Kai Li

Phay Marie Labeye

Mette J Ho

B Gaarde
email: mgaarde1@lsu.edu

Linda Young
email: young@anl.gov

Phay J Ho

Mette B Gaarde

Resonant propagation of x rays from the linear to the nonlinear regime

come

I. INTRODUCTION

Understanding the fundamental interaction between matter and high-intensity electromagnetic radiation has long been a vibrant scientific frontier. The nonlinear phenomena enabled by intense light sources in the optical, infrared, and microwave regions have been utilized to control electronic, nuclear, and spin transitions which has enabled breakthroughs across many fields of science, such as medical imaging, telecommunication, and the creation and manipulation of novel materials. So far it has been challenging to investigate coherent nonlinear effects at x-ray wavelengths due to their extremely small cross sections. However, modern x-ray free electron laser (XFEL) facilities delivering x-rays at unprecedented intensities up to 10 19 W/cm 2 now make it possible to also study nonlinear effects in x-ray region [START_REF] Emma | First lasing and operation of an ångstromwavelength free-electron laser[END_REF][START_REF] Ishikawa | A compact x-ray free-electron laser emitting in the sub-ångström region[END_REF]. Although the majority of x-ray nonlinear interaction studies have involved sequential multiphoton ionization processes in thin targets, leading to normally inaccessible charge states [START_REF] Young | Femtosecond electronic response of atoms to ultra-intense x-rays[END_REF][START_REF] Ho | Theoretical tracking of resonance-enhanced multiple ionization pathways in x-ray free-electron laser pulses[END_REF][START_REF] Rudek | Ultra-efficient ionization of heavy atoms by intense x-ray free-electron laser pulses[END_REF][START_REF] Rudenko | Femtosecond response of polyatomic molecules to ultra-intense hard x-rays[END_REF][START_REF] Fukuzawa | Deep inner-shell multiphoton ionization by intense x-ray free-electron laser pulses[END_REF], two-photon absorption [START_REF] Doumy | Nonlinear atomic response to intense ultrashort x rays[END_REF][START_REF] Tamasaku | X-ray two-photon absorption competing against single and sequential multiphoton processes[END_REF][START_REF] Ghimire | Nonsequential two-photon absorption from the k shell in solid zirconium[END_REF], second-harmonic generation [START_REF] Shwartz | X-ray second harmonic generation[END_REF] and optical-x-ray wavemixing [START_REF] Glover | X-ray and optical wave mixing[END_REF] have also been demonstrated in the x-ray regime. The ability of XFEL pulses to efficiently populate core-excited states, and thus induce a population inversion through a thick target, has enabled the observation of x-ray stimulated emission and x-ray stimulated Raman scattering (SRS) [START_REF] Rohringer | Atomic inner-shell x-ray laser at 1.46 nanometres pumped by an x-ray free-electron laser[END_REF][START_REF] Beye | Stimulated x-ray emission for materials science[END_REF][START_REF] Weninger | Stimulated electronic x-ray raman scattering[END_REF][START_REF] Yoneda | Atomic inner-shell laser at 1.5-ångström wavelength pumped by an x-ray free-electron laser[END_REF].

A different aspect of nonlinear laser-matter interaction is manifested in the propagation of an intense, short pulse through a dense medium with a resonance that is long-lived relative to the pulse duration. The longlived free-induction decay [START_REF] Bloch | Nuclear induction[END_REF][START_REF] Brewer | Optical free induction decay[END_REF] that persists after the pulse leads upon propagation to modifications in both the spectral, temporal, and spatial profile of the short pulse [START_REF] Mccall | Self-induced transparency[END_REF][START_REF] Mccall | Self-induced transparency by pulsed coherent light[END_REF][START_REF] Crisp | Propagation of small-area pulses of coherent light through a resonant medium[END_REF][START_REF] Chiao | Self-trapping of optical beams[END_REF][START_REF] Gibbs | Coherent on-resonance self-focusing of optical pulses in absorbers[END_REF][START_REF] Wright | Self-focusing of coherent pulses[END_REF] such as self-induced transparency (SIT) [START_REF] Mccall | Self-induced transparency[END_REF][START_REF] Mccall | Self-induced transparency by pulsed coherent light[END_REF], Burnham-Chiao oscillations in the temporal profile [START_REF] Burnham | Coherent resonance fluorescence excited by short light pulses[END_REF], and self focusing [START_REF] Chiao | Self-trapping of optical beams[END_REF][START_REF] Gibbs | Coherent on-resonance self-focusing of optical pulses in absorbers[END_REF][START_REF] Wright | Self-focusing of coherent pulses[END_REF], and these phenomena have been extensively investigated in the optical regime. More recently, reshaping of extreme ultraviolet pulses through collective effects has been demonstrated, using either high-harmonic generation [START_REF] Pfeiffer | Alternating absorption features during attosecond-pulse propagation in a laser-controlled gaseous medium[END_REF][START_REF] Liao | Beyond the single-atom response in absorption line shapes: probing a dense, laser-dressed helium gas with attosecond pulse trains[END_REF][START_REF] Bengtsson | Space-time control of free induction decay in the extreme ultraviolet[END_REF] or FEL sources [START_REF] Harries | Superfluorescence, free-induction decay, and four-wave mixing: Propagation of free-electron laser pulses through a dense sample of helium ions[END_REF] with synchronized infrared pulses as a means of imposing nonlinear processes. The x-ray regime presents a challenge given that inner-shell resonances typically have lifetimes of just a few femtoseconds, and that intense xray pulses based on self-amplified spontaneous emission (SASE) have durations of tens to hundreds of femtoseconds and exhibit poor temporal coherence. However, the recent demonstrations of sub-femtosecond, nearly transform-limited soft and hard x-ray pulses promise a new regime of coherent ultrafast x-ray matter interactions [START_REF] Marinelli | Experimental demonstration of a single-spike hard-x-ray free-electron laser starting from noise[END_REF][START_REF] Huang | Generating single-spike hard x-ray pulses with nonlinear bunch compression in free-electron lasers[END_REF][START_REF] Duris | Tunable isolated attosecond x-ray pulses with gigawatt peak power from a freeelectron laser[END_REF].

In this paper, we study the spectral, temporal and spatial reshaping of sub-and few-femtosecond, temporally coherent x-ray pulses as they propagate through an atmospheric density neon gas on resonance with the 1s → 3p transition. We introduce a method of simultaneously calculating the quantum dynamics of the neutral and singly ionized neon atoms, represented by few-level systems, and solving the three-dimensional Maxwell wave equation (MWE) to incorporate the collective response of the nonlinear medium. This approach allows us to treat all the different linear and nonlinear processes on an equal footing at both the microscopic and the macroscopic level. We find that stimulated Raman scattering (SRS) between core-excited states 1s -1 3p and 2p -1 3p occurs at high x-ray intensity [START_REF] Weninger | Stimulated electronic x-ray raman scattering[END_REF][START_REF] Sun | Propagation of a strong x-ray pulse: Pulse compression, stimulated raman scattering, amplified spontaneous emission, lasing without inversion, and four-wave mixing[END_REF], and that it is strongly enhanced when also including the 1s -1 -2p -1 (x-ray lasing) response of the ion. We theoretically observe and distinguish the growth of the SRS channel in the neutral relative to the stimulated emission in the ion [START_REF] Rohringer | Atomic inner-shell x-ray laser at 1.46 nanometres pumped by an x-ray free-electron laser[END_REF][START_REF] Weninger | Stimulated resonant xray raman scattering with incoherent radiation[END_REF], which due to their near degeneracy renders a clean decomposition of the two less experimentally relevant. We also find that at high, but reachable, intensities above 10 19 W/cm 2 a transmitted spectrum with unexpected spectral strength between the SRS and resonant peaks develops during propagation, and that this can be understood as a generalized Rabi-cycling in an intense multi-color field consisting of both initial and generated frequency components in the pulse. Finally, we investigate the spatial reshaping of the x-ray pulse and find that it is influenced both by SIT and by true self-focusing, and that this can be interpreted in terms of the intensity dependent, and therefore radially varying, modification of the amplitude and phase of the polarization field by the strong x-ray pulse. Finally, we show that all of the effects mentioned above still remain if we substitute the temporally coherent x-ray pulse with a SASE pulse.

This paper is organized as follows. Section II describes our theoretical approach to understanding linear and non-linear effects in x-ray propagation through a thick resonant medium. Section III discusses a number of results for propagation through a neon gas with radiation resonant with the strong core-level 1s → 3p transition: (A) effects of including various ionization continua on the transmitted spectral and temporal profiles, (B) spectral and temporal profiles as a function of intensity and pulse duration, and (C) self induced transparency and self-focussing as a function. In section III(D) we generalize the results in the previous sections obtained for coherent pulses to resonant propagation of SASE pulses. Finally in Section IV, we summarize our findings.

II. THEORETICAL APPROACH

In order to model the spatiotemporal reshaping of an x-ray pulse due to absorption and emission processes we have developed a versatile code based on coupling the 3D MWE to a TDSE solution for a few-level system that can describe both the neutral and ionized species of the system, extending previous x-ray propagation calculations in one spatial dimension [START_REF] Weninger | Stimulated electronic x-ray raman scattering[END_REF][START_REF] Harries | Superfluorescence, free-induction decay, and four-wave mixing: Propagation of free-electron laser pulses through a dense sample of helium ions[END_REF][START_REF] Sun | Propagation of a strong x-ray pulse: Pulse compression, stimulated raman scattering, amplified spontaneous emission, lasing without inversion, and four-wave mixing[END_REF][START_REF] Weninger | Stimulated resonant xray raman scattering with incoherent radiation[END_REF][START_REF] Lyu | Narrow-band hard-x-ray lasing with highly charged ions[END_REF]. For the TDSE solution, we use a hybrid approach in which the evolution of the states in the neutral is treated using a wavefunction-based formalism, whereas states in the ionized species are treated using a density matrix formalism. For the neutral, this includes coherent dipole couplings between relevant states, incoherent loss terms due to Auger decay of core-excited states, and incoherent loss terms due to direct ionization out of the relevant core and/or FIG. 1. Schematic of the energy levels, dipole couplings, and decay rates in the neon atom (left) and the neon ion (right). The dipole couplings d0,2 = 0.0077 a.u., d1,2 = 0.048 a.u.,d3,4 = 0.083 a.u. are taken from [START_REF] Weninger | Stimulated resonant xray raman scattering with incoherent radiation[END_REF] valence orbitals. For the ionized species, the population of each state is directly related to the ionization out of the equivalent orbital in the neutral, where the relationship is enforced via energy conservation in the field+gas system (see more details below), and the coherences in the ion are then allowed to develop in the field as it coherently couples different ionic states to each other. In the next couple of paragraphs, we outline this approach in more detail.

In this paper, we consider a 5-level neon system consisting of three states in the neutral and two in the ion, as illustrated in Fig. 1. These are the ground state, the 1s -1 3p core excited and the 2p -1 3p valence excited states in the neutral, at energies of 867.5 eV and 18.2 eV above the ground state, respectively, and the 2p -1 and 1s -1 ground and core excited state of the ion, at energies of 19.3 eV and 869.1 eV above the neutral ground state. The coherent and incoherent couplings between the states are illustrated in Fig. 1, including the magnitudes of the dipole couplings, in a.u. The Auger decay rates of the two core excited states have values Γ 2 = Γ 4 = 0.0099 a.u., and the ionization cross sections are σ 2p = 0.0084 Mbarn and σ 1s = 0.30 Mbarn. The transition dipoles d 0,2 , d 1,2 , d 3,4 from Ref. [START_REF] Weninger | Stimulated resonant xray raman scattering with incoherent radiation[END_REF] were originally calculated with [START_REF] Cowan | The Theory of Atomic Structure and Spectra[END_REF] and correspond to a peak absorption cross section for of 1.98 Mb for the 1s → 1s -1 3p neutral Ne transition. We do not include spontaneous emission as a means of initiating the SRS transitions; the spectral background provided by the combination of the initial pulse and the 2.4 fs exponential decay of the resonant polarization is sufficient to provide a seed, see also discussions in [START_REF] Weninger | Stimulated electronic x-ray raman scattering[END_REF][START_REF] Sun | Propagation of a strong x-ray pulse: Pulse compression, stimulated raman scattering, amplified spontaneous emission, lasing without inversion, and four-wave mixing[END_REF].

Our hybrid approach means we are solving a set of six coupled differential equations for the three wave function coefficients Ψ 0 , Ψ 1 , Ψ 2 , the two populations ρ 3,3 and ρ 4,4 , and the coherence between the ionic states ρ 3,4 , in atomic units:

i Ψ0 (t) =-i Γ 1s (t) 2 + Γ 2p (t) 2 Ψ 0 (t) + d 0,2 E(t)Ψ 2 (t) (1a) i Ψ1 (t) = 1 Ψ 1 (t) + d 1,2 E(t)Ψ 2 (t) (1b) i Ψ2 (t) = 2 -i Γ 2 2 Ψ 2 (t) + d 0,2 E(t)Ψ 0 (t) + d 1,2 E(t)Ψ 1 (t) (1c) ρ3,3 (t) =Γ 2p (t)|Ψ 0 (t)| 2 -2id 3,4 E(t) Im(ρ 3,4 (t)) (1d) ρ4,4 (t) =Γ 1s (t)|Ψ 0 (t)| 2 -Γ 4 ρ 4,4 (t) + 2id 3,4 E(t) Im(ρ 3,4 (t)) (1e) ρ3,4 (t) = i ( 4 -3 ) - Γ 4 2 ρ 3,4 (t) -id 3,4 E(t)(ρ 4,4 (t) -ρ 3,3 (t)). (1f)
In these equations, the red terms correspond to the phase and amplitude evolution driven by the energies and lifetimes of the different states; the blue terms are the terms driven by field-induced dipole couplings; and the green terms are responsible for incoherently populating the states of the ion via ionization of the neutral ground state.

First, note that we do not keep track of any coherence between the neutral and the ion, since we do not keep track of the photoelectrons. This is what allows us to treat them separately, each of them in their own formalism, for numerical efficiency. Second, since the only sources of decoherence in the neutral are from photoionization and Auger decays, this can be simply implemented, within the wave function formalism, as loss terms in the TDSE. On the contrary, there are several incoherent processes affecting the ion. The Auger decay from the core hole can also be implemented as a loss term, but the photoionization processes are source terms that increase incoherently the population in the ionic states. This cannot be implemented within the wave function formalism, which is why the ion has to be treated in the density matrix formalism. Indeed, the photoionization terms (green terms) will only increase the populations ρ 3,3 and ρ 4,4 in the ionic states, without affecting the coherence ρ 3,4 . Note that, since the ion dipole is proportional to the coherence, as long as the coherence is zero the ion neither absorbs nor emits light. The coherence in the ion can only build over time through its interaction with the field (blue term). This is the term that will be responsible for absorption or emission of light in the ion. The incoherent population transfer (green term) will not lead to any coherent emission of light. Finally, it is worth noting that the wave function formalism is less computationally demanding, especially for systems with large numbers of states.

The 3D MWE is solved in the frequency domain, by space marching through the gas in a frame that moves at the speed of light. The incident field is linearly polarized, and thus the only driven (and phase matched) response from the atoms is linearly polarized as well. In SI units (used in the remainder of this article), and with all frequency-dependent quantities also functions of the cylindrical coordinates r and z, the MWE along the field polarization takes the form:

∇ 2 ⊥ Ẽ(ω) + 2iω c ∂ Ẽ(ω) ∂z = - ω 2 0 c 2 P (ω) -i ω c ρ at σ(ω) Ẽ(ω). (2)
Here Ẽ(ω) is the electric field which contains all the frequencies of the incoming and generated field, and the polarization-field source term P (ω) = ρ at d(ω) is calculated from the single atom dipole moment d(ω), including both neutral and ion contribution, via the few-level TDSE solutions described above. The second term on the right-hand side represents the loss of energy from the field due to ionization, where σ(ω) is the effective cross section due to all ionization processes at frequency ω, and ρ at is the density of neutral atoms.

To ensure energy conservation in the field+gas system this energy loss from the field, which is implemented in a macroscopic sense in the MWE, has to match the energy absorbed at the microscopic level by the atoms in the TDSE through the time dependent ionization rates. The energy absorbed from the field is easily deduced from the cross section with Beer's law. However, we can't know how much energy is absorbed by the atoms before we solve the TDSE. We thus implemented a two step procedure to enforce energy conservation. First we evaluate how many photons each atom is expected to absorb, using the cross section, to compute a time dependent rate that is used to solve the TDSE. Then we extract from the TDSE the exact number of photons absorbed by each atom. Finally we adjust the cross section that we use in the MWE. In the following, we detail how this is implemented.

For each ionization channel i, the frequency dependent ionization cross section σ i (ω) is taken as an arctangent function centered around the ionization threshold energy ω th :

σ i (ω) = σ i 1 2 + 1 π arctan 2 ω -ω th Γ ion , (3)
where σ i is the ionization cross section, and Γ ion is the inverse lifetime of the ionic state populated by the ionization process, here it is equal to the Auger decay rate. The arctangent function is the integral of the lorentzian cross sections for each continuum states that can be populated by an ω photon, under the assumption that the ionized states all have the same lifetimes, and that the cross section is flat [START_REF] Breinig | Atomic inner-shell level energies determined by absorption spectrometry with synchrotron radiation[END_REF].

The corresponding time-dependent ionization rate Γ i (t) used in the TDSE is computed from this cross section σ i and from the electric field. In usual first order time dependent perturbation theory approaches, the ionization rate does not depend on time, and in the case of a monochromatic incident field of amplitude F the rate is proportional to F 2 . Here we suppose that we can incoherently sum the contributions to ionization from all incident frequencies, and we make the approximation that the rate adiabatically follows the slowly varying field envelope F (t). For a more rigorous derivation of the ionization rate, we refer the interested reader to the appendix.

Γ i (t) = α i F 2 (t) (4)
F (t) = 1 π ∞ 0 Ẽ(ω) e -iωt dω . (5)
To set the proportionality constant α i , we compute the expected number of absorbed photons per atom N ph by integrating Γ i (t) over the full pulse. Note that this supposes that the population in the ground state is almost constant. Then, we express N ph as the energy U i lost by the field at a given z position, divided by the number of atoms at z and by the energy of one absorbed photon:

N ph = ∞ 0 Γ i (t)dt = U i ρ at ω i (6)
U i = ε 0 c 2π∆z ∞ -∞ | Ẽ(ω)| 2 1 -e -ρatσi∆z dω . (7)
Using the fact that

∞ -∞ F 2 (t)dt = 1 π ∞ -∞ | Ẽ(ω)| 2 dω ( 8
)
we finally get

Γ i (t) = ε 0 c 2 ω i ρ at ∆z ∞ -∞ | Ẽ(ω)| 2 1 -e -ρatσi∆z dω ∞ -∞ | Ẽ(ω)| 2 dω F 2 (t) (9)
However, because of all the coherent and incoherent processes taking place simultaneously, the atoms might actually absorb less photons that the expected N ph , especially if the interaction with the field is nonlinear. To account for this, we keep track of the number of photons Ñph that are really absorbed within each ionization channel i described by the TDSE. This is computed by summing all the population that is transferred from the neutral to the ion during the TDSE propagation. The field+gas energy conservation is enforced by rescaling the ionization cross section σi that is used in the MWE:

σi (ω) = σ i (ω) Ñph N ph = σ i (ω) Ñph ρ at ω i U i , (10)
so that the number of photons macroscopically removed from the field by the MWE exactly corresponds to the number Ñph of photons that the atoms actually absorbed. Finally, the effective cross section σ(ω) in the MWE (Eq. ( 2)) is the sum of all the rescaled σi (ω) of all the different ionization channels. This procedure reproduces the experimental photoabsorption cross section with the 1s-continuum deduced by computing the quantity σ abs = ln[I incident (ω)/I trans (ω)]/nL, where n and L are the number density and pathlength of the target, respectively. We note that in the treatment of ionization described above, we are making the approximation that all frequencies in the pulse share a common envelope F (t). If considering a scenario in which different frequency components are substantially separated in time (for example an ionizing pump pulse followed by a non-ionizing probe pulse), one could instead use different envelopes for different frequency components in Eq. ( 4) and ( 5).

The initial electric field is a focused Gaussian in space, and in the frequency domain is defined as the Fourier transform of the initial pulse. At each plane in the propagation direction, the time-dependent dipole moment is calculated from both the neutral and the ionic systems, d(t) = d n (t) + d i (t), and the macroscopic polarization field is proportional to the dipole moment via the density (note that this is the initial density of the gas; the time-dependence of the neutral and ion density is already incorporated into the time dependent wave functions and coherences in (1) above). Finally, we use the calculated source terms on the right-hand side of Eq. 2 to propagate the electric field to the next plane in the propagation direction. This self-consistent approach ensures that both the linear and non-linear response of the atoms to the field is incorporated back into the propagating XUV (and IR) field, and we can therefore treat both linear and non-linear absorption and emission on an equal footing. Note that in contrast to this polarization-field-driven coherent exchange of energy between the field and the gas, the ionization loss term is incoherent.

The TDSE (1) is solved with a split-operator algorithm, where the propagation of the neutral wave function on the one hand and of the density matrix of the ion on the other hand are done in parallel. The MWE (2) is solved with a Crank-Nicolson algorithm. All Fourier transforms are computed with the FFTW package [START_REF] Frigo | The Design and Implementation of FFTW3[END_REF], and diagonalizations were performed using the LAPACK library [START_REF]Linear algebra package[END_REF].

III. DISCUSSION OF RESULTS

A. Role of Ionization Continua

We now use this theoretical machinery to explore resonant propagation in neon gas as a function of pulse intensity and duration. First, to give insight into the physics at play we examine the role of ionization in the combined atom-ion-EM-field system. The effects of non-resonant ionization can be negligible in some cases [START_REF] Sun | Propagation of a strong x-ray pulse: Pulse compression, stimulated raman scattering, amplified spontaneous emission, lasing without inversion, and four-wave mixing[END_REF], but here, where we access the deep 1s inner shell with ultrashort, broad bandwidth pulses, this is not the case. We specifically explore the effects of ionization by computing three scenarios: no ionization, ionization of the 1s shell and ionization of both 1s and 2p shells which directly provide population in the ion.

We consider a 0.25-fs (FWHM) Gaussian pulse centered on the 1s → 3p resonance at 867.5 eV with a peak intensity of 10 18 W/cm 2 (pulse energy ∼ 100 µJ) propagating through 3-cm of gaseous neon at atmospheric pressure. The beam is focused to the center of the cell with a confocal parameter of b = 12.35 cm corresponding to a focus of 6.12 µm (FWHM) at the cell center. For context, the resonance has a peak absorption cross section of σ = 1.98 Mb, such that Beer's law I(L) = I 0 e -σnL with σnL ≈ 150 is well in the optically-thick, self-inducedtransparency regime [START_REF] Mccall | Self-induced transparency by pulsed coherent light[END_REF][START_REF] Slusher | Self-induced transparency in atomic rubidium[END_REF]. (A more recent high accuracy determination of the resonance cross section is 30% higher [START_REF] Müller | Photoionization of ne atoms and ne + ions near the k edge: PrecisionSpectroscopy and absolute cross-sections[END_REF], but our observed trends and results are expected to be unchanged.) Here the pulse duration, τ p , is less than any relaxation or decoherence timescale (set by the Auger decay of 2.4 fs) such that the pulse has a spectral content that is much broader than the absorption line of the medium.

The simulation uses 30 points per period, T p = 4.67 attoseconds, and covers a range of 100 fs. For propagation from z = 0 -3 cm, the electric field and state populations in the time domain, and, the spectra in the frequency domain are written as output files for sequential propagation planes spaced by ∆z = 0.05 cm. The propagation calculations were performed at a stepsize of ∆z/30. The 100-fs time window covers the oscillating pulse tails and corresponds to a 0.04-eV grid in the frequency domain. This fine grid is sufficient to resolve the Ne absorption spectra which have a natural width of 0.27 eV.

In Fig. 2 and 3 we show time-domain and frequency domain results, respectively, for selected propagation distances of between 0 and 3 cm, for the three different ionization scenarios described above. The left column in Fig. 2 shows the results when no ionization continua are included. The amplitude of the electric field is shown in gray and the populations of two excited states in neutral neon, 1s -1 3p and 2p -1 3p, in red and blue, respectively. We observe a dramatic temporal reshaping of the pulse as it propagates through the medium. The response at the entrance slab of the medium (z=0) shows a population of the resonantly excited 1s -1 3p state of 8% at the end of the pulse, that subsequently decays with the 1s hole lifetime of 2.4 fs. The small blip associated with the 2p -1 3p state is related to the dressing of the states by the electric field, but does not correspond to real population transfer to the 2p -1 3p state.

As we continue to propagate into the medium, a ringing is evident as energy is exchanged between the two resonant levels and the propagating field. The third level, 2p -1 3p, plays little role. This situation is analogous to that shown in Sun et al. [START_REF] Sun | Propagation of a strong x-ray pulse: Pulse compression, stimulated raman scattering, amplified spontaneous emission, lasing without inversion, and four-wave mixing[END_REF] where both 2-and 3-level systems without ionization channels show similar ringing behavior for large-area pulses, i.e input π pulse. The results here show that the use of large area pulses (> π) is not required to achieve the oscillation in the x-ray regime -a feature long appreciated in the optical community [START_REF] Crisp | Propagation of small-area pulses of coherent light through a resonant medium[END_REF][START_REF] Burnham | Coherent resonance fluorescence excited by short light pulses[END_REF]. We also note that the population of the excited 1s -1 3p state is out of phase with the electric field demonstrating the energy exchange between the medium and field in this simple configuration. The time-domain ringing, and its evolution with propagation distance, can be understood by considering the evolution of the spectrum (see Fig 3(a,b)): As the pulse propagates a hole is eaten from its spectrum, which creates a beat (ringing) in the time domain between the spectral content above and below the resonance frequency. As the hole widens during propagation, the time scale of the ringing shortens with propagation distance, as observed going from z = 1.5-3.0 cm in the bottom two panels. (We note that the inclusion of higher Rydberg levels (3p, 4p, 5p, 6p) suppresses the aforementioned Burnham-Chiao time-domain ringing but otherwise merely leads to the expected multiple absorption dips in the transmitted spectrum.)

In the center column of Fig. 2 we show results when the 1s-ionization continuum and associated ion states are included, There is a dramatic difference compared to the left column. A 4% population of the 1s -1 ion state is observed in the first slab. The presence of the 1s -1 hole produces dipole coupling at the 1s -1 → 2p -1 transition frequency that is nearly resonant with corresponding transition in the neutral, 1s -1 3p → 2p -1 3p. This can also be observed in the spectrum in Fig. 3(a) as a two-order-ofmagnitude enhancement of the 850 eV signal after propagating only one step (0.05 cm). After propagation to 1.5 cm a fine temporal beating of the electric field appears that corresponds to ∆t = 2π/(ω 02 -ω 12 ) = 0.22 fs. The beating persists throughout propagation and is also manifested in the transmitted spectrum, where the stimulated Raman scattering (SRS) peak is prominent (see Fig. 3b). For the no-ionization case in the left column, the transmitted SRS peak is much lower, and the fine temporal beating is nearly absent in the temporal profile.

The right column of Fig. 2, shows propagation when both the 1s and 2p ionization continua are included. The addition of the 2p continua affects the temporal profile and populations to a lesser degree than the first inclusion of the 1s continuum. The magnitude of the temporal oscillations is slightly suppressed, but the main features shown in the center column are still apparent. The major difference is the more prominent appearance of the 2p -1 ion state, which enables absorption at this photon energy, thus suppressing the 850-eV peak and the associated temporal beating.

In the following we use as a default the five-level Ne system shown in Fig. 1 and the 1s-ionization continuum, i.e. the conditions of the central column in Fig. 2. This configuration illustrates the main physics involved in the coupled atom-ion-EM field system.

B. Spectral and temporal reshaping as a function of intensity

Now we consider the temporal and spectral reshaping of x-ray pulses upon propagation through the resonant neon medium as a function of intensity (10 16 -10 19 W/cm 2 ) for two different pulse durations (0.25 fs and 2.5 fs (FWHM). The 10 16 W/cm 2 represents the linear response of the system and matches simulations at lower intensities (See Fig. 9c andd for linearity of transmitted pulse energies versus intensity). Fig. 4 displays incident and transmitted spectral and temporal profiles for 0.25-fs Gaussian pulses incident at three peak intensities: 10 16 , 10 18 and 10 19 W/cm 2 . At all intensities, there is a sharp dip in the transmitted spectrum at 867.5 eV due to resonant absorption. Photon energies above 870 eV are absorbed via 1s-photoionization. The ultra-short 0.25 fs pulse corresponds to a transformlimited 7.3 eV (FWHM) bandwidth such that much of the radiation is off resonance and transmitted. A very small fraction of the incident pulse (∼10 -7 ) provides photons at ∼ 850 eV that seed a stimulated Raman transition (SRS) between the core-excited state (1s -1 3p) and valence-excited state (2p -1 3p), and on the corresponding x-ray lasing (XRL) transition in the ion (1s -1 ) and (2p -1 ), which is populated via the part of the initial spectrum above 870 eV. Fig. 5(a) and(c) show that at the intermediate intensities 10 17 W/cm 2 and 10 18 W/cm 2 , the stimulated Raman scattering (SRS) signal is exponen-tially amplified during propagation. At 10 18 W/cm 2 , the spectral strength of the SRS peak is comparable to the peak of the spectrum near 865 eV. Higher-order scattering processes such as four-wave mixing contribute to a signal at 833 eV, as was observed also in [START_REF] Sun | Propagation of a strong x-ray pulse: Pulse compression, stimulated raman scattering, amplified spontaneous emission, lasing without inversion, and four-wave mixing[END_REF].

At even higher intensity, seen in Fig. 4(c), the SRS peak becomes dominant in the transmitted spectrum and spectral content appears throughout the gap between the 867.5 eV resonance and the 849.3 eV SRS peak, Fig. 4c. We are not aware of this "intermediate" spectrum, between the SRS and resonance peaks, having been observed before, and we have further investigated its origin. Fig. 5(b) shows that the intermediate spectrum appears and then grows exponentially at longer propagation distances (z > 0.3 cm) as the SRS approaches saturation. While the SRS signal saturates early during propagation also at lower intensities (see Fig. 5(c)), it is only the combination of SRS intensity, spectral intensity near the resonance and high fields that gives rise to the intermediate spectrum. At this intensity, the Rabi frequencies associated with both the initial pulse and the generated SRS pulse, and the two resonant transitions, are on the order of 5-10 eV, and the intermediate spectrum can be though of as generalized Rabi sidebands produced by the strong multi-color field. This interpretation was verified with a separate calculation using two-color, high-intensity (10 19 W/cm 2 ) incident pulses, which was found to give rise to sidebands in the dipole spectrum spanning most of the frequency range between 850 and

eV (not shown).

The temporal pulse profile dramatically differs for the three intensities. At the lowest intensity, the ringing associated with on-resonant pulse propagation for low pulse areas appears. At the intermediate intensity, finer oscillations associated with the beating between the SRS and main peak are observed. At the highest intensity, 10 20 W/cm 2 , the transmitted pulse is relatively compressed, consistent with the broad spectral profile discussed above, and only a few oscillations of the the lowpulse-area ringing are visible.

Fig. 6 displays incident and transmitted spectral and temporal profiles for a 2.5-fs Gaussian pulse for the same three intensities. With the longer pulse, the spectral width (0.73 eV) is comparable to that of the natural width of the absorption resonance (0.24 eV) such that much more of the pulse is absorbed at the low intensity limit -providing a more efficient energy exchange between the field and medium. Hence, nonlinearities occur at lower intensity, giving rise to complex spectra. Already by 10 18 W/cm 2 the SRS resonance peak becomes the largest spectral feature. At the highest intensity 10 19 W/cm 2 , the rising edge of the pulse is strongly steepened, leading to a much higher peak intensity (∼10-fold), as previously observed computationally for the Ar system [START_REF] Sun | Propagation of a strong x-ray pulse: Pulse compression, stimulated raman scattering, amplified spontaneous emission, lasing without inversion, and four-wave mixing[END_REF][START_REF] Sun | Slowdown and compression of a strong x-ray free-electron pulse propagating through the mg vapors[END_REF].

C. Self-induced transparency and self-focusing

When a weak pulse enters a resonant medium, a fraction of the pulse energy is absorbed by creating excitations. After a few absorption lengths the pulse energy decays to zero according to Beer's law. Self-induced transparency (SIT) refers to the situation when electromagnetic fields pass through a medium with energy attenuation smaller and transit time longer than expected. This happens for a weak pulse with duration shorter than the decoherence timescale [START_REF] Mccall | Self-induced transparency by pulsed coherent light[END_REF][START_REF] Crisp | Propagation of small-area pulses of coherent light through a resonant medium[END_REF]. The excited dipoles remain in phase collectively after the pulse passes, and thus can radiate power back to the field coherently. As detailed in [START_REF] Crisp | Propagation of small-area pulses of coherent light through a resonant medium[END_REF] the propagation of a small-area pulse in an attenuating medium satisfies an area theorem. That is, the pulse area drops to zero exponentially. However, this does not mean the pulse energy decreases exponentially. The pulse reshapes itself to produce oscillating tails that have alternating phases -as seen, e.g., in Figs. 2 and4. The pulse area decreases due to the cancellation between tails, but the pulse energy remains constant.

As a preface to our more complex situation we first consider propagation through a simple two-level neon system (1s and 3p) with photoionization ignored as shown in Figs. 7 and8. We first consider the transmitted pulse energy. Fig. 7(a) shows the evolution of the pulse energy for different incident intensities and two pulse durations (2.5 fs and 0.25 fs). The black dashed line indicates energy absorption according to Beer's law for monochromatic light on resonance. The 2.5 fs pulse at 10 18 W/cm 2 is close to Beer's law, but deviates due to the off-resonant radiation. As is easily understood, the 0.25-fs pulses are attenuated much less than the Beer's law prediction due to substantially more off-resonant components -short pulses correspond to larger bandwidth and only photons within the absorption bandwidth are absorbed. When the intensity is increased to 10 20 W/cm 2 , population (Rabi) oscillations occur to create the self-induced transparency as shown in Figs. 4a,c.

Next, with our three-dimensional model we study resonant self-focusing of strong x-ray pulses as shown in Figs. 7(b),(c). Self-focusing occurs when the electromagnetic field induces a refractive index change in the medium which then reacts back on the field and affects its propagation. Fig. 7(b) shows the evolution of beam spot size (HWHM) for different pulse intensities. At 10 18 W/cm 2 , the beam size evolves according to the propagation of a Gaussian beam with 3.2 µm HWHM at the center of the gas cell. Self-focusing becomes apparent with increasing pulse intensity. The beam radius at 10 19 W/cm 2 deviates slightly from that at 10 18 W/cm 2 ; but at 10 20 W/cm 2 there is a dramatic decrease of the beam radius from 3.2 µm to 1.4 µm by the end of the gas cell. The corresponding on-axis electric field attenuation is shown in Fig. 7(c). At low pulse intensity, the on-axis electric field drops, but it grows by a factor of 2.1 at high peak intensity, 10 20 W/cm 2 .

To illustrate the mechanism behind the resonant x-ray self-focusing we show the evolution of the beam's radial and phase profile in Fig. 8 for 10 20 W/cm 2 where a large self-focusing effect is observed. The energy-integrated transverse beam profile is shown in Fig. 8a for a family of propagation distances, z=0-3 cm. The transverse beam profile is reshaped to a sharp peak in the center with the maximum field strength increased by a factor of 2× by the end of the gas cell.

The radial amplitude and phase profiles at the resonance energy, 867.5 eV, are shown in Figs. 8(b) and (c), respectively. The Gaussian beam profile develops higherorder transverse modes during propagation and the evolution is not as smooth as that of the energy-integrated profile. Rather than monotonically decreasing, the beam first flattens and grows, and then shrinks to a very sharp peak with a 4-fold increased maximum field by the end of the cell.

The evolution of the phase profile of the beam is shown in Fig. 8(c). For reference, the phase profile of a Gaussian beam, focused at z=1.5 cm, after propagation in vacuum for z=3 cm is shown as a black dashed line. This beam is defocusing at z=3, with an increasing phase as a function of radial position. For the beam propagating through Ne gas, initially the beam is focusing toward the center of the cell and shows a slightly decreasing phase for larger radial distances. On-axis, the relatively highintensity creates stronger SIT, while the off-axis fields are attenuated. There is a reshaping of the wavefront to be strongly focusing which induces energy flow inward and growth of the on-axis power density. Modification of absorption and phase due to changes of the imaginary and real parts of the nonlinear complex refractive index contribute to this pulse front reshaping. In summary, resonant x-ray self-focusing results from the lower intensity off-axis fields interacting differently than the high intensity on-axis fields as observed in the optical regime [START_REF] Gibbs | Coherent on-resonance self-focusing of optical pulses in absorbers[END_REF][START_REF] Wright | Self-focusing of coherent pulses[END_REF][START_REF] Marburger | Self-focusing: theory[END_REF]. It is worth noting that, the resonant selffocusing effect here has a different origin compared to the normal off-resonant optical Kerr effect, which results from higher-order susceptibility induced by strong fields. In fact, due to the rather small x-ray refractive index, the off-resonant Kerr effect would not be expected to occur until reaching much higher x-ray intensities.

For the five-level Ne, Ne + and 1s-continuum system the SIT and self-focusing display a much more complicated dependence on intensity and pulse duration. The attenuation of an ultra-short 0.25 fs pulse for several intensities are shown in Fig. 9(a). For peak intensities lower than 10 18 W/cm 2 stimulated Raman scattering is weak; the five-level system behaves similar to a two-level system, where non-Beer's law behavior is due to the ultra-short pulse duration. One difference is that 1s-photoionization causes the transmission to drop from 0.6 to 0.2. For the 2.5-fs pulse propagating through the five-level system, (Fig. 9(b), deviations occur at relatively low intensities. Comparison of the transmitted pulse energies as a function of incident intensity for 0.25-and 2.5-fs pulses are shown in Figs. 9(c) and (d). A dashed line is shown for the intensity associated with a π-pulse for the 1s → 3p transition. As can be seen from the Fig. 9, deviations from linear transmission occur at intensities greater than that required for a π-pulse -at the intensities associated with SIT and self-focusing.

We now turn our attention to self-focusing in the fivelevel system. At low intensities, the beam spot size for a five-level system, Fig. 10(a), behaves the same as in a two-level system, Fig. 8(b). However at 10 19 W/cm 2 beam size oscillations appear and a stronger self-focusing effect occurs. Surprisingly, the beam radius due to self-focusing does not necessarily decrease more rapidly with higher peak intensity -compare 10 19 W/cm 2 and 10 20 W/cm 2 . The x-ray radial evolution is clearly coupled to changes in the on-axis power density shown in Fig. 10(b). This non-monotonic behavior is the sign of highly nonlinear radiation-matter interactions. The strong interaction between the field and atom ensemble gives rise to more complex electronic transitions within the dressed atomic states which further induces polarization fluctuations and modifies the refractive index. Higher field strengths in the center of the beam lead to stronger SRS and less absorption. This mechanism exacerbates the self-focusing effect at 10 20 W/cm 2 leading to beam spot shrinkage to one third of the original size and an on-axis field strength 2.8× that of the incident pulse.

D. SASE pulse propagation

It is natural to compare the propagation of a Gaussian with a SASE pulse, since those are most readily available at XFELs. We simulate SASE pulses with typical parameters of τ = 10 fs pulse duration, bandwidth = ∆E/E = 0.005, and peak intensity of 10 19 W/cm 2 . These parameters correspond to a beam spot size of 2 µm, and pulse energy of 5 mJ. The pulse is produced by a computer simulation technique developed for generating superposed coherent and chaotic radiation [START_REF] Vannucci | Computer simulation of superposed coherent and chaotic radiation[END_REF]. The radiation field amplitudes at different frequency can be modeled as chaotic variables with Gaussian statistics. The ensemble averaged SASE pulse is assumed to have a Gaussian power spectrum with bandwidth ∆E E of the central photon energy. The field amplitude at different frequency are independent zero-mean Gaussian random variables whose variance is proportional to the corresponding power spectrum. The SASE pulse is first generated in the energy domain with 4096 points spaced by 0.04 eV. After the inverse Fourier transform to the time domain and normalization, a Gaussian envelope is used to create the pulse. This field amplitude and phase are then linearly interpolated to 524288 grid points to get the input field for the TDSE-MWE propagation code. As shown in the Fig. 11, the generated SASE pulse is comprised of coherent spikes with τ coh = h ∆E ∼ 1 fs duration, which is related to the total spectral bandwidth. The averaged 0.4 eV width spikes in the SASE energy spectrum correspond to h τ . The simulation results for propagation of the SASE pulse shown in Fig. 11 have features that closely resemble the Gaussian pulse. The input SASE x-ray pulse is absorbed and the SRS signal is generated at 850 eV. The SRS saturates upon propagation to 0.5 cm, and the SRS spectrum contains spikes associated with the incident SASE spectrum. Four-wave mixing signals with ∼ 10× smaller intensity are observed at 830 eV. The SASE pulse is delayed and reshaped (compare the red and blue pulse envelopes in Fig. 11b). Some very short ∼ 20 as (FWHM) bursts are generated on top of the SASE pulse.

The atomic inner-shell x-ray laser [START_REF] Rohringer | Atomic inner-shell x-ray laser at 1.46 nanometres pumped by an x-ray free-electron laser[END_REF] was also simulated using our code by tuning the incident energy above the ionization threshold. The simulation was carried out for a SASE pulse with 40-fs duration and 960-eV central photon energy propagating through a 500 torr Neon gas. The 0.3 mJ x-ray pulse was focused to 2 µm spot size to produce peak intensity around 10 17 W/cm 2 . These parameters correspond to the experiment performed at LCLS [START_REF] Rohringer | Atomic inner-shell x-ray laser at 1.46 nanometres pumped by an x-ray free-electron laser[END_REF][START_REF] Weninger | Transient-gain photoionization x-ray laser[END_REF]. We calculate a saturation length of 6 mm and 0.4% fractional XRL energy in agreement with the experiment.

IV. SUMMARY

In this paper, we investigated resonant propagation of ultra-short, high-intensity XFEL pulses in a gaseous medium from the linear to the nonlinear regimes. We solved the three-dimensional timedependent Schrödinger equation (TDSE) for the single atom response and the Maxwell wave equation (MWE) . Specifically, the propagation of XFEL pulses with photon energy resonant with the Ne 1s → 3p transition through an optically thick target was investigated. The stimulated Raman scattering (SRS) signal, i.e. the transition between core-excited (1s -1 3p) and valence-excited state (2p -1 3p), grows exponentially during propagation for intensities up to 10 18 W/cm 2 for 0.25-fs pulses. At higher intensities, spectral intensity not associated with atomic transitions appear during propagation due to the existence of two ingredients: significant SRS intensity and strong fields to induce Rabi-flopping sidebands. Stronger interactions occur with the longer 2.5-fs pulse, leading to strong spectral and temporal reshaping of a XFEL pulse. X-ray self-induced transparency and self-focusing are observed when the intensity is sufficient to induce a π-pulse on the resonant transition. The newly developed TDSE-MWE methodology is very general: readily applicable to SASE pulses and demonstrated to reproduce the off-resonant propagation that leads to the atomic x-ray laser [START_REF] Rohringer | Atomic inner-shell x-ray laser at 1.46 nanometres pumped by an x-ray free-electron laser[END_REF][START_REF] Weninger | Stimulated resonant xray raman scattering with incoherent radiation[END_REF]. Generalizing our formalism to an arbitrarily polarized field would be simple at the level of the MWE, where each polarization component would propagate independently. Generalizing the single-atom response would involve including more states, with different m-quantum numbers, and incorporating selection rules. Such extensions would be of interest for ultrafast circular dichroism studies. In summary, the understanding of resonant propagation at high x-ray intensities has potential applications for XFEL pulse shaping and is relevant for x-ray optics and transient absorption spectroscopy. An experiment devoted to quantifying these effects is feasible at various XFEL facilities by measuring the energy spectrum before and after propagation through the target gas. We anticipate investigating nonlinear resonant propagation effects experimentally in the near future.

FIG. 2 .

 2 FIG.2. The evolution of the electric field and state populations during resonant propagation in neon gas of a 0.25-fs (FWHM) x-ray pulse centered on the 1s → 3p Ne transition at 867.5 eV with peak intensity 10 18 W/cm 2 . Time dependent populations of the core-excited state 1s -1 3p, valence-excited state 2p -1 3p for neutral Ne and 1s and 2p hole states for Ne + are shown for propagation distances z=0, 1.5 and 3.0 cm. The profile of the electric field at each propagation distance is indicated in gray. The three columns represent the evolution without ionization channels (left), including 1s ionization (center) and including 1s and 2p ionization (right),

FIG. 4 .

 4 FIG. 4. Spectral and temporal profiles for the incident and transmitted pulse at various intensities. Incident pulse is 0.25 fs (FWHM) Gaussian centered at 867.5 eV (blue -dashed line). Transmission through 3 cm Ne gas at 1 bar pressure calculated using the five-level Ne, Ne + and 1s-continuum system (red -solid line). Upper row display spectra for peak intensity (a) 10 16 W/cm 2 , (b) 10 18 W/cm 2 and (c) 10 19 W/cm 2 . Lower row display temporal profiles:(d) 10 16 W/cm 2 , (e) 10 18 W/cm 2 and (f) 10 19 W/cm 2

FIG. 5 .

 5 FIG. 5. Spectral evolution during propagation through the five-level Ne, Ne + and 1s-continuum system at 1 bar.(a) Spectral evolution of a 0.25 fs, 10 18 W/cm 2 pulse. (b) Spectral evolution of a 0.25 fs, 10 19 W/cm 2 pulse. (c) Growth of the fraction of transmitted energy contained in the SRS+XRL peak (integrated between 846 -853 eV) for 10 16 ,10 17 , 10 18 W/cm 2 peak intensities.

FIG. 8 .

 8 FIG. 8. Pulse propagation through the Ne two-level system for 0.25-fs, 10 20 W/cm 2 pulses: radial intensity and phase profiles for a family of propagation distances from 0 to 3 cm. (a) Radial dependence of the beam intensity integrated over photon energy. (b) Radial dependence of the beam intensity at the resonance energy (867.5 eV) (c) Radial dependence of the phase profile. The black dashed line indicates the output phase profile for a Gaussian beam propagating in vacuum.

FIG. 9 .

 9 FIG. 9. Transmitted pulse energies for 0.25 and 2.5-fs pulses five-level Ne system with 1s photoionization. The solid black lines in (c) and (d) represent a linear relationship between the incident and transmitted pulse energies, i.e slope = 1.0. (a) Propagation of 0.25-fs pulse with peak intensities from 10 16 W/cm 2 to 10 20 W/cm 2 (b) Propagation of 2.5-fs pulse with peak intensities 10 16 W/cm 2 to 10 20 W/cm 2 . (c) Transmitted pulse energy after passage through 3 cm gas at various intensities for 0.25 fs pulses. (d) Same as (c) for 2.5 fs pulses.

FIG. 10 .

 10 FIG. 10. (a) Beam spot size of x-ray pulses with 0.25 fs pulse duration and different peak intensities ranging from 10 16 W/cm 2 to 10 20 W/cm 2 propagating through the five-level Ne, Ne + and 1s-continuum system. (b) The evolution of the corresponding on-axis power densities is shown in (b).

FIG. 11 .

 11 FIG. 11. (a) Input (blue) and output (red) spectrum of SASE pulses with 10 fs (FWHM) pulse duration and peak intensity 10 19 W/cm 2 propagating through the five-level system with photoionization. The beam spot size is 2 µm (FWHM) and pulse energy is 5 mJ. (b) The corresponding input and output x-ray temporal profile.

ACKNOWLEDGMENTS

L.Y. and K.L. are grateful for stimulating initial discussions with N. Rohringer and V. Majety. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Science, Chemical Sciences, Geosciences and Biosciences Division that supported the Argonne group under contract number DE-AC02-06CH11357, and the LSU group under contract number DE-SC0010431.

Appendix: Wigner-Weisskopf approach to ionization

Minimal coupling Hamiltonian

We consider a system with bound and continuum states, under the influence of a classical,linearly polarized, electric field E(t) that only couples the ground state |0 to all continuum states | k , with dipole matrix elements d k that we chose real.

Initially the system is in the ground state, and at time t the wave function writes

with the ground state energy 0 = ω 0 , and continuum state energy k = ω k -i Γ ion /2. Where Γ ion is the decay rate of the ion related to process such as Auger decay, etc. Inserting (A.1) in the Schrödinger equation gives

3) can be formally integrated to give

that we can then insert in (A.2) to get:

with

where we integrated over the angular degrees of freedom of k, and changed variable to energy.

Wigner-Weisskopf approximation

The Wigner-Weisskopf approximation relies on the fast decay of K(τ ) when τ > 0. This term can be seen as a "sum" (integral) of oscillating exponential of all frequencies that quickly decays to zero as soon as τ 1/ω 0 . This is sometimes interpreted as the "loss" of memory of the environment, i.e. that the process is Markovian. We can thus (i) consider that a 0 (t ) a 0 (t) in the integral since a 0 varies slowly at this time scale and (ii) extend the integral for τ → ∞, i.e. t → -∞. Note that in our case, the field is zero for t < 0 so that (ii) is not really an approximation here.

Applying this approximation to (A.6), we get

This is the famous Wigner-Weisskpof exponential decay.

Demonstration of the formula used in our program

To prove that Γ(t) is proportional to the envelope of the field, we first suppose that we can rewrite the time dependent electric field as:

where the envelope F (t) and phase φ(t) vary slowly at the 1/ω 0 timescale, so that when we insert this expression in (A.9) and (A.10), we can use the Wigner-Weisskopf approximation again and set F (t ) F (t) and φ(t ) φ(t) in the integral:

× e -i(ω -ω0+ω)τ 1 + e 2i(ωt+φ(t))

+ e -i(ω -ω0-ω)τ 1 + e -2i(ωt+φ(t))

The first term would be resonant in the emission case, we can safely neglect it. In the second term, we can neglect the term oscillating at 2ω, indeed, it will integrate to zero. This is similar to the rotating wave approximation. Finally, we are only interested in the real part of Γ(t). Indeed, the imaginary part would give the AC Stark shift.

We thus finally get

If we now have more than one frequency in the pulse, i.e. if we write the field E(t) = 1 2π ∞ -∞ dω Ẽ(ω). Then we can incoherently sum the contributions of all frequency, where each contribute proportionally to | Ẽ(ω)| 2 :

If we approximate the angular frequency ionization energy, we almost recover the formula given in (9) of the manuscript. We just need to suppose σρ at ∆z 1 in (9) and develop the exponential.


Mathieu Weil
email: mathieu.weil@cirad.fr

Keywords: Turmeric, Blanching, Curcuminoids, Essential oil, Microscopy

Effects of cooking and drying on colour, curcuminoids, essential oil and aroma compounds of Curcuma longa L. were assessed. Sliced fresh turmeric rhizomes were air-dried at 60 °C directly or after cooking at 95 °C for 3 or 60 min. Microscopic observations showed that curcuminoids and essential oil are located in different dedicated cells. Curcuminoids and essential oil of dried turmeric were both around 10 % db. After processing, curcuminoids were dispersed throughout the matrix. Drastic cooking and drying operations decreased chromatic values more than smooth cooking. Cooking had no impact on curcuminoid and essential oil contents and slightly modified aromatic profile of essential oils. Drying decreased the curcuminoid (< 38 %) and essential oil (< 13 %) contents. Turmeric starchy matrix preserves the curcuminoids and essential oil during the process. We recommend a preliminary smooth cooking step to reduce drying time, save energy consumption and preserve turmeric quality.

Impact of cooking and drying operations on colour, curcuminoids and aroma of Curcuma longa L.

Molika Yin a, c , Mathieu Weil a, b* , Sylvie Avallone a , Marc Lebrun a, b , Geneviève Conejero d , Sokneang In c , Philippe Bohuon a 1. Introduction Turmeric (Curcuma longa L.) belongs to the Zingiberaceae family and is widely distributed throughout tropical and subtropical regions of the world [START_REF] Kutti Gounder | Comparison of chemical composition and antioxidant potential of volatile oil from fresh, dried and cured turmeric (Curcuma longa) rhizomes[END_REF]. Curing turmeric is a significant postharvest processing operation that involves cooking fresh turmeric in boiling water. The goals of curing are to reduce microbial load, inactivate enzymes, avoid unpleasant odours, gelatinize the starch, and change the cell walls of the turmeric facilitating its permeability and reducing resistance to mass transfer which leads to an increase in the drying rate [START_REF] Jayashree | Comparison of quality of dry turmeric (Curcuma longa) produced by slicing and other curing methods[END_REF]. The main aim of drying is to reduce the moisture present in turmeric, from 2.3 -4.0 (kg kg -1 db) at the time of harvest to a safe value of 0.11 (kg kg -1 db) (International Organization for Standardization, 2015). Air drying is an alternative to sun drying, not only to increase drying rate and reduce drying time but also to better preserve the quality of the product [START_REF] Prathapan | Effect of heat treatment on curcuminoid, colour value and total polyphenols of fresh turmeric rhizome[END_REF]. The optimum drying conditions for best product quality were found to be air temperature of 55 -60 °C and air velocity of 2 m s -1 [START_REF] Singh | Effect of mechanical drying air conditions on quality of turmeric powder[END_REF]. High air temperature (> 60 °C) caused degradation of volatile compounds and curcuminoids. Traditionally, the rhizomes are dried directly or cooked and then dried. We question the value of using these operations in a combined way, as cooking transfers water inside the product and drying will have to remove the water from the cooked rhizome. In the traditional process, whole rhizomes are cooked for a long time at high temperatures. Moreover, numerous studies have demonstrated the impact of processes on the quality of turmeric powder. However, they did not focus on the impact of each unit operation (cooking and drying) on turmeric's quality.

In our study, we applied, as an alternative, short cooking time to sliced rhizomes and microscopic observation and biochemical contents evolution during the process were studied. Our objective was to assess the impact of cooking and drying in controlled conditions on the quality of Curcuma longa L. The quality characteristics considered in this study were colour, curcuminoids and aromatic profiles. The impact of cooking on drying kinetics was studied. Mass balances of dry matter, water and curcuminoids were assessed. Microscopic analysis was realized to localize essential oil, starch and curcuminoids and to assess process effects.

Materials and methods

2.1.

Materials Solvents (ethanol and dichloromethane), sodium sulfate, homologous series of C8 -C20 nalkanes standards, and HPLC standards (curcumin, demethoxycurcumin and bisdemethoxycurcumin) were obtained from Sigma-Aldrich (Saint Quentin Fallavier, France). The polytetrafluoroethylene (PTFE) membranes were obtained from Sartorius (Palaiseau, France). The water used in all the experiments was purified water (milli-Q reference water purification). Turmeric, Curcuma longa L., used in this study was a product of Thailand which was purchased (4 kg) from an Asian grocery store (Wei Sin) in Montpellier, France, on the 20 th of February 2020. Fresh turmeric rhizomes were stored at 4 °C until being processed. Before processing, the fresh turmeric rhizomes were cleaned and washed thoroughly under running water to remove adhering soil and mud, spray residues, roots and other foreign materials. Then, the excess water was drained and the cleaned turmeric rhizomes were sliced manually to a thickness of 5 mm. The sliced turmeric is called "fresh" turmeric.

2.2.

Processing experiments 2.2.1. Cooking Two-unit operations were applied to obtain different samples (Figure 1). The sliced fresh turmeric (F) was divided into two lots of 420 g for cooking experiments and one lot of 360 g for direct drying assessment. Cooking treatment consisted of soaking sliced turmeric in a nylon net in hot water at a ratio of 1:10 w/w material to water at two different conditions: FC1: 95 °C/3 min and FC2: 95 °C/60 min. The starch could be completely gelatinized in these cooking conditions. After cooking, the cooked turmeric was immediately soaked in an ice water bath for 1 min to stop the cooking process, before being drained. About 60 g of each sample was collected for further analysis.

Drying and drying kinetics

A hot air dryer, developed in our laboratory, was used for drying the sliced turmerics. In the vertical drying chamber, 360 g of sliced turmerics were spread on-grid rack (0.25 m long × 0.25 m wide × 0.06 m high). Hot air (60 ± 1 °C, RH 40 ± 2 %) was circulated downwards through the layer of sliced turmerics by a high capacity fan. The drying times are different for each treatment i.e. 7 h 14 min, 4 h 29 min and 3 h 38 min for fresh turmeric (F), FC1 (cooking at 95 °C/3 min) and FC2 (cooking at 95 °C/60 min), respectively. Airspeed was measured thanks to an anemometer (ALMEMO ® 2690-8A, Ahlborn Mess, Germany). The air velocity was just high enough (2.1 ± 0.1 m s -1 ) to have no significant effect on temperature when passing through the layer of sliced turmerics. Monitoring by weighing was carried out continuously during the drying process, every 10 min for the first hour and then every 15 min. The water content, which was measured on a dry basis (noted X) as a function of time, was estimated in line, using the mass reading of the sieve. Water content kinetics X (t) were fitted with a cubic smoothing spline (Matlab ® Version 5.2, The Mathworks Inc., USA). The drying rate (dX/dt) was calculated as the direct analytical derivative of the cubic smoothing spline function on X (t) .

Microscopic analysis

Thick cross-sections (100 -135 µm) were obtained from fresh (F), cooked (FC), cooked and dried (FCD) and dried (FD) turmeric rhizomes using a microtome with a vibrating blade (Thermo Scientific, Microm HM650V, Walldorf, Germany) and then dipped in 10 mM phosphate buffer saline (7 mM Na2HPO4, 3 mM NaH2PO4, 120 mM NaCl, 2.7 mM KCl). Starch and terpenes were respectively stained dipping cross-sections in Lugol solution (2 g KI and 1 g I2 dissolved in 100 ml distilled water) for 5 min or NADI solution (0.001% 1-naphthol, 0.001 % N,N-dimethyl-p-phenylenediamine dihydrochloride and 0.4 % ethanol in 100 mM sodium cacodylate-HCl buffer (pH 7.2) for 45 min [START_REF] David | Coloration différentielle dês inclusions lipidique et terpeniques dês pseudophylles du pin maritime au moyen du reactif Nadi[END_REF]. Rhizomes tissues were observed with a wide-field microscope Eclipse Ni-E (Nikon Instruments Inc., NY, USA). The pictures were obtained with 4x 0.2 NA, 10x 0.45 NA or 20x 0.75 NA Plan-APO objectives under transmitted light. The autofluorescence of curcuminoids was visualized with a multiphoton microscope (Zeiss 880, Jena, Germany, infra-red laser Chameleon Ultra II, Coherent, CA, USA) and an objective 20X Plan APO 1.0 NA. The spectral detector of this microscope was used at 720 nm (UV-like excitation) to obtain emission spectra of pure curcuminoids (curcumin, demethoxycurcumin and bisdemethoxycurcumin) between 400 and 690 nm, and to visualize the fluorescence of these molecules in the rhizome tissues [START_REF] Talamond | Secondary Metabolite Localization by Autofluorescence in Living Plant Cells[END_REF].

2.4.

Sample preparation for colour, curcuminoids and essential oil measurements The samples resulting from the different processing operations (approximately 50 g of each treatment) were frozen by using liquid nitrogen and then ground for 10 s at 10 000 rpm in a mill (Retsch Grindomix GM200, Retsch Gmbh, Germany) for immediate analyses of dry matter and colour. The samples for further essential oil and curcuminoids analyses were put in glass bottles and frozen at -80 °C. The samples dedicated to curcuminoids analysis were made on the sliced rhizome.

Analytical methods 2.5.1. Dry matter and water contents

The dry matter content (means ''dry matter free of essential oil") was obtained by drying 1 g of ground turmeric in an aluminium cup in the oven (Gefran 800, Italy) at 105 °C for 30 h (i.e. until constant weight). The mean relative deviation of repeatability was ± 4.4 % (n = 4). Water content expressed on a dry basis was deduced from essential oil and dry matter contents.

Colour measurements

Colour values i.e. lightness (L*), redness (a*) and yellowness (b*) of ground turmeric samples were determined using a chromameter (Minolta CR-400, Minolta, Osaka, Japan). The illuminant was D65, and an incidence angle of 0 ° was used. Each data point was the mean of three replications measured on the surface of the samples at randomly selected positions. Chroma value (C*) was calculated following this equation: C* = √𝑎𝑎 * 2 + 𝑏𝑏 * 2 . The mean relative deviation of repeatability was 3.5 %, 4.2 %, 5.8 % and 5.4 %, respectively for L*, a*, b* and C* (n = 6).

Curcuminoid contents

Approximately 0.3 g of sliced turmeric was mixed with 30 mL of 60 °C ethanol (99.8 %) and homogenized for 2 min at 30 000 rpm (IKA T10 basic Ultra-Turrax, Prolabo, France). The samples were heated for 30 min at 60 °C [START_REF] Sogi | Effect of extraction parameters on curcumin yield from turmeric[END_REF]. After cooling, the extracts were diluted 1/10 with ethanol and filtered on 0.45 µm PTFE Minisart SRP4 membrane (Sartorius, Palaiseau, France). Curcuminoids were analysed by high-performance liquid chromatography (Agilent System 1200 series, Massy, France). The column was a polymeric ACES C18 (250 × 4.6 mm, 5 µm particle size, Inc Wilmington NC) and the injection volume was 5 µL. The quantification of curcuminoids was carried out according to the method of [START_REF] Sepahpour | Comparative analysis of chemical composition, antioxidant activity and quantitative characterization of some phenolic compounds in selected herbs and spices in different solvent extraction systems[END_REF] with small modifications. The elution was done isocratically with a mixture of acetonitrile and 0.1 % acetic acid (40:60) at a flow rate of 1.0 mL/min and the temperature of the column was set at 25 °C. Chromatograms were recorded over 30 min period with a UV-visible photodiode array detector (Agilent Technologies 1200 series) at the wavelength of maximum absorption of the curcuminoids in the mobile phase (i.e. 425 nm). The curcuminoids were identified by their retention time and spectrum. External calibration was realized weekly with standard solutions of the pure chemicals in ethanol in the range of 1 to 50 mg/L. The curcuminoid contents were expressed in g/100g of initial dry weight basis (g/100g db). The mean relative deviation of repeatability was 12.5 %, 11.2 %, 16.8 % and 13.5 %, respectively for curcumin, demethoxycurcumin, bisdemethoxycurcumin and total curcuminoids (n = 4).

Essential oil content

The essential oil content was determined using a method adapted from the international official standard method ISO 6571:2008 (International Organization for Standardization, 2008). The modification in the applied method was the elimination of xylene. Approximately 20 g of ground turmeric samples were weighed and transferred to 1 L of a round bottom flask, then 250 mL of distilled water was added and about 10 pieces of pumice stones were added to homogenize boiling. It was heated at medium heat for 4 h and the condensed vapour was separated. The essential oil present at the uppermost layers was collected and put in a vial containing sodium sulfate and then stored at -20 °C for later essential oil compounds analysis by gas chromatography-mass spectrometry (GC-MS). The essential oil content was expressed in mL/100g of initial dry weight basis (mL/100g db). The mean relative deviation of repeatability was ± 4.7 % (n = 4).

Identification of essential oil compounds Separation on a polar column

An Agilent 6890 series GC (Agilent Technologies, Palo Alto, CA, USA) equipped with a DB-WAX UI column (60 m × 250 μm, 0.25 µm phase film thickness, Agilent J&W GC column) coupled to an Agilent 5973 mass spectrometer detector (Agilent Technologies) was used. Hydrogen was used as carrier gas at 1.5 mL/min at a constant flow. Column temperature program was 100 °C to 200 °C at the rate of 2 °C/min, then to 250 °C at the rate of 10 °C/min. The sample injection volume was 1.0 µL with a split ratio of 100:1. The samples (essential oils) were diluted 1/10 with dichloromethane before injecting. The retention indices were calculated using a homologous series of n-alkanes C8 -C20.

Separation on a non-polar column

On the same GC-MS with the same column program temperature, a DB-5MS column (60 m × 250 μm, 0.25 μm phase film thickness, Agilent J&W GC column) was used. The retention indices were calculated using a homologous series of n-alkanes C8 -C20.

Identification

The aroma compounds separated on both polar and apolar columns were identified by comparing their mass spectrum to those available in commercial libraries (NIST 14 and PubChem). The mean relative deviation of repeatability was ±13.2 % (n = 8).

2.5.6. Statistical analysis Differences in the mean values of colour (L*, a*, b* and C* values), curcuminoid contents, essential oil content and its composition were tested by analysis of variance (ANOVA); the significance of differences between samples was determined using Duncan's test. The level of significance was P ≤ 0.05.

Results and discussion

Characterization of fresh turmeric 3.1.1. Localization of essential oil, starch and curcuminoids

The distribution of essential oil, starch and curcuminoids was represented in Figure 2 and The curcuminoids were distributed throughout the whole rhizome; two cells containing curcuminoids were never adjacent (Figure 3.I). Curcuminoids occupied the entire cell volume in fresh turmeric (Figure 3.I). They were recognizable by their orange colour, but as they had the property of fluorescing in UV, they could be identified more specifically. Curcuminoids had the property of being fluorescent under UV light; they emitted in yellow (Figure 3). The observed fluorescence was obtained between 500 and 650 nm. The maximum fluorescence emission of the three curcuminoids (curcumin, demethoxycurcumin and bisdemethoxycurcumin) was observed between 500 and 600 nm (Figure 3.III).

Curcuminoid and essential oil contents

The major curcuminoid compound found in our fresh turmeric was curcumin (6.24 g/100g db) followed by bisdemethoxycurcumin (5.70 g/100g db) and demethoxycurcumin (2.92 g/100g db). Moreover, the total curcuminoid contents (the sum of the three curcuminoids) in the turmeric was 14.86 g/100g db (equal to 1.95 g/100g wb). This value was higher than the values found by [START_REF] Hirun | Turmeric (Curcuma longa L.) drying: An optimization approach using microwave-vacuum drying[END_REF] (9.39 g/100g db), [START_REF] Govindarajan | Turmeric-Chemistry, technology, and quality[END_REF] (2 -9 g/100g db) and [START_REF] Garg | Variation in the rhizome essential oil and curcumin contents and oil quality in the land races of turmeric Curcuma longa of North Indian plains[END_REF] (0.61 -1.45 g/100g wb). The essential oil content of our fresh turmeric was 10.72 mL/100g db (equal to 1.32 mL/100g wb). This value was higher than that reported by [START_REF] Govindarajan | Turmeric-Chemistry, technology, and quality[END_REF] who found the content of essential oil up to 6.3 % db. However, the value was in agreement with [START_REF] Garg | Variation in the rhizome essential oil and curcumin contents and oil quality in the land races of turmeric Curcuma longa of North Indian plains[END_REF] who stated that the essential oil contents of Curcuma longa rhizomes collected from the sub-Himalayan Tarai region of India were in the range of 0.16 -1.94 mL/100g wb. The disparity of the content of active phytochemicals could be due to environmental factors of each growing location, plant development stage, planting period, harvesting season and turmeric varieties [START_REF] Hirun | Turmeric (Curcuma longa L.) drying: An optimization approach using microwave-vacuum drying[END_REF][START_REF] Monton | Acceleration of turmeric drying using convection and microwave-assisted drying technique: An optimization approach[END_REF].

3.2.

Impacts of the unit processing operations 3.2.1. Impact of cooking on colour values, curcuminoid content, essential oil content and drying curve By mass balances (data not shown), we measured low water gain (whether for short or long cooking time); the water gain was lower than 3.0 % (kg water per 100 kg initial product). Moreover, very little dry matter and curcuminoids were transferred from the turmeric to the cooking water (not significantly different from zero). Thus, the cooking process only slightly increased the water amount, which must be removed during drying. Cooking had a slight impact on colour values (Figure 4). Our results showed that both cooking conditions had no impact on L*; however, smooth cooking significantly decreased a* value (8.3 %), while drastic cooking significantly decreased a* (6.1 %), b* (9.3 %) and C* (8.6 %) values. Heat treatment of the rhizome before dehydration can inactivate oxidative enzymes and limit the browning of the products. The browning of the turmeric during drying could be due to the Maillard reaction. However, rhizomes subjected to immersion cooking should have limited Maillard reactions because some of the reducing sugars could have diffused into the cooking water. Overall, drastic cooking impacted b* and C* colour values more than smooth cooking. Cooking had no impact on curcuminoid content (Figure 4). This may be because curcuminoid is an oil-soluble pigment, practically insoluble in water at acidic and neutral pH, soluble in alkali. Moreover, it is stable at high temperatures and in acids, but unstable in alkaline conditions and the presence of light [START_REF] Lee | Curcumin and its Derivatives: Their Application in Neuropharmacology and Neuroscience in the 21st Century[END_REF]. Curcumin, demethoxycurcumin, bisdemethoxycurcumin and total curcuminoid contents of fresh and cooked turmerics (F, FC1 and FC2) ranged from 6.00 -6.24, 2. 92 -3.31, 5.35 -5.71 and 14.71 -15.02 g/100g db, respectively. Cooking had no impact on essential oil content (Figure 5) as evidenced by the absence of a significant difference (P ≤ 0.05) in the results obtained for essential oil in samples F, FC1 and FC2. The essential oil contents of fresh and cooked turmerics (F, FC1 and FC2) ranged from 10.70 -11.06 mL/100g db). The results of microscopic analysis (Figure 2.II and III) also confirmed that the essential oils were not affected by cooking. The impact of cooking on the drying curves was shown in Figure 6. Cooking saved 38.1 % and 49.8 % of drying time for FC1 (cooking at 95 °C/3 min) and FC2 (cooking at 95 °C/60 min), respectively. About 59 min, 48 min and 45 min were required to obtain a 50 % reduction in the initial water content of fresh turmeric (F) and cooked turmerics i.e. FC1 and FC2, respectively. Drying kinetics presented a classical behaviour: an intense water loss during the initial stage and slowly at a later stage. This can be attributed to the fact that the moisture from the surface is easily evaporated while it takes time for the moisture to be removed from the interior. The comparison of the drying curves showed a much higher initial drying rate (5.05 ± 0.25 kg.kg -1 .h -1 for FC2 and 5.01 ± 0.23 kg.kg -1 .h -1 for FC1) in cooked turmerics than in fresh ones (F: 4.83 ± 0.33 kg.kg -1 .h -1 ). ). The different dash types on curves correspond to different trials. [START_REF] Govindarajan | Turmeric-Chemistry, technology, and quality[END_REF] observed that when turmeric rhizomes are cooked, the starch granules gelatinize as a result of the heat treatment, which facilitates drying and increases the rate of dehydration; as a result, the total drying time is reduced. The results of the microscopic analysis confirmed that cooking for 3 min at 95 °C was enough to completely gelatinize the starch (Figure 2.VI). The amyloplasts were clearly visible in fresh (F) turmeric (Figure 2.V) while they were no longer visible in cooked (FC1) turmeric (Figure 2.VI). Moreover, cooking combined with drying caused the partial exit (dispersion) of the curcuminoids from their dedicated cells; curcuminoids were then found throughout the whole rhizome and seem to be partly adsorbed on the starch cells (Figure 3.II). The degradation of the cell walls allowing curcuminoid exited could also explain the easier water removal when turmeric was cooked prior to drying. Cooking at 95 °C/3 min is the optimum cooking condition for turmeric sliced

Impact of drying on colour values, curcuminoid content and essential oil content

Drying did have a marked impact on colour and curcuminoid contents (Figure 4). There were significant differences between samples F/FD, FC1/FC1D and FC2/FC2D considered in pairs for all values (a*, b* and C*) except for L*. The greatest differences were observed between sample F and FD with reductions of 17.3 %, 14.8 % and 15.3 % for a*, b* and C* values, respectively. Similar results were described by [START_REF] Bambirra | Influence of post harvest processing conditions on yield and quality of ground turmeric (Curcuma longa L.)[END_REF]. In the absence of heat treatment, a product with lower intensity of a* and b* was obtained. These results indicated that cooking the turmeric prior to dehydration is essential to obtain a product with higher intensity of a* and b*. There were significant differences (P > 0.05) between the F/FD, FC1/FC1D and FC2/FC2D samples considered in pairs for all curcuminoids and total curcuminoids with the exception of FC1/FC1D and FC2/FC2D for bisdemethoxycurcumin content. Direct drying significantly decreased curcumin, demethoxycurcumin and bisdemethoxycurcumin contents 29.5 %, 32.3 % and 27.4 %, respectively while drying with previous cooking significantly decreased curcumin and demethoxycurcumin contents 24.5 % and 34.6 -38.8 %, respectively. These results were in agreement with [START_REF] Madhusankha | Analysis of curcumin content in Sri Lankan and Indian turmeric rhizomes and investigating its impact on the colour[END_REF] who indicated that there was an evident and clear relationship between the curcuminoid content and the colour values. When the curcuminoid content decreased, a*, b* and C* values also decreased. Curcumin, demethoxycurcumin, bisdemethoxycurcumin and total curcuminoid contents of our dried turmerics (FD, FC1D and FC2D) ranged from 4. 40 -4.68, 1.98 -2.07, 4.14 -4.54 and 10.51 -11.29 g/100g db, respectively. These values were comparable to those found by [START_REF] Monton | Quantitation of curcuminoid contents, dissolution profile, and volatile oil content of turmeric capsules produced at some secondary government hospitals[END_REF]. These authors found that: the contents of curcumin, demethoxycurcumin, bisdemethoxycurcumin and total curcuminoids were 6.61 -7.67, 2.76 -3.33, 2.64 -3.47 and 12.02 -14.36 % w/w, respectively. However, our turmerics contained a higher amount of curcuminoids than those found by Monton et al. (2019a) who reported that their turmeric powders contained 2.3 -3.6 % curcumin, 1.0 -1.6 % demethoxycurcumin, 1.5 -2.3 % bisdemethoxycurcumin and 4.8 -7.3 % (w/w) total curcuminoids. Drying did have an impact on essential oil content (Figure 5) as significant visible differences (P > 0.05) were observed between samples F/FD, FC1/FC1D and FC2/FC2D considered in pairs (relative loss of 8.8 -13.2 %). The method of drying usually has a significant effect on the contents of the essential oil [START_REF] Kutti Gounder | Comparison of chemical composition and antioxidant potential of volatile oil from fresh, dried and cured turmeric (Curcuma longa) rhizomes[END_REF]. The essential oil is present in the specific cells and ducts present in the meristematic region of the rhizome. These oleaginous cells are damaged during the cooking of the rhizome and the exposure of the essential oil to the atmosphere induces a loss by volatilization during drying [START_REF] Díaz-Maroto | Effect of Drying Method on the Volatiles in Bay Leaf (Laurus nobilis L.)[END_REF]. The essential oil content of our dried turmerics (FD, FC1D and FC2D) ranged from 9.60 -9.76 mL/100g db. These values were higher than that found by [START_REF] Monton | Quantitation of curcuminoid contents, dissolution profile, and volatile oil content of turmeric capsules produced at some secondary government hospitals[END_REF] (7.00 -8.00 % v/w) and [START_REF] Monton | Acceleration of turmeric drying using convection and microwave-assisted drying technique: An optimization approach[END_REF] (5.20 -8.50 % v/w). The factors that might affect the content of curcuminoids and essential oil were seasonal variation, environmental condition, post-harvest handling, storage, manufacturing and microbial contamination [START_REF] Monton | Quantitation of curcuminoid contents, dissolution profile, and volatile oil content of turmeric capsules produced at some secondary government hospitals[END_REF].

3.3.

Impacts of ''full" processes In this study, a "full" process referred either to a single drying or to a process including cooking before drying (Figure 1). The impacts of the "full" processes on colour and curcuminoid content (Figure 4), essential oil content (Figure 5) and aroma profile (Table 1 andFigure 7) were described.

Impact of the ''full" processes on colour and curcuminoids content

The impact of the "full" processes on colour and curcuminoid content were illustrated in Figure 4. There were significant differences (P > 0.05) between samples F/FD, F/FC1D and F/FC2D considered in pairs for all colour values (L*, a*, b* and C*). When turmeric was cooked for short time (95 °C/3 min; FC1D) prior to the drying, the colour values decreased less than during drying alone. However, when drying was preceded by drastic cooking (95 °C/60 min; FC2D), the colour values decreased more than during drying alone (FC1D>FD>FC2D). By comparing the three finished products (FD, FC1D and FC2D), the results showed that there were no significant differences (P ≤ 0.05) between FD and FC1D for all colour values, while FC2D had more impact on b* and C* values as compared to FD and FC1D. The "full" processes did have a marked impact on curcuminoid content (Figure 4). There were significant differences (P > 0.05) between samples F/FD, F/FC1D and F/FC2D considered in pairs for all curcuminoid contents and total curcuminoids except that there were no significant differences (P ≤ 0.05) between samples F/FC1D and F/FC2D considered in pairs for bisdemethoxycurcumin content. The greatest differences were observed between sample F and FD with reductions of 29.5 %, 32.3 %, 27.4 % and 29.3 % for curcumin, demethoxycurcumin, bisdemethoxycurcumin and total curcuminoids, respectively. These results were in agreement with [START_REF] Suresh | Effect of heat processing of spices on the concentrations of their bioactive principles: Turmeric (Curcuma longa), red pepper (Capsicum annuum) and black pepper (Piper nigrum)[END_REF] who stated that curcumin loss from heat processing of turmeric was in the range of 27 -53 %, with maximum loss in pressure cooking (at high temperature) for 10 min. However, [START_REF] Bambirra | Influence of post harvest processing conditions on yield and quality of ground turmeric (Curcuma longa L.)[END_REF] indicated that heat treatment (up to 100 °C) had no effect on curcuminoids in turmeric. Therefore, an appropriate treatment (not too high nor not too low) to dry the turmeric may play a vital role in preserving curcuminoids in the turmeric [START_REF] Hirun | Turmeric (Curcuma longa L.) drying: An optimization approach using microwave-vacuum drying[END_REF]. The ground turmeric obtained without heat treatment was the one of worst quality respect to curcuminoid pigments and good quality of ground turmeric can be obtained by cooking in plain water as cooking favours diffusion of pigments from cells to adjacent tissue, contributing to a better pigment homogenization [START_REF] Bambirra | Influence of post harvest processing conditions on yield and quality of ground turmeric (Curcuma longa L.)[END_REF]. There were no significant differences (P ≤ 0.05) between the three finished products (FD, FC1D and FC2D) for curcuminoid content as compared between them. These results were in accordance with [START_REF] Prathapan | Effect of heat treatment on curcuminoid, colour value and total polyphenols of fresh turmeric rhizome[END_REF] who found no significant differences in curcuminoid contents between turmeric samples submitted to different cooking and drying conditions.

Impact of the ''full" processes on essential oil content

The "full" processes did have a marked impact on essential oil content (Figure 5). There were significant differences (P > 0.05) between samples F/FD, F/FC1D and F/FC2D considered in pairs for essential oil content (relative loss of 9.0 -10.0 %). This relative loss was lower than that of the traditional drying method which could result in the loss of volatile oil up to 25 % by evaporation and in the destruction of some light-sensitive oil constituents [START_REF] Ararsa | Effect of processing methods on quality of turmeric (Curcuma longa L.) finger rhizome growing in Teppi, Sheka zone, Ethiopia[END_REF]. Though, there were no significant differences (P ≤ 0.05) between the three finished products (FD, FC1D and FC2D) for essential oil content as compared between them. These results were in agreement with those found by [START_REF] Jayashree | Processing of turmeric (Curcuma longa) by different curing methods and its effect on quality[END_REF] who reported that there were no significant differences for essential oil content once the duration of cooking in boiling water increased up to 60 min.

3.3.3. Impact of the ''full" processes on aroma composition Fresh sample (F) was used as a reference and 15 aroma compounds representing 95.0 % of the total essential oil were identified. Among these, 10 major compounds, all of which present at a rate greater than 7.0 %, represented 92.1 % (Table 1) of the total essential oil. The major compounds in fresh turmeric were sesquiterpenes i.e., turmerone (37.9 %), aRturmerone (16.8 %) and β-turmerone (16.1 %). These turmerones have similar chemical structures, physical properties and molecule weights, although they have different tastes [START_REF] Kao | Supercritical CO2 extraction of turmerones from turmeric and high-pressure phase equilibrium of CO2+turmerones[END_REF] and they are believed to be an intermediate for the formation of zingiberene and sesquiphellandrene [START_REF] Asghari | Curcuminoid and essential oil components of turmeric at different stages of growth cultivated in Iran[END_REF]. α-terpinolene (8.1 %), β-sesquiphellandrene (4.0 %), α-zingiberene (3.7 %), α-phellandrene (1.9 %), caryophyllene (1.6 %), α-curcumene (1.3 %) and eucalyptol (0.7 %) were also identified. [START_REF] Raina | Rhizome and leaf oil composition of curcuma longa from the lower Himalayan region of northern India[END_REF] reported that the major components of essential oils from turmeric were α-turmerone (44.1 %), β-turmerone (18.5 %) and aR-turmerone (5.4 %), while [START_REF] Naz | Chemical analysis of essential oils from turmeric (curcuma longa) rhizome through GC-MS[END_REF] reported that the most abundant components of the oils were aR-turmerone (25.3 %), α-turmerone (18.3 %) and β-turmerone (12.5 %). The aroma composition of the fresh turmeric rhizome oils varies with species, origin, agricultural system, climate, or maturity. The impact of the "full" processes on the aroma profile was shown in Figure 7. There were significant differences (P > 0.05) between samples F/FD, F/FC1D and F/FC2D considered in pairs for all molecules, except for turmerone and α-phellandrene. For aR-turmerone and βturmerone, the greatest differences were observed between sample F and FD with relative loss of 21.6 % and 9.8 %, respectively. For α-curcumene, the greatest difference (30.8 %) was observed between sample F and FC1D. For α-terpinolene, β-sesquiphellandrene, αzingiberene, caryophyllene and eucalyptol, the greatest differences were observed between sample F and FC2D with a reduction in the range of 19.7 -46.3 %. Cooking slightly impacted the concentration of aroma compounds (Figure 7). Smooth cooking significantly decreased 14.9 %, 21.7 % and 19.7 % of aR-turmerone, α-curcumene and eucalyptol, respectively. Drastic cooking significantly decreased aR-turmerone, α-terpinolene, caryophyllene and eucalyptol with a reduction of 6.5 %, 43.4 %, 37.2 % and 19.7 %, respectively. Conversely, cooking (smooth and drastic) significantly increased turmerone up to 19.0 %. The major aroma compounds in our fresh and cooked turmerics (F, FC1 and FC2) were turmerone (37.9 -44.6 %), aR-turmerone (14.0 -16.8 %) and β-turmerone (16.1 -16.8 %) (data not shown). Drying also had a slight impact on the concentration of aroma compounds (Figure 7) as there were significant differences (P > 0.05) between samples F/FD, FC1/FC1D and FC2/FC2D considered in pairs. Direct drying significantly decreased aR-turmerone, β-turmerone and αcurcumene with relative loss of 21.6 %, 9.8 % and 14.5 %, respectively. Drying with previous smooth cooking significantly decreased turmerone, β-turmerone, β-sesquiphellandrene and α-zingiberene with relative loss of 13.9 %, 14.2 %, 23.3 % and 24.4 %, respectively. Drying with previous drastic cooking significantly decreased aR-turmerone (8.3 %), β-turmerone (8.2 %) and β-sesquiphellandrene (15.4 %). The changes in the percentage of each aroma compound may be due to the complete release of these compounds during cooking and drying processes and another reason may be due to oxidation or rearrangement of less stable compounds to the more stable compounds [START_REF] Kutti Gounder | Comparison of chemical composition and antioxidant potential of volatile oil from fresh, dried and cured turmeric (Curcuma longa) rhizomes[END_REF]. The major aroma compounds in our dried turmerics (FD, FC1D and FC2D) were turmerone (40.9 -44.4 %), aR-turmerone (14.6 -15.8 %) and β-turmerone (16.1 -17.1 %) (data not shown).

The major aroma compounds of dried turmeric from Thailand found by Monton et al. (2019a) were aR-turmerone (43 -49 %), turmerone (13 -16 %) and β-turmerone (17 -18 %) and by [START_REF] Thongphasuk | Effects of gamma-irradiation on free radicals, active components and toxicity of Turmeric rhizomes[END_REF] were aR-turmerone (32 %), α-turmerone (16 %), and βturmerone (13 %).

Conclusion

Our original approach combining microscopy and biochemistry brought interesting results. Curcuminoid and essential oil were found in different dedicated cells. After processing, curcuminoids were dispersed throughout the matrix. This starchy matrix seems to protect essential oil and curcuminoid to a certain extent. Drastic cooking (95 °C/60 min) and drying operations impacted b* and C* colour values more than smooth cooking (95 °C/3 min).

Cooking (either smooth or drastic) had no impact on curcuminoid and essential oil contents and only a slight impact on essential oil composition. Drying significantly decreased curcuminoid (between 24.5 -38.8 %) and essential oil (between 8.8 -13.2 %) contents.

There is a real interest in combining cooking and drying unit operations. Indeed, cooking turmeric before drying saved 38.1 to 49.8 % of the drying time. Cooking at 95 °C/3 min is the optimum cooking condition for turmeric sliced rhizome (5 mm thick) because it is a good compromise between drying time and the quality of turmeric considering colour and aroma. It is also less energy-consuming than cooking at 95 °C/60 min. For further studies, we should focus on the impact of unit operations (cooking, drying and grinding) on the functional and sensory qualities of turmeric.

Figure 1 .

 1 Figure 1. Processes applied to turmeric. F: fresh turmeric; C: cooking (C1: 95 °C/3 min; C2: 95 °C/60 min); D: drying (60 °C, 40 % RH).

Figure 3 .

 3 The essential oils (Figure2.I) were particularly abundant in the cortex. They were more concentrated under the epidermis and in the endoderm than in other parts. Starch was present in amyloplasts throughout the whole rhizome and was most abundant in the cortex (Figure2.IV).

Figure 2 .

 2 Figure 2. Visualization of essential oil, starch and curcuminoids distribution in turmeric rhizome by widefield microscopy. (I) essential oil distribution (blue Nadi colouration) in a cross-section of turmeric rhizome, (II) essential oil distribution in the epidermis of fresh (F) turmeric, curcuminoid distribution (natural orange colour) in the epidermis of fresh (F) turmeric, (III) essential oil distribution in the epidermis of cooked (FC1) turmeric, curcuminoid distribution in epidermis cooked (FC1) turmeric, (IV) starch distribution (black Lugol colouration) in a cross-section of turmeric rhizome, (V) starch distribution in the cortex of fresh (F) turmeric, curcuminoid distribution in the cortex of fresh (F) turmeric and (VI) starch distribution in the cortex of cooked (FC1) turmeric. am, amyloplast; c, cortex; cur, curcuminoids cell; en, endodermis; ep, epidermis; gel, gelatinized starch; st, stele; bar = 500 µm (I and IV) or 50 µm (II, III, V and VI).

Figure 3 .

 3 Figure 3. Visualization of curcuminoids distribution by multiphotonic microscopy. (I) curcuminoids distribution in the cortex of fresh turmeric, (II) curcuminoids distribution in the cortex of cooked and dried (FC2D) turmeric and (III) Emission spectra of curcuminoids between 400 and 700 nm, excitation 720 nm, bar = 50 µm.

Figure 4 .

 4 Figure 4. Impact of processes on (A) colour values (L*, a*, b* and C*) and (B) curcuminoid content (g/100g initial dry weight basis). FR: frozen rhizome; F: fresh turmeric; C: cooking (C1: 95 °C/3 min; C2: 95 °C/60 min); D: drying (60 °C, 40 % RH). The error bars represent the standard error (n = 6). Within the same parameters, the values followed by the same superscript letters are not significantly different (P ≤ 0.05).

Figure 5 .

 5 Figure5. Impact of processes on essential oil content (mL/100g initial dry weight basis). F: fresh turmeric, C: cooking (C1: 95 °C/3 min; C2: 95 °C/60 min); D: drying (60 °C, 40 % RH). The error bars represent the standard error (n = 4). Within the same parameters, the values followed by the same superscript letters are not significantly different (P ≤ 0.05).

Figure 6 .

 6 Figure 6. Drying curves of fresh and cooked turmerics. (A) water content (X) on a dry basis as a function of time (t) and (B) drying rate (dX/dt) as a function of X. Drying curves recorded by an air dryer (60 °C, RH 40 %, air velocity at 2.1 ± 0.1 m s -1 ). The different dash types on curves correspond to different trials.

  thick) because it is a good compromise between drying time and the quality of turmeric in terms of b* and C* colour chromatic values and it is also less energy-consuming than cooking at 95 °C/60 min.

Figure 7 .

 7 Figure 7. Composition of essential oil of fresh and processed turmerics. F: fresh turmeric; C: cooking (C1: 95 °C/3 min; C2: 95 °C/60 min); D: drying (60 °C, 40 % RH). The error bars represent the standard error (n = 8). Within the same parameters, the values followed by the same superscript letters are not significantly different (P ≤ 0.05).

Table 1

 1 Major aroma compounds in Curcuma longa L. fresh turmeric (F) essential oil. ± 95 % confidence interval. RI a is retention indices relative to C8-C20 n-alkanes (Experimental value) and RI b is retention indices from NIST and PubChem (literatures).

	Aroma compounds	DB-Wax UI column RI a RI b	DB-5MS column RI a RI b	Area (%)
	Turmerone	2198	2245	1682	1650	37.8 ± 3.4
	aR-Turmerone	2274	-	1678	1637	16.8 ± 1.9
	β-Turmerone	2261	-	1711	1680	16.1 ± 0.8
	α-Terpinolene	1307	1283	1096	1079	8.1 ± 1.4
	β-Sesquiphellandrene	1771	1772	1527	1515	4.0 ± 0.6
	α-Zingiberene	1724	1724	1497	1488	3.7 ± 0.7
	α-Phellandrene	1197	1167	1016	998	1.9 ± 0.3
	Caryophyllene	1610	1595	1425	1419	1.6 ± 0.4
	α-Curcumene	1773	1777	1482	1473	1.3 ± 0.3
	Eucalyptol	1240	1213	1041	1022	0.7 ± 0.1
	Total					92.1 ± 1.0
	Mean values (n = 8)

ACKNOWLEDGEMENTS

The authors are grateful to the French Government Scholarships Cofounded by the Ministry of Education, Youth and Sports of Cambodia, CIRAD -UMR QualiSud and Cambodia Higher Education Improvement Project (Credit No. 6221-KH) for financial support of this study. We also would like to thank Delpech Charlotte, Ollier Léa, Boulanger Renaud, Meot Jean-Michel and Servent Adrien (CIRAD -UMR QualiSud) for their contribution to this study.

CONFLICTS OF INTEREST

There are no conflicts of interest to declare.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Novelty Impact Statement

Our study brings us to propose better practices for turmeric processing. A smooth cooking (95 °C/3 min) before drying is recommended to preserve the quality of the turmeric as the curcuminoids and essential oil are preserved during cooking. This protection could be linked to the starchy structure of the turmeric matrix.