WO2007088173A1

WO2007088173A1 - Apparatus and method for characterizing a system for counting elementary particles

Info

Publication number: WO2007088173A1
Application number: PCT/EP2007/050928
Authority: WO
Inventors: Antoine Souloumiac
Original assignee: Commissariat A L'energie Atomique
Priority date: 2006-01-31
Filing date: 2007-01-31
Publication date: 2007-08-09
Also published as: FR2896877A1; EP1979736A1

Abstract

Method of characterizing a system that can emit elementary particles, characterized in that it comprises the following steps: a. exciting (20) the system (60) at different predetermined instants, b. for each of the excitations, b.1. detecting (30) the elementary particles (70) emitted by the system (60), b.2. measuring (40) the response time of just the first of said detected particles, c. determining (50) a first empirical distribution function (Formula (I)) or a first empirical probability density (Formula (II)) for the response times measured in steps (b.2), d. determining (50) a second empirical distribution function (Formula (III)) or a second empirical probability density (Formula IV)) for the response time of each of the particles detected in steps (b.1) on the basis of an equation (EQ) which relates the second empirical distribution function (Formula (II)) to said first empirical distribution function (Formula (I)).

Description

Apparatus and method for characterizing an elementary particle counting system

The invention relates to a method and an apparatus for characterizing a system capable of emitting particles. Methods and apparatus based on the principle of the invention are generally used in various fields such as molecular fluorescence, luminescence, and more generally the counting of photons or elementary particles correlated in time.

By way of non-limiting example, the invention can advantageously be applied to a technique known by the acronym acronym TCSPC

("Time Corelated Single Photon Counting").

In this case and for the field of fluorometry for example, this technique generally consists of subjecting the system to be studied (one or more molecule (s) according to this example) to periodic laser excitations. Following each of these excitations, the response time of each fluorescence photon emitted by the excited system is measured and a histogram of the response times is then constructed in order to estimate a probability density.

The probability density thus obtained is then used to characterize the system, that is to say, in this example, identify the molecule (s). The estimation of this probability density often consists in determining only exponential function time constants.

Indeed, it is assumed that said probability density can be fully defined by means of such functions.

More specifically, FIG. 1 illustrates an apparatus typically used in the context of the TCSPC technique. Of course, such a device is not limited to this technique. As can be seen, it comprises a detector 1 which detects, or said otherwise captures, the photons 2 emitted by a system 3 excited by an excitation source. The detector 1 delivers a signal to a converter 4 which in turn delivers another signal that can be processed by a computer 5.

The converter 4 is always a converter known as the "TAC converter" (TAC is the acronym for "Time to Amplitude Converter" in the English language). It is recalled that such a converter operates in a very particular way.

Indeed, it operates in such a way that only the first of the photons detected is considered in the following of the method implemented by the apparatus.

In other words, once the signal relative to the first photon arrived at the converter, it is no longer sensitive input. Such an operation is based on the principle that the converter must be initialized before or after each laser excitation, so that after a certain number of excitations, the apparatus can measure the response time of all the first photons.

The computer 5 receives from the converter 4 signals from which it determines, among other things, the aforementioned exponential functions for defining said probability density.

In particular, it determines the number of functions and the values of the time constants of each of these functions. A known problem is that the performance of this type of apparatus degrades when the time constants decrease and become lower than a predetermined threshold which depends on the device.

Such degradation is notably due to the fact that the latter has a non-negligible random measurement time with respect to photon response times.

Therefore, a value of the photon response time provided by such a device substantially corresponds to the actual response time of the photon _/ added to the random measurement time, which distorts the result. In order to solve this problem, the methods implemented in these devices are improved in particular by constructing two histograms instead of one as before.

A first histogram takes into account what is commonly called the "device function". In order to construct this histogram, the method is implemented without exciting the system so that no fluorescence photon is emitted.

An estimate of the probability density of the measurement time of the apparatus can then be obtained.

A second histogram is constructed conventionally, that is, using the conventional technique described above; the system is excited so that there is fluorescence.

The time constants of the exponential functions are then estimated by minimizing a difference between the second histogram and the first histogram convoluted by the exponential functions comprising the constants not yet estimated. Another problem related to this apparatus, in particular to the use of the TAC converter, is that the probability density of the response time of all the photons detected does not correspond to the probability density of the response time of the first photons considered in isolation. i.e., the probability density of the response times measured by the converter.

In particular, at each excitation, this isolation of the first photon biases the measurements of the response time and therefore, at a more global level, it biases the estimation of the fluorescence time constants.

Indeed, to a certain extent this amounts to a sorting operation consisting in systematically selecting short response times to the detriment of long response times.

Note that such a problem is commonly referred to as the "stacking effect" or in the English language "stack-up".

Several solutions have been proposed to limit this problem. A first very practiced solution consists in using the apparatus A under conditions in which the average number of emitted photons has a very low value (for example a value less than 0.02 photons detected by excitation). For this purpose, the power of the excitation is reduced.

Under these conditions, the probability that more than one photon is emitted is low and the distortion due to the converter becomes negligible.

However, a disadvantage is that the process takes a long time to implement since nearly 98% of the excitations do not generate any photon emission and are therefore useless. A second solution [1], which makes it possible to use higher intensities, consists in correcting the histogram of the response times measured by the converter TAC.

More specifically, this solution uses a correction formula to be applied to said histogram.

More precisely, the proposed method consists in: measuring the response time of the first photon detected during a plurality of excitations (operation performed by the TAC converter), constructing a histogram from the measured response times, histogram using the correction formula, and deduce from this histogram the probability density of the response times of all the photons detected at each excitation (deduce in particular the time constants of the exponential functions which define the probability density in question ). However, this solution still has disadvantages.

In particular, the construction of a histogram results in a loss of information and therefore measurement accuracy on said time constants, in particular.

Moreover, such a solution necessarily rests on the principle according to which the number of photons detected following an excitation follows a law of

Fish, which disadvantageously limits its applications.

An object of the invention is to overcome the disadvantages at least described above. For this purpose, there is provided according to the invention a method of characterizing a system capable of emitting elementary particles, characterized in that it comprises the following steps: a. excite the system at different predetermined times, b. for each excitation, detect the elementary particles emitted by the system, b.2. measuring the response time of the first only of said detected particles,

vs. determine a first empirical distribution function (Φ (t)) or a

first empirical probability density (φ (t)) of the response times measured in steps (b.2), d. determine a second empirical distribution function (F (ή) or a

second empirical probability density (f (t)) of the response time of each of the particles detected in steps (b.l) on the basis of an equation that relates the second empirical distribution function {F (ή)} to said first

empirical response function (Φ (t)).

Preferred but nonlimiting aspects of this method are the following: the first empirical distribution function Φ (/) is determined by an expression of the following type:

N EXC ^{~~ N} 0 where, Nεxc the total number of excitations, No the number of excitations which did not lead to any particle detection, N (t) the number of particles whose measured response time is less than t ; the equation is a function of a parameter λ to be estimated, this parameter translating a dependence between an intensity of the excitations and a law of probability of the number of particles detected after each excitation;

the parameter λ is estimated according to an expression of the following type:

where g is a predetermined function which depends on the probability law of the number of particles detected after each excitation;

the function g is defined by an expression of the following type:

g (A) = Σ a _π Λ ⁿ

with a coefficient and n the number of particles detected;

the second empirical distribution function F (ή of the response time of each of the particles detected as a result of the different excitations has an expression of the following type:

F (/) = l-ig ^"! Ga) - \ g (λ) ~ g (o) jΦ (o λ

the second empirical probability density f (t) of the response time of each of the particles detected as a result of the different excitations is deduced from the second empirical distribution function Fit) by derivation with respect to time in the equation; the probability law being a Poisson law of parameter λ, the function g is defined by: n≈ + ro in n-0 ^{1 Ll} the estimation of the parameter λ and the second empirical distribution function F (t) of the response time of each of the particles detected following the different excitations have an expression of the type:

λ / M

the distribution law being a Binomial distribution of parameters (M, ρ = ',), Ia

function g is defined by:

the estimation of the parameter λ and the second empirical distribution function F (t) of the response time of each of the particles detected following the different excitations have an expression of the type:

A method for characterizing a system that can emit elementary particles is also proposed, characterized in that it comprises the following steps: a. excite the system at different predetermined times, b. for each of the excitations, bl detect elementary particles emitted by the system, b.2. measuring the response time of the last only one of said detected particles,

Λ c. determine a first empirical distribution function Φ (t) or a

first empirical probability density cp (t) of the response times measured in steps (b.2), d. determine a second empirical distribution function F (J) or a

second empirical probability density f (t) of the response time of each of the particles detected in steps (b.l) on the basis of an equation which relates the second empirical distribution function F (t) to said first function of

empirical distribution Φ (t).

Preferred but non-limiting aspects of the two abovementioned methods are as follows:

the particles are photons; the system comprises at least one fluorescent substance and the particles are photons emitted by the substance;

the system is an optical fiber and the particles are photons.

Furthermore, according to the invention, there is provided an apparatus capable of implementing the method of the invention according to any one of the abovementioned aspects considered alone or in a suitable combination.

In addition, a computer program of a system that can emit elementary particles when it is excited is proposed, the program being loadable on a computer and comprising a set of instruction codes suitable for - determine a first empirical distribution function Φ (t) or a

first empirical probability density cp (t) of the response time of each of the first particles detected during, respectively, a series of excitations of the system; and - determining a second empirical distribution function F (t) or a

second empirical probability density f (t) of the response time of all the particles detected during, respectively, the different series excitations, on the basis of an equation which links the second distribution function

Thus, the invention offers numerous advantages over the state of the art.

In particular, a first advantage of the method is that it is quick to implement.

Indeed, this method being insensitive to particle stacking effects, it makes it possible to use high intensity excitation, unlike some of the aforementioned techniques according to which intensity must be strongly limited, for example laser , so as to ensure that the system does not emit more than one particle by excitation.

A second advantage is that the method is simple to implement. Indeed, 3a correction of the first distribution function or the first probability density of the response times of the first photons is implemented in particular on the basis of a simple equation and whose variables are easily determinable. Another advantage of the above is that the method according to the invention applied in particular to the TCSPC technique does not require additional components compared to some of the aforementioned apparatuses of the state of the art.

It does not generate additional cost. The method can be applied both in real time directly on the measurement chain, a posteriori and / or remotely, for example by running on a PC type computer the program according to the invention.

A fourth and not least advantage is that the accuracy obtained on the estimation of the response times is remarkable. It follows that the invention allows a characterization of the extremely fine system.

The Applicant considers that such a performance comes in particular from the fact that the correction equation is directly applied to an empirical probability density or by equivalence to an empirical distribution function, and not, for example to a histogram which depletes in information and therefore limits the accuracy.

Other aspects, objects and advantages of the invention will appear better on reading the following description of the invention, with reference to the appended drawings, in which: FIG. 1 schematically shows an apparatus of the prior art, the FIG. 2 schematically shows an apparatus according to the invention, FIG. 3 is a flowchart of a method of the invention that can be implemented in the apparatus of FIG. 2.

Referring now to Figure 2, there is illustrated an apparatus adapted to implement a method according to the invention. The apparatus, designated by reference numeral 10, comprises an excitation source 20, a detector 30, a time-amplitude converter 40 (TAC), and a computer 50 or other calculation means known per se as a processor, a DSP

(acronym for "Digital Signal Processing" in English) or an FPGA (acronym for "FuIl Programmable Gâte Array" in English).

The device can operate as follows.

The source 20 excites at a predetermined intensity a system 60, for example one or more fluorescent molecule (s).

The excited system 60 emits more or fewer particles 70, in the example it is photons, depending on the intensity.

It should be noted here that the latter may have a value such that the number of particles emitted is greater than 1.

Assuming for example that there is emission of several particles, they arrive randomly on the detector 30. This then generates at least one signal as soon as the first photon has arrived, which signal is delivered to the time-amplitude converter. 40.

The converter 40 cooperates with the computer 50 to measure the response time of the first only of said detected particles.

It should be noted that the response time in question here corresponds to a conventional definition, namely a time difference between the moment when the system is excited and the time when it is detected at the detector 30 (FIG. 2).

Thus, in the case of a fluorescent molecule, the response time is defined by the time difference between an absorption of a wave 80 emitted by a laser 20 and the subsequent emission of the photon in question. In order to be able to have several measured response times, these steps, including the excitation step, are repeated a predetermined number of times.

From these measured response times, the computer 50, using the computer program, is then able to determine, preferably, a

first empirical distribution function Φ (t).

Note that instead, one can choose to determine a first density

of empirical probability φ (t), given the obvious link between a probability density and a distribution function.

For reasons of simplification in the reading below, the method of the invention will be described based on the search for the first function of

empirical distribution Φ (1), only.

Of course, those skilled in the art will obviously know how to adapt this

method in the case of a search for the empirical probability density f (t), the latter being equivalent by simple derivation to said distribution function F (t).

For reasons that have already been mentioned above (stacks

in particular) the first distribution function Φ (t) is to some extent distorted, in any case not sufficiently precise.

In order to correct this problem, notably related to the omission of the effects on the measurements related to the particular operation of the converter 40, said first function is corrected from an EQ equation based on a precise modeling of this converter 40. Prior to the description of the following steps of the method of the invention implemented by the computer, the principles on which the invention is based will be described.

In particular _/ the accurate modeling of the converter 40 based on a principle that the converter 40 operates as if it systematically considered the minimum of a random number of independent response time.

In other words, after each excitation, a random number N of photons reaches the detector after independent and identically distributed response times ti, ... tN, and the converter only measures the minimum response time among these times. , ... tN.

Such a theoretical model can therefore be written in the following form:

where "min" denotes a minimum function and τ is the response time measured by the converter.

The reader's attention is drawn to the fact that the index values are not meant to represent a particular order of arrival of the particles.

Thus, the index 1 of ti does not mean that the response time ti is the shortest of all these times. It is possible that the variable τ corresponds to tio, for example.

From the aforementioned model, it is then possible to establish the aforementioned correction equation EQ by proceeding to the following development.

We notice :

- F (t) the corrected distribution function also designated here by second distribution function. This is particularly the distribution function of the response time of all the particles detected at the detector 30; this distribution function therefore takes into account what you are looking for, namely all the times ti to tN and not only the time τ. Φ {t) the uncorrected distribution function, also called the first distribution function, the minimum response time τ, or in other words the response time measured by the converter TAC. Φ (t) is therefore an observed quantity since it is estimated from the measurements of this converter,

Pχ {N = ή) a probability of random variable N, N being an integer, n the number of detected particles and λ a parameter that reflects the influence of the intensity of the excitation on the distribution of the number of photons detected.

For example, it translates that a laser whose power increases involves the detection of more photons on average. Noticing now that:

Pr (min (t, t ₂ t _N )> τ) = l- Φ (τ) where Pr denotes a probability, a development within the reach of those skilled in the art makes it possible to quickly obtain an expression for the first function of distribution Φ (f), namely:

_Φ (t) ≈ i - Y ^* P _Λ (N = n) π _ _{F (t)} y

This equation implies a relation of the type: F (t) = C (Φ (t)) (EQ) where C is a correction function.

It will be seen later that the same function C can be used to determine the estimate F (t) of the distribution function F (t) from the estimate Φ (/) of the distribution function Φ (t) of random variable the minimum of the response times measured by the converter 40, namely:

F {t) = C (Φ (ή) (EQ)

We will also see that, obviously, this same function C or the EQ equation offers the possibility of determining the corrected probability density.

f (t), also called second probability density, by a simple derivation with respect to time.

We will then obtain an equation connecting the corrected density f (t) to Ia

first probability density φ (t) of random variable the minimum of the response times measured by the converter 40.

To return to the principle of the invention, the probability of the number of particles n detected after each excitation can be written:

P _λ {N = n) = a _n k 'lg (λ) with gβ) = Σa _n λ ⁿ π = 0 where an is a coefficient greater than or equal to zero, we obtain: g (A) - g ( A (1 - F (t))) ⁽ '8 (A) - S (O)

Thus defined, g is a strictly increasing function.

It is therefore possible according to this last equation to obtain an expression of the second distribution function F (t) of the response time of all the particles detected in the form of the preceding equation EQ:

F (t) = 1 - ig ¹ [g (λ) - (g (λ) - g (0)) Φ (t) J It can be seen that the equation thus obtained connects the second distribution function F (t) to the first random variable distribution function Φ (t) with the minimum of the response times measured by the converter 40.

The value of the parameter λ can be estimated using a relation of the following type:

^{λ {}} gβ)

Indeed, by counting the frequency of the number of excitations which did not give rise to a particle emission, one can know the term to the left of equality. . And, g being a predetermined function or known in advance, the term g (0) is perfectly determinable.

It follows that the parameter λ can be estimated by proceeding by adjusting the two parts on both sides of the equality.

In a first case, the law (L) of the distribution relative to the number of photons detected may be a Poisson law.

The function g can then be written:

A "g (λ) ≈ e ^λ because Pτ _λ (N = n) = e- ^λ -

with n still corresponding to the number of particles detected.

From there, the function of distribution of the minimum response times of the particles can be written:

1 _ p - ^AF <t) Φ (t) = ~ _A ^κ} 1 - e ^"Λ and by inversion of this expression we obtain the expression of F (t):

(0] As can be seen again here, this equation is in the form of the equation EQ and advantageously connects the second distribution function F (t) of the response time of all the detected particles, which allows to go back to a reliable characterization of the system, at the first distribution function Φ (t) of random variable the minimum of the measured response times, that is to say the function of distribution of the times measured by the converter.

In a second case, the law (L) of the distribution relative to the number of

Detected photons can be a Binomial distribution of parameters (M, p = ^" ').

In this case, the function g can be written as:

Similarly, the function of distribution of the minimum response times of the particles is written:

and by inverting this equation, we obtain the distribution function F (t) of the response times of all the detected particles:

Having described the principles on which the method of the invention is based, will now be described, by way of non-limiting example, the sequence of steps of the method implemented in the computer. Since, like any calculator, that of the invention does not deal with theoretical variables (such as the variable Φ (t)), but with estimates of these variables, we will designate them by the term "empirical".

For example, the empirical distribution function Φ (t) in the calculator will correspond to the theoretical distribution function Φ (t).

In said nonlimiting example, there is: N _EXC a total number of excitations,

N ₀ the number of excitations that did not cause any particle detection, N (t) the number of particles whose measured response time is less than t,

Whatever the distribution law chosen (Poisson, etc.), the method of the invention comprises the successive steps illustrated in FIG.

In a step 100, the parameter λ is determined, if it is not already known. For this purpose, we can use the equation:

In a step 200, the first empirical distribution function is determined as announced at the beginning of the description by:

M EXC ^1W 0 In a step 300, the second empirical distribution function, or corrected function, is determined using the equation:

(3) As a reminder, this equation has the general form described above:

F (ή = C (Φ (0)

In the case where we choose a Poissoniene distribution law, we have in particular:

In the case where we choose a Binomial distribution law, we have in particular:

Of course, the invention is not limited to the embodiment shown above and the drawings.

In particular, it is possible to use other probability laws of the number of photons or particles detected after an excitation (for example: a geometrical, uniform, or degenerate law).

In addition, the method and apparatus of the invention can be clearly adapted to the case where only the response time of the last of the detected particles is measured. In particular, instead of determining the probability of the minimum response time, the probability of the maximum among these times will be determined. This case corresponds to a use of a time-amplitude converter of the type of a TAC taking into account not first I, but the last photon arrived after excitation.

Furthermore, the method and the apparatus of the invention may relate to different types of particles.

We have seen in particular that they are well adapted to fluorometry based on the TCSPC technique.

In this particular case, the particles are therefore fluorescence photons.

And it will be advantageous to perform a correction of the device function in addition to the method.

This method and apparatus can also be used in techniques other than TCSPC.

In general, they can be used for X-rays, gamma rays, or other corpuscular rays even, especially since it includes a step in which only the first of the corpuscles detected is measured, ie that is to say from the moment that is used after the detector a component having an operation of the type of the converter TAC.

Moreover, the method according to the invention can advantageously be implemented in an optical reflectometer in the time domain.

This is precisely a device designated by the acronym OTDR for "Optical Time Domain Reflectometry".

In a manner known per se, this apparatus emits in particular photon pulses for measuring the distance between the injection point and a defect creating a reflection by calculating the time that the impulse to return to its starting point after reflection on this interface.

Moreover, as will be understood the invention is not limited to a method and an apparatus comprising all the steps presented above since the excitation of the system to the supply, in graphic form for example , of the composition of this system.

In particular, the principles of the invention can be applied directly to the data representing the detected elementary particles.

Bibliographical references

[1] PB Coates: "The correction of the Photon Pile-up in the Measurement of Radiative Lifetimes' / Sd Instrum, Phjx} s E _1, 1968, 1, Series 2, pp 878-879......

[2] Walker J. G.: 'Iteractive correction for' stack-up 'in single-photon lifetime measurement', 2002 Elsevier Science B.V./Optics Communications 201 271-277.

[3] Davis C. C. and King T. A., 'Correction Methods for Photon PiIe-Up in Lifetime Determination by Single-Photon Counting', /. Phys. A 3, 1970, pp 101-109.

Claims

1. A method of characterizing a system capable of emitting elementary particles, characterized in that it comprises the following steps: a, exciting (20) the system (60) at different predetermined times, b. for each of the excitations, b.l. detect (30) the elementary particles (70) emitted by the system

(60), b.2. measuring (40) the response time of the first only of said detected particles,

vs. determining (50) a first empirical distribution function (Φ (t)) or

a first empirical probability density (φ (t)) of the response times measured in steps (b.2), d. determining (50) a second empirical distribution function (F (t)) or

Λ a second empirical probability density (f (t)) of the response time of each of the particles detected in steps (b1) on the basis of an equation (EQ) which relates the second empirical distribution function (F (ή) at said first

empirical distribution function (Φ (t)).

2. Method according to claim 1, characterized in that the first empirical distribution function (Φ (0) is determined by an expression of the following type:

N _Erc -N, where, NEXC the total number of excitations, No. the number of excitations which did not result in any particle detection, N (t) the number of particles whose measured response time is less than t.

3. Method according to one of the preceding claims, characterized in that the equation (EQ) is a function of a parameter λ to estimate, this parameter translating a dependence between an intensity of the excitations and a law (L) of the probability of number of particles detected after each excitation.

4. Method according to claim 3, characterized in that the parameter λ is estimated from an expression of the following type:

where g is a predetermined function which depends on the probability law (L) of the number of particles detected after each excitation.

5. Method according to claim 4, characterized in that the function g is defined by an expression of the following type:

^{g (λ) =} Σ n = Û * "*"

with a coefficient, and n the number of particles detected.

6. Method according to one of the preceding claims, characterized in that the second empirical distribution function F (t) of the response time of each of the particles detected following the different excitations has an expression of the following type: / (0 = 1- h- {g (λ) - (g (λ) - _g (0)) b (ή} AT

7. Method according to one of the preceding claims, characterized in that the second empirical probability density (fit)) of the response time of each of the particles detected following the different excitations is deduced from the second empirical distribution function F (t). ) by time derivative in the equation (EQ).

8. Method according to one of claims 4 to 7, characterized in that the probability law (L) is a Poisson distribution of parameter λ, the function g is defined by:

S (Λ) = e- ^> = Σ ^

9. Method according to claim 8, characterized in that the estimate of the parameter λ and the second empirical distribution function (F (I)) of the response time of each of the particles detected following the different excitations have an expression of the type: s, IN _rγι

10. Method according to one of claims 4 to 7, characterized in that the law of

distribution (L) being a Binomial distribution of parameters (M, p = "^' ), the

function g is defined by:

11. Method according to claim 10, characterized in that the estimate of the parameter λ and the second empirical distribution function (F (t)) of the response time of each of the particles detected following the different excitations have an expression of the type:

12. A method of characterizing a system that can emit elementary particles, characterized in that it comprises the following steps: a. exciting (20) the system (60) at different predetermined times, b. for each of the excitations, b.l. detect (30) the elementary particles (70) emitted by the system

(60), b.2. measuring (40) the response time of the last of only said detected particles, vs. determine (DO) a first empirical distribution function (Φ (t)) or

a first empirical probability density (φ (t)) of the response times measured in steps (b.2), d. determining (50) a second empirical distribution function (F (O) ° ^u

empirical distribution function (Φ (t)).

13. Method according to one of the preceding claims, characterized in that the particles are photons.

14. Method according to one of the preceding claims, characterized in that the system comprises at least one fluorescent substance and in that the particles are photons emitted by the substance.

15. Method according to one of the preceding claims, characterized in that the system is an optical fiber and the particles are photons.

Apparatus for characterizing a system capable of emitting elementary particles, comprising:

Means for exciting (20) the system (60) at different predetermined times, Means for each of the excitations, detecting (30) the elementary particles (70) emitted by the system (60), and measuring (40) the response time of the first only one of said detected particles, characterized in that it further comprises - means for determining (50) a second empirical distribution function (F (I)) or a second empirical probability density

(f (t)) the response time of each of the detected particles on the basis of an equation (EQ) which relates the second empirical distribution function (F (ή) to said first empirical distribution function

(Φ (t)).

17. Apparatus according to claim 16, characterized in that it constitutes a fluorometry apparatus.

18. Apparatus according to claim 16, characterized in that it constitutes an optical reflectometer in the time domain.

19. Computer program for characterizing a system that can emit elementary particles when excited, the program being loadable on a computer and including a set of instruction codes adapted for

- determine a first empirical distribution function Φ (t) or a

first empirical probability density φ (t) of the response time of each of the first particles detected during, respectively, a series of excitations of the system; and determining a second empirical distribution function F {t) or a

empirical F (t) to said first empirical distribution function Φ (t).