Super resolution time domain spectroscopy method and device for sample characterization

1. A method for determining a set of physical parameters of a sample, comprising the steps of:

A Retrieving a measured sample temporal trace Es(t),

the measured sample temporal trace Es(t) having been obtained by Time Domain Spectroscopy, by illuminating a sample (S) by an excitation beam (EB) periodically emitting electromagnetic pulses with a period T and presenting a comb frequencies, and detecting an electromagnetic field coming from the sample as a function of time by a coherent detection, a time duration on which the sample temporal trace is measured being tmax, with tmax<T,

B retrieving a measured reference temporal trace Eref(t), the measured reference temporal trace Eref(t) having been obtained by illumination and detection in the same conditions than in step A but without the presence of the sample,

C determining an widened reference temporal trace, called Eref0(t), extending on the period T and obtained by affecting a zero value to instants for which no measurement have been performed, and determining a discrete Fourier transform {tilde over (E)} ref0 (ω) of the widened reference temporal trace calculated on a time window equal to T,

D determining a modeling of an impulse response of the sample in the frequency domain, depending on the set of physical parameters (pi), called sample frequency model {tilde over (E)} model {pi}(ω), from the Fourier Transform of the widened reference temporal trace {tilde over (E)} ref0 (ω) and a physical behavior model of the sample,

E applying an optimization algorithm on the set of physical parameters (pi) comprising the sub steps of:

E 1 initializing physical parameters (pi),

realizing iteratively the sub steps of:

E 2 calculating an inverse discrete Fourier transform of the sample frequency model {tilde over (E)} model {pi}(ω), called estimated sample temporal trace E est {pi}(t),

E 3 calculating an error function (ε er {pi}) from the difference between the measured sample temporal trace Es(t) and the estimated temporal trace Eest(t),

until obtaining a set of values (pi opt ) of physical parameters minimizing said error function.

2. The method as claimed in claim 1 , wherein the excitation beam (EB) is in the THz domain, having a frequency comprised between 100 GHz to 30 THz.

3. The method as claimed in claim 1 , wherein the maximum time delay tmax is chosen in order to include more than 95% of the energy of the measured reference temporal trace.

4. The method as claimed in claim 1 , wherein the sample frequency model {tilde over (E)} model {pi}(ω) consists in the multiplication of the Fourier Transform Eref0(ω) by a transfer function T(ω) characterizing the sample behavior.

5. The method as claimed in claim 4 , wherein the transfer function T(ω) depends on a complex refractive index n(ω).

6. The method as claimed in claim 5 , wherein the square of the complex refractive index called permittivity ε(ω) follows a Drude-Lorentz model for each spectral line, a spectral line being characterized by a set of three parameters, an amplitude (M), a width called damping rate (γ), and a central frequency (ω0).

7. The method as claimed in claim 1 , wherein the error function is defined as:

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8. A computer program adapted to implement the steps of claim 1 .

9. A non-transitory computer readable storage medium having stored thereon a computer program according to claim 8 .

10. A characterization device for characterizing a sample (S), said device comprising:

a memory (MEM) storing a measured sample temporal trace Es(t) and a measured reference temporal trace Eref(t),

the measured sample temporal trace Es(t) having been obtained by Time Domain Spectroscopy, by illuminating the sample (S) by an excitation beam (EB) periodically emitting electromagnetic pulses with a period T and presenting a comb frequencies, and detecting an electromagnetic field coming from the sample as a function of time by a coherent detection, a time duration on which the sample temporal trace is measured being tmax, with tmax<T,

the measured reference temporal trace Eref(t) having been obtained by illumination and detection in the same conditions than for measured sample temporal trace Es(t) but without the presence of the sample,

a processing unit (PU) configured to:

determine an widened reference temporal trace, called Eref0( t ), extending on the period T and obtained by affecting a zero value to instants for which no measurement have been performed, and determine a discrete Fourier transform {tilde over (E)} ref0 (ω) of the widened reference temporal trace calculated on a time window equal to T,

determine a modeling of an impulse response of the sample in the frequency domain, depending on a set of physical parameters (pi), called sample frequency model {tilde over (E)} model {pi}(ω), from the Fourier Transform Eref0(ω) of the widened reference temporal trace and a physical behavior model of the sample,

apply an optimization algorithm on the physical parameters (pi) comprising the steps of:

initializing physical parameters

realizing iteratively the sub steps of:

calculating an inverse discrete Fourier transform of the sample frequency model {tilde over (E)} model {pi}(ω), called estimated sample temporal trace E est {pi}(t),

calculating an error function (ε er {pi}) from the difference between the measured sample temporal trace Es(t) and the estimated temporal trace Eest(t), until obtaining a set of values (p opt i) of physical parameters minimizing said error function.

11. A spectrophotometer (Spectro) comprising:

a characterization device as claimed in claim 10 ;

a measuring device (MeD) comprising:

a source (LS) configured to illuminate a sample (S) by the excitation beam (EB),

a detector (D) configured to detect the measured sample temporal trace Es(t) and the measured reference temporal trace Eref(t).