration step is used for the definition of the next step’s fitting window and for the new initial guess of the Levenberg–Marquardt algorithm (Fig. 2). The abort criterion of the iteration was achieved when the difference of the maximum and the minimum t of the last three steps was less than one thousandth of the average t

ARTICLE IN PRESS ¨ bach et al. / Optics and Lasers in Engineering 47 (2009) 75–79 J. Bru

1.8

77

7

c0 = 1/40 c0 = 1/20 c0 = 1/10 c0 = 1/6

1.6

c0 = 1/40 c0 = 1/20 c0 = 1/10 c0 = 1/6

6

1.4 Lifetime τ (μs)

Lifetime τ (ms)

5 1.2 1 0.8 0.6

3 2

0.4

Fig. 2. Flow chart of the algorithm applying the iterative fitting window. The index i denotes the iteration step, i ¼ 0 corresponds to the initial guess.

4

1

0.2 0

2

4

6

Iteration step 7

1.8

c0 = 1/40 c0 = 1/20 c0 = 1/10 c0 = 1/6

6

c0 = 1/40 c0 = 1/20 c0 = 1/10 c0 = 1/6

1.6

0

2

4

6

Iteration step

Fig. 4. Convergence behaviour of two single shots for Y2 O3 : Eu at T ¼ 773 K (left plot) as well as at T ¼ 1073 K (right plot). Step 0 corresponds to the initial guess t0 ¼ c0  Dt.

1.4 Lifetime τ (μs)

Lifetime τ (ms)

5

4

3

Table 1 Convergence behaviour of two single shots for Mg4 FGeO6 : Mn at T ¼ 295 K (upper tabular) as well as at T ¼ 973 K (lower tabular)

1.2 1 0.8 0.6

2

0.4 1 0.2 0

2

4

6

0

Iteration step

2

4

6

Iteration step

Fig. 3. Convergence behaviour of two single shots for Mg4 FGeO6 : Mn at T ¼ 295 K (left plot) as well as at T ¼ 973 K (right plot). Step 0 corresponds to the initial guess t0 ¼ c0  Dt.

1 c0 ¼ 40

1 c0 ¼ 20

1 c0 ¼ 10

c0 ¼ 16

t0 t1 t2 t3 t4 t5

(ms) (ms) (ms) (ms) (ms) (ms)

1.0000 3.1194 3.5589 3.5896 3.5912 3.5913

2.0000 3.4189 3.5809 3.5912 3.5913 3.5913

4.0000 3.6125 3.5914 3.5914 3.5914

6.6666 3.7139 3.5973 3.5913 3.5913

t0 t1 t2 t3 t4 t5 t6

ðmsÞ ðmsÞ ðmsÞ ðmsÞ ðmsÞ ðmsÞ ðmsÞ

0.25000 1.18410 0.79163 0.78520 0.78415 0.78399

0.50000 0.80930 0.79156 0.78520 0.78415 0.78399

1.0000 0.81134 0.79156 0.78520 0.78415 0.78399

1.66666 0.72764 0.77239 0.78231 0.78511 0.78415 0.78399

Step 0 corresponds to the initial guess t0 ¼ c0  Dt.

of these steps: maxðti ; ti1 ; ti2 Þ  minðti ; ti1 ; ti2 Þo103 ½13ð i

t þ ti1 þ ti2 Þ

(7)

In several pre-studies, the settings c1 ¼ 1 and c2 ¼ 4 yielded good results, hence these specifications were employed for all evaluations of the present work. The optimisation of these settings is a trade-off between high accuracy and high precision, as the signal-to-noise ratio is better at early times of the waveform, when it is under stronger influence of parameters other than the temperature (e.g. the laser pulse energy) [18].

4. Results and discussion For different initial guesses, Tables 1 and 2 show the evaluated lifetimes of the consecutive iteration steps for four different single shots. Additionally, these results are plotted in Figs. 3 and 4. In all cases, the evaluation converged within less than seven steps. Comparing the final values of t, the differences caused by the variation of the initial constant c0 and thus the error due to the variation of the observation length Dt, is in the order of 104 t.

Depending on the gradient of the respective lifetime/temperature characteristics, this implies an evaluation-induced systematic error of the temperature reading of approximately 0.01–0.001 K. Regarding the discretisation, a quartered or quadrupled number of samples within the observation window did not change the results significantly. In contrast to this, the accuracy obtained by applying a non-iterative, predefined fitting window according to Eqs. (2) and (3) can be estimated by looking at the different values 1 of t1 : Taking the initial constant c0 ¼ 20 as a reference, the initial 1 1 constants c0 ¼ 10 and 40 mimic a halved and a doubled observation length Dt for a predefined fit. This is a realistic scenario, since most detection systems only allow a change of the observation length by a factor of 2. According to Tables 1 and 2, t1 differs in the order of 0:1t and even t. This implies an evaluation-induced systematic error of the temperature determination of several 10 K. Additionally, the shot-to-shot standard deviation for a predefined fitting window with constant Dt and constant c0 was compared to the shot-to-shot standard deviation of an iteratively adapted fitting window, whereas the bounds of the predefined 1 4 window were set to t 1 ¼ t 0 þ 10  Dt and t 2 ¼ t 0 þ 10  Dt according to Eqs. (2) and (3). Evaluating 250 single shots for both phosphors

ARTICLE IN PRESS ¨ bach et al. / Optics and Lasers in Engineering 47 (2009) 75–79 J. Bru

78

at both temperatures each, the standard deviation of t was in the same order for both fitting methods. The absolute standard deviation converted into temperature scale was approximately 2 K. Therefore, the employment of an iterative fitting window is preferable due to much better accuracy at similar precision. For Mg4 FGeO6 : Mn at T ¼ 973 K and for an initial constant of 1 c0 ¼ 40 t is overshooting shortly after the start of the iteration (Fig. 3). Hence, for this special case a comparison of the signal and the fitted curves is presented for the first four iteration steps Table 2 Convergence behaviour of two single shots for Y2 O3 : Eu at T ¼ 773 K (upper tabular) as well as at T ¼ 1073 K (lower tabular) 1 c0 ¼ 40

1 c0 ¼ 20

1 c0 ¼ 10

c0 ¼ 16

t0 t1 t2 t3 t4 t5 t6

(ms) (ms) (ms) (ms) (ms) (ms) (ms)

0.2500 0.6699 0.7925 0.8121 0.8153 0.8153 0.8153

0.5000 0.7612 0.8069 0.8139 0.8153 0.8153

1.0000 0.8359 0.8171 0.8157 0.8153 0.8153

1.6666 0.8781 0.8220 0.8158 0.8153 0.8153

t0 t1 t2 t3 t4 t5 t6

ðmsÞ ðmsÞ ðmsÞ ðmsÞ ðmsÞ ðmsÞ ðmsÞ

1.0000 3.3277 3.8090 3.7428 3.7543 3.7550 3.7552

2.0000 3.3545 3.8096 3.7428 3.7543 3.7550 3.7552

4.0000 3.7954 3.7478 3.7545 3.7550

6.6666 4.8036 3.9742 3.7846 3.7480 3.7545 3.7550

Step 0 corresponds to the initial guess t0 ¼ c0  Dt.

10−1

10−2

10−3

10−1

10−2

10−3 0

2

4 6 Time (μs)

8

10

0

Signal Fit

Step 3

100

10−1

10−2

2

4 6 Time (μs)

8

10

Signal Fit

Step 4

100

Intensity

Intensity

Signal Fit

Step 2

100

Intensity

Intensity

Signal Fit

Step 1

100

(Fig. 5). Generally the opposite being the case (Fig. 1), the gradient of this signal is lower at the beginning of the waveform. This is due to the convolution of the phosphorescence intensity decay with the transfer function of the detection system that is primarily determined by the low pass character of the input resistance of the oscilloscope. At short time scales, as it is the case in Fig. 5, a significant influence on the signal emerges. Due to the 1 choice of the initial constant (c0 ¼ 40 ) and thus due to the small value for the initial guess t0 , the first fitting window is located at a very early position of the waveform where a very low slope and thus a high lifetime t1 is fitted. Corresponding to Eqs. (4) and (5), the next fitting window is located at a much later position of the signal decay and thus a much lower lifetime t2 is evaluated. During the following steps, t is diminishing further on, until the abort criteria is accomplished. In contrast to this, for longer time scales with steeper gradients at the beginning of the waveform, the fit is generally converging monotonically (Figs. 3 and 4). In some rare cases, the abort criterion is never accomplished, since t starts to oscillate from step to step. However, it appeared that only small-scale oscillations including two iteration steps and no long-scale drifts occur after 10–15 steps. Therefore, in these cases the iteration was aborted after reaching 30 steps and the final lifetime t was set to the algebraic average of the last 10 iterations. Applying this procedure, the results did not differ significantly from those where no oscillation emerged. Finally, the improved accuracy obtained by applying the iterative fitting window compared to a predefined one is demonstrated by the evaluation of a whole temperature/lifetime characteristic of Mg4 FGeO6 : Mn (Fig. 6). Whereas the employment of a predefined fitting window (upper plot) results in a

10−1

10−2

10−3

10−3 0

2

4 6 Time (μs)

8

10

0

2

4 6 Time (μs)

8

10

Fig. 5. Temporally resolved phosphorescence intensity as well as the fitted curves for the iteration steps 1–4 (Mg4 FGeO6 : Mn, T ¼ 973 K). Further steps are not presented, since the changes from step to step become too marginal.

ARTICLE IN PRESS ¨ bach et al. / Optics and Lasers in Engineering 47 (2009) 75–79 J. Bru

5. Conclusions

101

101

An algorithm providing a non-ambiguous reduction of multiexponentially decaying phosphorescence waveforms to one single lifetime was characterised in detail. In this context, the length and location of the fitting window do not depend on the adjustable length of the observation window, but only on the lifetime t itself. Thus, an iterative evaluation of t is necessary. Compared to less elaborate considerations applying a predefined fitting window depending on an ambiguous setting of the phosphorescence signal’s detection length, the evaluation-induced systematic error of temperature determination could be reduced by the factor of 103 at similar precision. Besides the phosphor thermometry, this technique might be useful for numerous other disciplines, where multi-exponential decay characteristics are investigated, e.g. the fluorescence spectroscopy of organic molecules. Future work could be focussed on the fitting constants c1 and c2 ; and therefore on the relation of t and the fitting window with regard to a balance of high precision and high accuracy, since these optimal settings generally vary for different phosphor materials.

100

Acknowledgements

Lifetime τ (ms)

100

10−1

10−2

10−3

10−4

Lifetime τ (ms)

79

300

400

500

600 700 800 Temperature (K)

900

1000

The authors gratefully acknowledge financial support of the DFG (Deutsche Forschungsgemeinschaft), GRK 1114 and would like to thank Matthias Mu¨ller, Ho¨sbach, Germany, for helpful discussions.

10−1

10−2 References

10−3

10−4

300

400

500

600 700 800 Temperature (K)

900

1000

Fig. 6. Comparison of the temperature/lifetime characteristic evaluated by employment of a predefined fitting window (upper plot) and an iterative fitting window (lower plot). While the upper plot shows significant steps due to the change of the observation window, the lower plot has a continuous shape. For each measurement point, 100 single shots were evaluated and averaged.

discontinuous calibration curve due to the inherent change of the time base of the observation system, the lower plot obtained by an evaluation of the same data under application of an iterative fitting window shows a continuous shape. For the iteratively defined fitting window, the data was evaluated by employing 1 1 1 1 different initial constants (c0 ¼ 40 ; 20 ; 10 ; 6). However, as the four plots did not differ in any kind of way, they were not presented separately.

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