Fabien Kéfélian

Senior Member, IEEE Philippe Gallion

Locking and noise properties of multisection semiconductor lasers with optical injection Application to Fabry-Perot and DFB cavities

Keywords: Injection locked oscillators, distributed feedback lasers, optical noise, semiconductor lasers

Locking and noise properties of multisection semiconductor lasers with optical injection Application

to Fabry-Perot and DFB cavities

Locking and noise properties of multisection semiconductor lasers with optical injection

Application to Fabry-Perot and DFB cavities I. INTRODUCTION Optical injection locking is a powerful technique to improve the performance of diode lasers and realize new techniques in telecommunications [START_REF] Kobayashi | Optical FM signal amplification and FM noise-reduction in an injection locked AlGaAs semiconductor laser[END_REF] [2] [START_REF] Simpson | Bandwidth enhancement and broadband noise reduction in injection-locked semiconductor lasers[END_REF] [4] [START_REF] Chow | All-optical modulation format conversion and multicasting using injection-locked laser diodes[END_REF]. It is also a very interesting phenomenon in terms of physics and non-linear dynamics [START_REF] Wang | Squeezed light from injection-locked quantum well lasers[END_REF] [START_REF] Jahanpanah | Theory of laser-amplifier injection locking[END_REF] [8] [START_REF] Goulding | Excitability in a quantum dot semiconductor laser with optical injection[END_REF] and can be used as a tool for determining fundamental laser parameters [10] [11]. Optical injection locking was first experimentally demonstrated more than forty years ago in gas lasers [START_REF] Stover | Locking of laser oscillators by light injection[END_REF] and its study in semiconductor lasers started in the early 1980's [START_REF] Kobayashi | Injection locking in AlGaAs semiconductor laser[END_REF] [START_REF] Otsuka | Theoretical studies on injection locking and injection-induced modulation of laser diodes[END_REF] (see [START_REF] Van Tartwijk | Semiconductor lasers with optical injection and feedback[END_REF] for an extended bibliography on the history of optical injection). Contrary to conventional lasers, semiconductor lasers allow a significant fraction of the light to escape and are consequently more sensitive to external perturbations. Moreover, the important dependence of the refractive index on the carrier density in semiconductor lasers has been shown to significantly affect the injection locking properties [START_REF] Lang | Injection locking properties of a semiconductor laser[END_REF]. The first works on injection locking in semiconductor lasers were mainly dedicated to the study of the range of locking, the stability [START_REF] Henry | Locking range and stability of injection locked 1.54 µm InGaAsP[END_REF] [START_REF] Mogensen | Locking conditions and stability properties for a semiconductor laser with external light injection[END_REF], and the noise properties [START_REF] Gallion | Contribution of spontaneous emission to the linewidth of an injection-locked semiconductor laser[END_REF] [START_REF] Schunk | Noise analysis of injection-locked semiconductor injection lasers[END_REF] [START_REF] Spano | Frequency and intensity noise in injection-locked semiconductor lasers: theory and experiments[END_REF]. Their results have demonstrated the interest of this technique for frequency stabilization and linewidth narrowing. The analyze of the modulation properties of a locked diode laser has shown the possibility to reduce the relaxation oscillations [START_REF] Petitbon | Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser[END_REF] [23], extend the modulation bandwidth, generate phase modulation and to reduce the chirp [START_REF] Spano | Frequency and intensity noise in injection-locked semiconductor lasers: theory and experiments[END_REF] [24] [START_REF] Wieczorek | Improved semiconductor-laser dynamics from induced population pulsation[END_REF]. The influence of the gain compression has been also investigated [START_REF] Li | Static and dynamic properties of injection-locked semiconductor lasers[END_REF] and has been shown to be a factor contributing to stabilization of the locking [START_REF] Lidoyne | Locking range, phase noise and power spectrum of an injection-locked semiconductor laser[END_REF].

Optical injection in semiconductor lasers has been mainly studied using simple rate equations for the outgoing electric field and the carrier density. Some authors have derived these rate equations from the full wave-equation [START_REF] Spencer | Laser with a transmitting window[END_REF] [START_REF] Hadley | Injection locking of diode lasers[END_REF], or the round trip time traveling wave amplifier model [START_REF] Van Tartwijk | Semiconductor lasers with optical injection and feedback[END_REF], but many works have used a modified version of the rate equation for isolated diode laser by phenomenologically introducing the external field with a coupling factor or feed-in rate. Consequently, the expression of the feed-in rate has sometimes not been set clearly or justified correctly. Moreover, for lasers exhibiting spatial hole burning or containing Bragg gratings, such as DFB or DBR lasers, rate equations cannot be derived without first using a full longitudinal treatment. DFB lasers with external feedback have been the subject of several theoretical studies [START_REF] Favre | Theoretical analysis of external optical feedback on DFB semiconductor lasers[END_REF] [START_REF] Tromborg | Transmission line description of optical feedback and injection locking for Fabry-Perot and DFB lasers[END_REF], but few results have been published concerning injection locking in DFB lasers. Experimental studies [START_REF] Hui | Injection locking in distributed feedback semiconductor lasers[END_REF] [33] [START_REF] Troger | Novel measurement scheme for injection-locking experiments[END_REF] have for example demonstrated that, contrary to a Fabry-Perot laser, a DFB laser exhibits a symmetric locking range for weak injection, due to the possibility of having a stable oscillation in a mode with higher threshold gain than the free mode. However, an investigation of the locking range as a function of the DFB laser fundamental parameters is still missing.

Different mathematical tools can be used for the full longitudinal treatment of a multisection laser [START_REF] Vankwikelberge | CLADISS -a longitudinal multimode model for the analysis of the static, dynamic, and stochastic behavior of diode lasers with distributed feedback[END_REF] [36] [START_REF] Hess | Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers. i. theoretical formulation[END_REF] [38] [START_REF] Radzianus | Modeling self-pulsating DFB lasers with an integrated phase tuning section[END_REF] [START_REF] Bandelow | Impact of gain dispersion on the spatio-temporal dynamics of multisection lasers[END_REF]. Among them, the Green's function method is well appropriate to derive analytical expressions. The Green's function method was introduced by C. Henry [START_REF] Henry | Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers[END_REF] to analyze the spontaneous emission of semiconductor amplifiers and multisection lasers. This work triggered numerous successful studies concerning the dynamics and noise properties of DFB and more generally of multisection lasers [START_REF] Makino | Transfer-matrix analysis of the intensity and phase noise of multisection DFB semiconductor lasers[END_REF] [START_REF] Marani | Spontaneous emission in non-hermitian optical-systems -distributed-feedback semiconductor-lasers[END_REF] [START_REF] Peirera | Green's functions theory for semiconductor-quantum-well laser spectra[END_REF] [45] [START_REF] Loudon | Theory of spontaneous emission noise in multisection semiconductor lasers[END_REF]. B. Tromborg et al. used this method to study the effect of spatial hole burning, nonuniform current injection and nonlinear gain on the linewidth of DFB lasers [START_REF] Tromborg | Theory of linewidth for multielectrode laser diodes with spatially distributed noise sources[END_REF], as well as analyzing the stability and the noise and modulation properties [START_REF] Pan | Modulation characteristics of tunable DFB/DBR lasers with one or two passive tuning sections[END_REF] [49] [START_REF] Chen | Static, dynamic, and noise analysis of multisection DFB lasers using frequency-domain transmission line model[END_REF]. The coherence collapse conditions were also derived using the same formalism [START_REF] Tromborg | Nonlinear injection locking dynamics and the onset of coherence collapse in external cavity lasers[END_REF]. The influence of the structure of multisection lasers on the phase-amplitude coupling factor and the spontaneous emission rate have been studied with Green's function by Duan et al. [START_REF] Duan | Dynamic and noise properties of tunable multielectrode semiconductor lasers including spatial hole burning and nonlinear gain[END_REF]. An effective phase-amplitude coupling parameter has been defined and including the non linear gain new explanations to the rebroadening of the spectrum at high power has been given. More recently, Green's function has been used for new studies of the external optical feedback, concerning the threshold of the coherence collapse [START_REF] Grillot | Feedback sensitivity and coherence collapse threshold of semiconductor dfb lasers with complex structures[END_REF] and the spectrum of external cavity lasers [START_REF] Detoma | The complex way to laser diode spectra: example of an external cavity laser strong optical feedback[END_REF].

In this paper, we present a theoretical study of the injection locking in a multisection laser. This work extends a previous work of B. Tromborg et al. [START_REF] Tromborg | Transmission line description of optical feedback and injection locking for Fabry-Perot and DFB lasers[END_REF] which used the concept of effective coefficients of reflection and transmission. Two questions are at the source of our investigation: For DFB lasers, which "round trip time" (or which Q factor) can be used for the expression of the feed-in rate? For symmetric lasers, is there a difference between an injection through the facet from which the emission is observed (front facet), and an injection through the rear facet? The purpose of the paper is to obtain a general equation of motion for the electric field envelope in both cases of injection through the front and rear facet, and to hereby derive the noise properties of the injection locked laser.

The paper is organized in the following way, in Section II, starting from the equation of propagation and using the Green's function method, we derive an expression for the Fourier transform of the electric field at the output of the laser as a function of the external injected fields to both facets and a Langevin force representing spontaneous emission. The characteristics of the laser are taken into account using effective reflection and transmission coefficients. In Section III and IV, the equation of motion of the temporal complex envelope of the electric field is calculated, using a Taylor expansion of the effective coefficients of the cavity, for respectively backward and forward injection. For each direction, the rate equations are linearized to give the locking relation and to determine the power spectral densities of the noises in matrix form. In Section V we apply our results, concerning the feed-in rate value, to the Fabry-Perot cavity and discuss the comparison to the classical expression of R. Lang. After, the parameter of injection for a DFB laser with an output facet anti-reflection coated are numerically studied. Finally, the conclusions are summarized in Section VI. We consider the general configuration of Fig. 1.The section between z = 0 and z = l represents the laser (Fabry-Perot, DFB or multisections). The internal electric field E is decomposed into right and left traveling fields E + and E -. At each interface we define a reflection and two transmission coefficients: at z = 0, r 1 , t 12 (into the laser) and t 21 (out of the laser) ; at z = l, r 2 , t 23 (out of the laser) and t 32 (into the laser). The laser can generally emit light via both facets, but all along the paper we consider that only the field E + out emitted through the right facet (z = l) is observed and studied. The right facet is consequently called the front facet, and the left facet is called the rear facet. We assume that an electric field E - injR can be injected through the front facet, using a coupler or a circulator, or that an electric field, E + injL , can be injected through the rear facet. The starting point is the equation of propagation of the electric field established using the Maxwell's equations in a macroscopic medium:

II. ELECTRIC FIELD IN MULTISECTION LASER WITH EXTERNAL INJECTION

∇ 2 E( r, t) -µ 0 σ ∂ E ∂t ( r, t) - 1 c 2 ∂ 2 E ∂t 2 ( r, t) = µ 0 ∂ 2 P ∂t 2 ( r, t) (1)
with r = (x, y, z) the position, P the electric polarization, µ 0 and 0 the magnetic permeability and electric permitivity of the vacuum, c the light velocity, and σ the conductivity. P can been expressed as a function of the electric field introducing the susceptibility χ. The spontaneous electric polarization, due to quantification of the field, is included using a random function, P s :

P ( r, t) = 0 χ( r, t) * E( r, t) + P s ( r, t) (2)
Susceptibility and conductivity can be taken into account together through the relative dielectric constant r : r ( r, ω) = 1 + χ( r, ω) -j σ( r, ω)

0 ω (3)
with ω the Fourier angular frequency.

The Fourier transform of the electric field is a solution of the Helmoltz equation:

∇ 2 E( r, ω) + ω 2 c 2 r ( r, ω) E( r, ω) = F ( r, ω) (4)
with F ( r, ω) = -ω 2 µ 0 P s ( r, ω). F is a Langevin force whose correlation coefficient is given by the theorem of fluctuation dissipation [START_REF] Landau | Statistical physics[END_REF]:

F ( r , ω) F * ( r, ω ) =2D F F * δ( r -r , ω -ω ) (5)
D F F * ( r, ω) = 2πω 3 c 3 0 n m ( r)g m ( r)n sp (6)
n sp = 1 1 -e ω-Ecv k B T (7)
with T the temperature, k B the Boltzmann constant, h the Planck constant, = h/2π, E cv the energy difference between the quasi Fermi levels, n m = e{ r } the material refractive index, g m = ω cnm m{χ} the material gain, and n sp the inversion factor.

We assume that, in each section i of the laser, the electric field oscillates in the fundamental transverse mode Φ i (x, y) polarized along u with the complex constant of propagation k i (z, ω) satisfying [START_REF] Agrawal | Semiconductors lasers[END_REF]:

∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ω 2 c 2 r (x, y, z, ω) Φ i (x, y) = k 2 i (z, ω)Φ i (x, y) (8)
Consequently E( r, ω) can be written as E(z, ω)Φ i (x, y) u with E(z, ω) satisfying:

∂ 2 ∂z 2 + k 2 (z, ω) E(z, ω) = f (z, ω) (9)
where f (z, ω) = F (x, y, z, ω)Φ(x, y) udxdy Φ(x, y)Φ(x, y)dxdy [START_REF] Liu | Characterization of fundamental parameters of a semiconductor laser with an injected optical probe[END_REF] represents the spontaneous emission in the transverse mode.

Solving the inhomogeneous differential equation ( 9) using the Green's function method (Appendix A), the Fourier transform of the forward traveling component of the electric field at the front facet of the laser can be expressed as a function of the injected fields and a Langevin force:

E + (l -, ω) = t L t 12 E + injL (0 -, ω) + r L t 32 E - injR (l + , ω) + F L [1 -r 2 r L ] (11)
where r L and t L are the equivalent reflection and transmission coefficients of the active region between z = 0 - and z = l -for a backward traveling wave and F L is a Langevin force representing the total spontaneous emission at the output of the laser. Expressions of r L , t L and F L are derived in Appendix A.

III. DYNAMIC EQUATIONS AND NOISE PROPERTIES OF LASER DIODE WITH BACKWARD INJECTION

We consider in this section the injection of a backward traveling wave E - injR through the front facet, i.e. backward injection.

A. Expansion of the electric field equation

For injection through the front facet [START_REF] Van Exter | Effect of optical injection on bias voltage and spectrum of a semiconductor laser[END_REF] becomes:

E + (l -, ω) r L = r 2 E + (l -, ω) + t 32 E - injR (l + , ω) + F L (ω) r L (12)
r L depends on ω, but also on the carrier and photon densities N (z) and P(z), via r . Since N and P depend on z, r L is a functional, i.e. a function of functions.

P(z) is the transversely average photon density in the active layer (A.L.):

P(z) = 2 0 nn g ca (z) n gm Γ ω 0 A act |E(z)| 2 (13)
where nn g ca (z) is the value of the product of the material refractive index and the group refractive index n gm = n m + ω∂n m /∂ω transversally average on the active layer:

nn g ca (z) = 1 A act A.L. n m ( r)n gm ( r)|Φ(x, y)| 2 dxdy (14)
Γ is the confinement factor of the transverse mode in the active layer and A act the transverse area of the active layer [START_REF] Agrawal | Semiconductors lasers[END_REF].

Without injection, when spontaneous emission is neglected, the static solution (ω s , N s (z), P s (z)) fulfills the condition of oscillation:

r 2 r L (ω s , N s (z), P s (z)) = 1 (15)
When the laser is submitted to optical injection, the static parameters of the laser are modified. Once locking is achieved, the central frequency of the slave laser becomes equal to the central frequency of the master laser ω i .

Assuming that the function 1/r L is slowly varying in a region around (ω s , N s , P s ), and that the detuned state with deviation ∆N and ∆P is included in this region, the value of 1/r L can be determined using a Taylor expansion:

1 r L (ω, N , P) ≈ 1 r Ls + ∂ 1 r L ∂ω s (ω -ω s ) + l 0 δ 1 r L δk (z) ∂k ∂N s (∆N (z))dz + l 0 δ 1 r L δk (z) ∂k ∂P s (∆P(z))dz (16)
δ1/r L is the functional derivate of 1/r L introduced by B. Tromborg et al. in [START_REF] Tromborg | Theory of linewidth for multielectrode laser diodes with spatially distributed noise sources[END_REF].

δ 1 r L = l 0 δ 1 r L δk (z)δk(z)dz (17)
where δ(1/r L )/δk represents the infinitesimal variation of 1/r L due to an infinitesimal variation δk of the function k.

B. Temporal envelope of the electric field

The complex envelope around ω i of the forward traveling component of the intracavity electric field at the front facet is defined by:

A + (t) = 1 2π ∞ 0 E + (l -, ω)e j(ω-ωi)t dω (18)
The envelope of the injected field, A i (t), is defined similarly in l + .

Using the expansion of 1/r L in the equation ( 12), the equation of motion of A + can be obtain by inverse Fourier transform:

dA + dt (t) = -j(ω s -ω i )A + (t) + A + (t) C N |∆N (t) + A + (t) + C P |∆P(t) + f R t 32 A i (t) + F A (t) (19)
with

f R = r Ls f D = j ∂ 1 r L ∂ω -1 s (20) C N (z) = - δ 1 r L δk (z) ∂k ∂N s f R (21)
C P (z) = - δ 1 r L δk (z) ∂k ∂P s f R (22)
We use here a notation similar to the notation of Dirac in quantum mechanics:

f | g = l 0 f (z )g(z )dz (23)
The derivates of k are given by:

∂k ∂ω = 1 v g + j 2 ∂g ∂ω (24)
∂k ∂N = j 1 + jα H 2 ∂g ∂N (25)
with α H = -2k ∂n/∂N ∂g/∂N the Henry factor, g the modal gain, n the modal refractive index and v g the group velocity. The first term on the right hand side of ( 19) represents the rotation of A + due to the detuning, the second and third ones represent the in phase and in quadrature contributions due to the modification of the gain and index induced by the carrier and photon density modifications. The functions C N and C P determine the variation of the global gain and index of the cavity due to the local variation in z of the carrier and photon densities. They take into account longitudinal spatial hole burning [START_REF] Pan | Spectral linewidth of DFB lasers including the effects of spatial holeburning and nonuniform current injection[END_REF] and spectral hole burning [START_REF]Gain nonlinearities in semiconductor lasers: theory and application to distributed feedback lasers[END_REF]. The fourth term represents the contribution of the injected field. F A is the Langevin force representing the contribution of the spontaneous emission to the envelope evolution:

F A (t) = f D 2π ∞ 0 F L (ω)e j(ω-ωi)t dω (26)
The autocorrelation of F A is derived from ( 6), ( 10), ( 26) and (98) (Appendix A):

F A (t)F * A (t ) = R sp σ 2 p δ(t -t ) (27)
with

R sp = 4ω 3 0 nn g | |Z L | 2 |f D | 2 ngn sp | |Z L | 2 |Z + 1 (0)| 2 c 3 Φ(x, y)Φ(x, y)dxdy 2 |Z - 2 (l)| 2 (28)
where the Z functions are defined in Appendix A. R sp is the average rate of spontaneous emission into the mode, taking into account the effect of the transverse mode [START_REF] Petermann | Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding[END_REF] and the longitudinal mode [START_REF] Henry | Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers[END_REF] [47] [START_REF] Tromborg | Transmission line description of optical feedback and injection locking for Fabry-Perot and DFB lasers[END_REF] on the standard relation of Einstein R sp = v g gn sp . σ p is a constant such that p = σ 2 p |A| 2 is the total number of photons in the laser cavity:

σ 2 p = 2 0 ω s nn g (z)|Z L (z)| 2 dz (29)
with

nn g (z) = n( r)n g ( r)|Φ(x, y)| 2 dxdy (30)
and

ng(z) = n m ( r)g m ( r)|Φ(x, y)| 2 dxdy (31)
C. Optical phase and power equations

The mean optical power I emerging from the front facet of the laser is given by:

I(t) = T 2 σ 2 I |A + (t)| 2 (32)
where σ 2 I = 2 0 cn g (l -), n g is the modal group refractive index and

T 2 = 1 -|r 2 | 2 .
The incident optical power I in (t) is similarly given by:

I in (t) = σ 2 I n g (l -) |A in (t)| 2 (33)
The rate equations for the output power I(t) and the phase of A + , φ(t), are determined by multiplication of ( 19) by 4A + * 0 cn g (l -)T 2 and separation into real and imaginary parts:

dI dt (t) =2I(t) e {C N } |∆N (t) + 2I(t) e {C P } |∆P(t) + 2T 2 I(t)I in (t)|f R | cos(θ(t) + ψ R ) + F I (t) (34)
dφ dt (t) =(ω s -ω i ) + m {C N } |∆N (t) + m {C P } |∆P(t) + T 2 I in (t) I(t) |f R | sin(θ(t) + ψ R ) + F φ (t) (35)
where

θ(t) = φ in (t) -φ(t), φ in is the phase of A i and ψ R = arg {f R }.
The carrier density is governed by the rate equation:

dN (z, t) dt =J(z, t) -R(N (z, t))
-v g g(N (z, t), P(z, t))P(z, t)

+ F N (z, t) (36)
where J is the density of injected carriers, R is the function describing the spontaneous recombination of the carriers and F N (z, t) is the Langevin force associated to the fluctuation of carriers due to the interaction between the field and the spontaneous electric polarization [START_REF] Henry | Quantum noise in photonics[END_REF].

The non zero Langevin forces diffusion coefficients are [START_REF] Lax | Classical noise IV:Langevin methods[END_REF]:

2D II = 2R sp I 0 T 2 σ I σ p 2 (37)
2D φφ = R sp 2p 0 ( 38
)
2D N N (z) = 2 [v g g 0 (z)n sp P 0 (z) + R(N (z))] A act ( 39
)
2D IN (z) = -2v g g 0 (z)n sp P 0 (z)T 2 σ I σ p 2 (40)

D. Calculation of the locking range

Locked states (θ 0 , I 0 , N 0 (z), P 0 (z)) are solutions of ( 34), ( 35) and ( 36) with all noise and derivative terms set to zero. When locking is achieved, the phase difference and angular frequency detuning are related by:

ω i -ω s = η b |f R | 1 + α 2 ef f sin(θ 0 + ψ R -tan -1 (α ef f )) (41)
where

η b = Iin0T2 I0/T2 , η 2
b is the ratio between the optical power injected through the front facet and the forward traveling intracavity optical power at the front facet. It is important to point out here that, for a locked state, it is possible to define a stationary mean value of the difference of the slave and master phases, but not of the individual phases which remain unstationary random functions. A generalized Henry factor has been used:

α ef f = m {C N } |∆N + m {C P } |∆P e {C N } |∆N + e {C P } |∆P (42)
taking into account the impact of the longitudinal distribution of the carrier and photon densities, and the gain compression. The imaginary part of C P , corresponding to a modification of the refractive index induced by the photon density modification, is generally negligible. The maximum detuning is:

max{ω i -ω s } = η b |f R | 1 + α 2 ef f (43)
Since α ef f is not a material parameter but rather a structural parameter, it depends in fact on the condition of injection. Consequently, the exact locking range must be calculated self-consistently. Stability of the solutions inside the locking range can be discussed using the Hurwitz criterion on the small signal equations.

E. Determination of the power spectral densities of noise

The longitudinal mode Z L (z) depends on ω, P and N . Consequently P(t) cannot be directy related to I(t)

using the group velocity. The following expression, determined by Tromborg et al. [START_REF] Tromborg | Theory of linewidth for multielectrode laser diodes with spatially distributed noise sources[END_REF], is used for P(t) to take into account these dependences:

|δP =(1 -M P ) -1 |P 0 δI I 0 + |H I 2I 0 dδI dt + |H φ dφ dt + M |δN (44)
Expression for the different functions and operators are recalled in Appendix B. The rate equations of ( 34), [START_REF] Vankwikelberge | CLADISS -a longitudinal multimode model for the analysis of the static, dynamic, and stochastic behavior of diode lasers with distributed feedback[END_REF] and [START_REF] Hansmann | Transfer matrix analysis of the spectral properties of complex distributed feedback laser structures[END_REF] are linearized around the static point (I 0 , I in0 , θ 0 , N 0 (z)). (δI, δI in , δθ, δN (z)) is the deviation to the static point induced by noise. Due to the unstationarity of the phase noise, δφ or δφ in cannot be defined. Using the following vectorial notations [START_REF] Genest | Microwave signals generated by optical heterodyne between injection-locked semiconductor lasers[END_REF]:

X(Ω) =      δI(Ω) φ(Ω) δN (Ω, z)      (45)
X in (Ω) =      δI in (Ω) φ in (Ω) δN in (Ω, z)      (46)
F (Ω) =      F I (Ω) F φ (Ω) F N (Ω, z) + δJ(Ω, z)      (47)
the expression of the noises can be expressed by:

X(Ω) = ([M (Ω)] + [D]) -1 ([C]X in (Ω) + F (Ω)) (48)
Noting

k c = η b |f R | cos(θ 0 + ψ R ) k s = η b |f R | sin(θ 0 + ψ R )
the coefficients of the matrix [M ], [D] and [C] are given by:

m 11 = -2 C N r | ∆N 0 -2 C Pr | ∆P 0 -2 C Pr | (1 -M P ) -1 |P 0 -jΩ C Pr | (1 -M P ) -1 |H I + jΩ m 12 = -jΩ2I 0 C Pr | (1 -M P ) -1 |H φ m 13 = -2I 0 C N r | -C Pr | M m 21 = - 1 I 0 C Pi | (1 -M P ) -1 |P 0 - jΩ 2I 0 C Pi | (1 -M P ) -1 |H I m 22 = -jΩ C Pi | (1 -M P ) -1 |H φ + jΩ m 23 = -C N i | -C Pi | M m 31 = v g ∂g ∂P P 0 (z) + v g g 0 (z) × (1 -M P ) -1 |P 0 I 0 + jΩ |H I 2I 0 m 32 = v g ∂g ∂P P 0 (z) + v g g 0 (z) × (1 -M P ) -1 |H φ jΩ m 33 = ∂R ∂N + v g g N P 0 (z) + v g ∂g ∂P P 0 (z) + v g g 0 (z) M + jΩ d 11 = -k c d 12 = -2I 0 k s d 21 = - k s 2I in0 d 22 = -k c c 11 = I 0 I in0 k c c 12 = -2I 0 k s c 21 = k s 2I in0 c 22 = k c ( 49
)
The power spectral densities can be expressed using an Hermitian product:

S X (Ω)δ(Ω -Ω ) = X(Ω)X(Ω ) † (50)
The coefficients of S X correspond to the power spectral densities of the intensity, phase and carrier density noises for the diagonal terms and to the respective interspectral power densities for the non diagonal ones. Using [START_REF] Pan | Modulation characteristics of tunable DFB/DBR lasers with one or two passive tuning sections[END_REF],

S X (Ω) =([M ]+[D]) -1 [C]S Xin (Ω)[C] † ([M ]+[D]) -1 † + ([M ] + [D]) -1 S F (Ω)([M ] + [D]) -1 † (51)
with

S F (Ω) = 2      D II 0 D IN (z) 0 D φφ 0 D IN (z) 0 D N N (z) + S J (Ω, z)      (52)
The linewidth of the injection locked spectrum can be determined from the value of S φ (Ω)/Ω 2 in Ω = 0. However, for very small values of k c , S φ (Ω)/Ω 2 is not constant on a sufficiently wide range of frequencies around zero and consequently the power spectrum of the injection locked laser can no longer be represented by a Lorentzian lineshape.

F. Comments

The coefficients which depend directly on the injected fields are d ij and c ij . c ij indicates how the noise of the injected field X in is coupled to the intrinsic sources of noise of the slave laser F , resulting in an equivalent internal source of noise [C]X in (Ω) + F (Ω). d ij indicates how the injection modifies the response of the laser to the source of noise, which is represented by the coefficients m ij . It is interesting to note that the coefficient d ij and c ij are independent on the structure, they are identical to the Fabry-Perot case [START_REF] Spano | Phase noise in semiconductor lasers: a theoretical approach[END_REF]. Inversely, coefficients m ij , which, without injection, determine the noise properties of the laser, depend on the structure of the laser. The terms involving a longitudinal integration take into account the effect of the spatial hole burning and non uniform current injection. The gain compression effect appears through C Pr and ∂g/∂P. The use of the expansion of the carrier density noise [START_REF] Peirera | Green's functions theory for semiconductor-quantum-well laser spectra[END_REF], involved new transfer of noise through the modification of the longitudinal mode represented by the operator M P and M and the functions H(z) and H I (z). This effect leads to an additional coupling between the rate equations of intensity, phase and carrier density which appears specially in the factor m 12 and m 32 , which would be null otherwise. Thus, factor m 12 corresponds to a contribution of the frequency noise to the intensity noise. Such a contribution is generally induced only by injection locking through the factor d 12 but is here also self-induced. It is due to the fact that the frequency noise induces a noise on the longitudinal distribution of the photon density and consequently a noise on the output power through the gain compression. Factor m 32 represents a contribution of the frequency noise on the carrier density noise, the frequency noise induces a noise on the longitudinal distribution of the photon density and consequently a noise on the carrier density through the photon-carrier coupling of the gain. Moreover the influence of those two parameters increase with the frequency Ω and consequently can become important for the calculation of the phase noise at frequency offset above 1 GHz. Finally, the gain curve influence is also present through the factor H(z) which is important for emission far from the maximum of the gain curve.

IV. DYNAMIC EQUATIONS AND NOISE PROPERTIES OF LASER DIODE WITH FORWARD INJECTION

We consider in this section the injection of a forward traveling wave E + injL through the rear facet, i.e. forward injection.

A. Temporal envelope of the electric field

For injection through the rear facet, [START_REF] Van Exter | Effect of optical injection on bias voltage and spectrum of a semiconductor laser[END_REF] becomes:

E + (l -, ω) = r L r 2 E + (l -, ω) + t L t 12 E + injL (0 -, ω) + F L (ω) (53)
Using the following first order expansion:

t L r L (ω, N , P) = t L r L (ω s , N s , P s ) + ∂ t L r L ∂ω s (ω -ω s ) + l 0 δ t L r L δk s ∂k ∂N ∆N (z) dz + l 0 δ t L r L δk s ∂k ∂P ∆P(z) dz (54)
the equation of motion of the temporal envelope of the forward traveling component of the intracavity electric field at the front facet is established by inverse Fourier transform:

dA + dt (t) = -j(ω s -ω i )A + (t) + A + (t) [ C N | ∆N + C P | ∆P ] + f T t 12 A i (t) + C t N ∆N + C t P ∆P t 12 A i (t) + f T f t d A i dt (t) + j f T f t (ω i -ω s )t 12 A i (t) + F A (t) (55)
where

f T = f D t Ls C t N = f R δ t L r L δk ∂k ∂N s C t P = f R δ t L r L δk ∂k ∂P s r L t L f t = j ∂ t L r L ∂ω -1 s (56)
The two first terms of the right hand side of ( 55) are identical to the case of backward injection, and the four following terms represents the contribution of the injected field. The feed-in rate f T t 12 is different from the backward case.

Four additional contributions from the injected field appear in the equation. Those contributions come from the effect of the single pass through the cavity by the injected field. The fourth terms represents the modification of the single pass gain and phase rotation due to the deviation of the carrier and photon densities. The fifth takes into account the delay induced by the single pass trip by introducing a contribution of the derivate of the envelope of the injected field, and the sixth comes from the detuning of the injected frequency from the natural frequency of the cavity.

B. Optical phase and power equations

The equations of motion for the optical phase and the optical power are derived using the same method as for the backward case:

dI dt (t) =2 [ C N r | ∆N + C Pr | ∆P ] I(t) + 2 |f T | T 1 T 2 I in (t)I(t) cos (θ(t) + ψ T ) + 2 C t N ∆N + C t P ∆P T 1 T 2 I in (t)I(t) × cos θ(t) + arg C t N ∆N + C t P ∆P -2 f T f t (ω i -ω s ) T 1 T 2 I in (t)I(t) sin (θ(t) + ψ t ) + f T f t dI in dt (t) T 1 T 2 I in (t) I(t) cos (θ(t) + ψ t ) -2 f T f t dφ in dt (t) T 1 T 2 I in (t)I(t) sin (θ(t) + ψ t ) + F I (t) (57)
dφ dt (t) = [ C N i | ∆N + C Pi | ∆P ] -(ω i -ω s ) + |f T | T 1 T 2 I in (t) I(t) sin(θ(t) + ψ T ) + C t N ∆N + C t P ∆P T 1 T 2 I in (t) I(t) × sin θ(t) + arg C t N ∆N + C t P ∆P + f T f t (ω i -ω s ) T 1 T 2 I in (t) I(t) cos (θ(t) + ψ t ) + 1 2 f T f t dI in dt (t) T 1 T 2 I in (t)I(t) sin (θ(t) + ψ t ) + f T f t dφ in dt (t) T 1 T 2 I in (t) I(t) cos (θ(t) + ψ t ) + F φ (t) (58)
where ψ t = arg{f T /F T }. Due to the involvement of the derivative of the temporal envelope of the injected wave, corresponding contributions appear in the equations of motion of the phase and the power.

C. Calculation of the locking condition

Locked states are solution of equations ( 36), ( 57) and ( 58) with all noise and derivative terms set to zero. When locking is achieved, phase difference and angular frequency detuning are related by:

ω i -ω s = η f 1 + α 2 ef f 1 -η f 1 + α 2 ef f f T ft cos(θ 0 + ψ t -tan -1 (α ef f )) × |f T | sin(θ 0 + ψ T -tan -1 (α ef f )) + |f ∆ | sin(θ 0 + arg(f ∆ ) -tan -1 (α ef f )) (59)
where:

f ∆ = C t N ∆N + C t P ∆P (60)
and

η f = Iin0T1 I0/T2 , η 2
f is the ratio between the optical power injected through the rear facet and the forward traveling internal optical power at the front facet. Because f ∆ depends on P and N , the exact expression of the locking range has to be determined self-consistently.

It could appear surprising to obtain a different expression of the locking range for the backward and the forward injections. In the case of a symmetric laser (T 1 = T 2 ), if we consider the light emitted by the rear facet, the existence of two different expressions seems to allow solutions where for example the forward traveling components of the intracavity field is locked but not the backward one. This is in fact not the case, because the injection modifies the longitudinal distribution of the optical power inside the cavity, and leads to an asymmetry of the emitted power, even for symmetric laser, as was shown in [START_REF] Hadley | Injection locking of diode lasers[END_REF] for a Fabry-Perot cavity. Consequently it is consistent to obtain different expressions for the forward and backward injections.

D. Power spectral densities of noise

Using the notations:

k c = η f |f T | cos(θ 0 + ψ T ) (61)
k s = η f |f T | sin(θ 0 + ψ T ) ( 62
)
χ t = f T f t ( 63
)
χ c = χ t cos (θ 0 + ψ t ) (64)
χ s = χ t sin (θ 0 + ψ t ) (65)
ξ N c = η C t N cos arg{C t N } + θ 0 ( 66
)
ξ N s = η C t N sin arg{C t N } + θ 0 ( 67
)
ξ Pc = η C t P cos arg{C t P } + θ 0 ( 68
)
ξ Ps = η C t P sin arg{C t P } + θ 0 ( 69
)
with the same method than for the backward case, the power spectral densities of the intensity, phase and carrier density noises are given by the following c ij and d ij coefficients and 48 :

d 11 = -k c -[ ξ N c | ∆N 0 + ξ Pc | ∆P 0 ] -ξ Pc | (1 -M P ) -1 |2P 0 -jΩ ξ Pc | (1 -M P ) -1 |H I + χ s (ω i -ω s ) d 12 = -2I 0 {k s + [ ξ N s | ∆N 0 + ξ Ps | ∆P 0 ] +jΩ ξ Pc | (1 -M P ) -1 |H φ + χ c (ω i -ω s ) d 13 =2I 0 ξ N c | + 2I 0 ξ Pc | M d 21 = 1 2I 0 k s + [ ξ N s | ∆N 0 + ξ Ss | ∆P 0 ] -ξ Ps | (1 -M P ) -1 |P 0 - jΩ 2 ξ Ps | (1 -M P ) -1 |H I + χ c (ω i -ω s ) d 22 =k c + ξ N c | ∆N 0 + ξ Pc | ∆P 0 + jΩ ξ Ps | (1 -M P ) -1 |H φ -χ s (ω i -ω s ) d 23 = ξ N s | + ξ Ps | M (70) c 11 = I 0 I in0 {k c + [ ξ N c | ∆N 0 + ξ Pc | ∆P 0 ] -χ s (ω i -ω s )} + jΩχ c c 12 = -2I 0 {k s + [ ξ N s | ∆N 0 + ξ Ps | ∆P 0 ] +χ c (ω i -ω s ) + jΩχ s } c 21 = 1 2I in0 {k s + [ ξ N s | ∆N 0 + ξ Ps | ∆P 0 ] +χ c (ω i -ω s ) + jΩχ s } c 22 =k c + [ ξ N c | ∆N 0 + ξ Pc | ∆P 0 ] -χ s (ω i -ω s ) + jΩχ c (71)

E. Comments

In difference with the backward case, the coefficients c ij and d ij , depend on the structure of the laser. The coefficients m ij remain identical to the backward case since they represent the noise properties of the laser without injection. It can be also noticed that all the coefficients c ij and d ij , excepted d 13 and d 23 , contain a component proportional to the frequency Ω. This contribution is particularly important for the characterization of the phase and intensity noise spectrum at large frequencies. For frequency closed to f D , the Taylor expansion ( 16) and [START_REF] Detoma | The complex way to laser diode spectra: example of an external cavity laser strong optical feedback[END_REF] should be extended to higher orders in (ω -ω s ) leading to higher order temporal derivates in [START_REF] Landau | Statistical physics[END_REF] and polynomes in Ω instead of linear terms in the expression of c ij and d ij .

V. APPLICATION TO FABRY-PEROT AND DFB LASERS

A. Injection locking of a Fabry-Perot

We discuss in this section the application of the equations previously derived to a Fabry-Perot cavity. The field is considered to be uniform along the cavity, the emission to be near the maximum of the gain curve and the gain compression to be negligible. These hypothesis are fulfilled with a Fabry-Perot laser with facet reflectivities above 0.5, operating not too far from the threshold [START_REF] Thedrez | Reassessment of standard rate equations for low facet reflectivity semiconductor lasers using traveling wave rate equations[END_REF].

For the case of backward injection:

f D = 1 τ in = v g 2l (72)
C N = 1 + jα H 2l v g ∂g ∂N (73) r Ls = 1 r 2 (74) f R = v g 2lr 2 (75)
Consequently, from [START_REF] Gallion | Contribution of spontaneous emission to the linewidth of an injection-locked semiconductor laser[END_REF], the equation of motion of A + is:

dA + dt (t) = -j(ω s -ω i )A + (t) + 1 + jα H 2 v g ∂g ∂N ∆N (t)A + (t) + v g 2l t 32 r 2 A i (t) + F A (t) (76)
This equation is similar to the well known equation used by R. Lang [START_REF] Lang | Injection locking properties of a semiconductor laser[END_REF]. However, the feed-in rate used by

Lang is c/(2nl) whereas in our case the feed-in rate is c/(2ln g r 2 ). We can notice two differences, the use of the group index and the involvement of the facet reflectivity r 2 . The phenomenological argument of Lang was that the injected field adds t 32 A i every time it hits the irradiated facet at a time interval equal to the cavity round trip time. However, the injected field is added only to the backward traveling wave reflected by the front facet with a coefficient of reflection r 2 . Consequently when the forward traveling wave is considered, the feed-in rate is equal to the inverse of the round trip time divided by r 2 . In many papers on injection locking, the 1/r 2 is missing. This problem is analog to the difference which has been noticed [START_REF] Wang | Locking bandwidth equations for electrically and optically injection-locked oscillators[END_REF] between Kurokawa's [66] and Adler's [START_REF] Adler | A study of locking phenomena in oscillators[END_REF] locking bandwidth due to the existence of different Q factors, the resonator unloaded Q, the loaded Q and the external Q.

The expression of the feed-in rate derived in [START_REF] Van Tartwijk | Semiconductor lasers with optical injection and feedback[END_REF] using the round trip time traveling wave amplifier model is equal to our expression. The expression of the feed-in rate given in [START_REF] Schunk | Noise analysis of injection-locked semiconductor injection lasers[END_REF] seems also to agree with our expression, however it is not clear whether it was established for the total intracavity electric field or for the forward traveling component.

For the envelope of the total intracavity electric field at the front facet, the feed-in rate is c(1 + r 2 )/(2ln g r 2 ), it is in agreement with the value obtained in [START_REF] Spencer | Laser with a transmitting window[END_REF] which uses for the mirror the model of a dielectric plan where a surface current is induced by the electric field instead of the boundary conditions used in this article.

In the forward case:

t Ls = 1 √ r 1 r 2 (77)
f T = 1 τ in √ r 1 r 2 (78)
f t = 2 τ in (79)
C t N = - 1 + jα H 4l √ r 1 r 2 ∂g ∂N v g (80)
and ( 55) becomes:

dA + dt (t) =j(ω s -ω i )A + (t) + 1 + jα H 2 v g ∂g ∂N ∆N A + (t) + t 12 τ in √ r 1 r 2 A i (t) - (1 + jα H )t 12 4 √ r 1 r 2 ∂g ∂N v g ∆N A i (t) -j t 12 2 √ r 1 r 2 (ω s -ω i )A i (t) + t 12 2 √ r 1 r 2 d A i dt (t) + F A (t) (81)
As previously noticed, the total feed-in rate can be decomposed in three terms, κ in = κ in1 + κ in2 + κ in3 :

κ in1 = 1 τ in √ r 1 r 2 (82)
κ in2 = - (1 + jα H ) 4 √ r 1 r 2 ∂g ∂N v g ∆N (83)
κ in3 = -j ω s -ω i 2 √ r 1 r 2 (84)
κ in1 is the main term of κ in . It is identical to the backward case only for a symmetric laser. The others terms of κ in can become noticeable when the cavity is particularly long or when locking is achieved with very large detuning.

Finally, the contribution of the derivative of the envelope of the injected electric field can become noticeable when the variation of A i during a round trip time is non negligible, for example when the master laser is modulated at high frequency which is an important point in telecommunication applications.

B. Numerical applications for DFB laser with an anti-reflexion coating

We will now present numerical results for the parameters f R , f T χ t for DFB lasers with an anti-reflection coated front facet.

1) Parameters of injection of a DFB laser with an anti-reflection coated facet: For the Fabry-Perot cavity, f D represents the inverse of the round-trip time of the wave between the two reflective facets. However, in a DFB laser, the definition of the round-trip time is not straightforward due to the distributed reflexion added to the facet reflexions. Consequently 1/f D represents an effective round-trip time for a laser containing distributed reflexions.

The resolution of the coupled equations for a DFB laser [START_REF] Agrawal | Semiconductors lasers[END_REF] with coupling coefficient of the grating κ, Bragg wave number β B and grating phase Ω B gives:

r L = (κ + δρ 1 ) tanh(γl) + jγρ 1 -(κρ 1 + δ) tanh(γl) + jγ e -jΩ (85)
with δ = k -β B , the complex detuning from the Bragg wave number, ρ 1 = r 1 e j(Ω B -2β B l) and γ = √ κ 2 -δ 2 , the complex wave number of the longitudinal envelope. For an anti-reflection coated front facet laser, the resonnance condition is 1/r Ls = 0, consequently:

tanh(γl) = jγ κρ 1 + δ (86)
The expression of f R is calculated from ( 85) and ( 86):

f R = -j v g l (κl) 2 -(γl) 2 (1 + ρ 2 1 ) + 2ρ 1 γ κ (γl)[ρ 1 + j(κl)(1 + ρ 2 1 )] + κl + 2jρ 1 (γl) 2 (87)
For forward injection, the additional parameter f t is derived from t L /r L , calculated from the solution of the coupled equations:

t L r L = γe jβ B l -jκ * e -jΩ sinh(γl) + r 1 [cosh(γl) -jδ sinh(γl)] (88)
We will now present numerical results for the parameter of injection in forward and backward configuration. (r = -0.565) is 1.8. Consequently, for normalized coupling coefficients of the grating lower than 1.5, the feed-in rate of the DFB laser is higher than for a Fabry-Perot laser, in the limit of a rear facet reflectivity lower than 0.95.

For higher coupling coefficient of the grating and intermediate values of the rear facet reflectivity, the feed-in rate of the DFB is lower than for a Fabry-Perot laser. For strong coupling the influence of the boundary conditions is weaker due to the predominance of the distributed reflexion, and consequently the feed-in rate is weaker. Moreover the feed-in rate tends toward zero for very high coupling coefficients regardless of the rear reflectivity.In contrast, for weak coupling, the value of the rear reflectivity and the position of the lasing wavelength have an important influence on the feed-in rate.

On Fig. 2(right) are shown the normalized modulus of the primary feed-in rate (f T ) for forward injection as a function of the normalized coupling coefficient to the grating and for four different values of the reflectivity of the rear facet. For a cleaved facet Fabry-Perot laser, the normalized feed-in rate is equal to 1.8. Consequently, the primary feed-in rate of the DFB laser becomes lower than for a Fabry-Perot laser for coupling coefficients higher than 1.7. The effect of the mode hopping for r 1 =0.775 and 0.565 is indicated by the two arrows and is weaker than for the backward case. 3) Lasing mode in the forbidden band: We have previously studied the case of a DFB laser with an anti-reflection coated front facet and a rear facet with a real reflectivity. The case of a purely imaginary rear reflectivity is also interesting because in this case the principal mode of the laser is in the center of the forbidden band [START_REF] Henry | Locking range and stability of injection locked 1.54 µm InGaAsP[END_REF]. Fabry-Perot laser. These results are important since they demonstrate that in choosing appropriately the value of the rear reflectivity, the feed-in rate, and consequently the locking range, can become extremely high or extremely small.

We do not find the same large range of values for the primary feed-in rate f R in the forward injection. However Fig. 5 shows that it increases with the reflectivity which is inverse to the case of a real reflectivity. Moreover, the sensitivity to the coupling coefficient of the grating is higher.

Finally, the results presented for the parameter χ t exhibit, as for the real reflectivity case, values lower than for the cleaved facet Fabry-Perot laser. It is interesting to note that for every value of κ.l, there is a value of r 1 for which χ t is zero.

4) Influence of the phase of the coefficient of reflection: We have shown that results for real and purely imaginary reflectivity are strongly different. Consequently we will now study the impact of the argument of the rear reflectivity on the different parameters considered.

VI. CONCLUSION

We have presented a theory for the injection locking of multisection lasers taking into account spatial hole burning, the non linear gain and the direction of injection. The contribution of the spontaneous emission has been treated with the Green's function method and the contribution of the injected fields using effective reflection and transmission coefficients accounting for longitudinal distribution of the carrier and photon densities. Two different equations of evolution for the complex envelope of the forward component of the intracavity electric field at the front facet have been derived, corresponding to the two possible directions of injection. This analysis shows that backward injection corresponds to the classical equation of R. Lang, whereas forward injection leads to additional contributions. Part of those additional contributions has been previously found in the equation derived by Tromborg et al. [START_REF] Tromborg | Transmission line description of optical feedback and injection locking for Fabry-Perot and DFB lasers[END_REF], but the contribution of the detuning was missing and it was not pointed out that those contributions were related to the facet of injection. The paper provides analytical expressions of the noise power spectral densities including mathematical tools presented in [START_REF] Tromborg | Theory of linewidth for multielectrode laser diodes with spatially distributed noise sources[END_REF]. Those expressions are interesting for accurate characterization of the phase noise spectrum of injection locked laser, especially for forward injection and for injection of modulated light. Moreover, they show that injection locked lasers can exhibit asymmetric noise properties, which has been experimentally demonstrated in laser submitted to external feedback [START_REF] Mink | Asymetric noise and output power in semiconductor lasers with optical feedback near threshold[END_REF]. The application of our general equation to Fabry-Perot laser shows that the expression of the feed-in rate considered in the literature is sometimes incomplete.

Numerical results show that DFB lasers can be designwith very large feed-in rate for backward injection, or with large feed-in rate for forward injection and very small one for backward injection. with Z 0 is a solution of the homogeneous equation determined by the boundary conditions:

E + (0 + , ω) =r 1 E -(0 + , ω) + t 12 E + injL (0 -, ω) (91)
E -(l -, ω) =r 2 E + (l -, ω) + t 32 E - injR (l + , ω)

E + injL (0 -, ω) and E - injR (l + , ω) are the projection of the external field on the transverse mode and polarization vector of the cavity in, respectively, 0 -and l + : E inj (z, ω) = E ext (x, y, z, ω)Φ(x, y) udxdy Φ(x, y)Φ(x, y)dxdy (93)

We choose Z 2 satisfying the left boundary condition without injection. Consequently, the left boundary condition gives:

Z + 0 (0)-r 1 Z - 0 (0) =(r 1 Z - 1 (0)-Z + 1 (0)) l 0 f (ω, z ) Z 2 (z ) W dz + t 12 E + injL (0 -) (94)
If Z 1 satisfies Z 1 (0) = 0 and dZ 1 (0)/dz = 1 such that W = (1 + r 1 )Z - 2 (0), since Z 0 is a solution of the homogeneous equation, Z 2 satisfies the left boundary condition and E(l -, ω) = Z 0 (l):

Z + 0 (l)Z + 2 (l) -Z - 0 (l)Z - 2 (l) =Z + 1 (0) l 0 f (ω, z ) Z 2 (z ) W dz + Z - 2 (0)t 12 E + injL (0 -) (95)
Using now the right boundary condition and defining the reflectivity r L and the transmission coefficient t L :

r L = Z + L (l -)/Z - L (l -) (96)
t L = Z + L (0 + )/Z - L (l -) (97)
F L (ω) = Z + 1 (0) Z - L (l) l 0 f (z , ω)Z L (z )dz (98)
we obtain the following equation for E + (l -, ω)

E + (l -, ω) =r 2 r L E + (l -, ω) + t L t 12 E + injL (0 -, ω)

+ r L t 32 E - injR (l + , ω) + F L (ω) (99)

Fig. 1 .

 1 Fig. 1. Laser submited to the injection by both facets

Fig. 2 .

 2 Fig. 2. Normalised feed-in rate for the backward injection (left) and forward injection (right)

Fig. 3 .Fig. 4 .

 34 Fig. 3. χt as a function of the coupling coefficient of the grating

Fig. 5 .

 5 Fig. 5. Parameters as a function of the reflectivity for two different coefficients (a) threshold gain, (b) forward feed-in rate, (c) backward feed-in rate, (d) χt forward parameter

Fig. 6 .

 6 Fig. 6. Influence of the phase of the reflectivity on the parameter of injection (a) threshold gain and Bragg detuning, (b) forward feed-in rate, (c) backward feed-in rate, (d) χt forward parameter

  APPENDIX A RESOLUTION OF THE INHOMOGENEOUS HELMOTZ EQUATION WITH GREEN'S FUNCTION Knowing two independent solutions Z 1 et Z 2 of the homogeneous Helmotz equation, i.e. two solutions with WronskianW = Z 1 dZ 2 /dz -Z 2 dZ 1 /dz(89)different from zero, the general solution of the inhomogeneous equation can be expressed as:E(z, ω) =Z 0 (z) + l z f (z, ω ) [Z 1 (z)Z 2 (z ) -Z 2 (z)Z 1 (z )] W dz(90)

H

  PHOTON LONGITUDINAL DENSITYEquation[START_REF] Peirera | Green's functions theory for semiconductor-quantum-well laser spectra[END_REF] is the expansion of the photon density established by Tromborg et al.[START_REF] Tromborg | Theory of linewidth for multielectrode laser diodes with spatially distributed noise sources[END_REF]. In this appendix, we briefly remind the method and the expression of the main parameters. The longitudinal mode is expended at first order in ω, P and N :Z L (z) = Z L0 (z) + ∂Z L ∂ω (ω -ω s ) + Z N (z)| δN + Z P (z)| δP (100) with δZ L (z) δk(z ) ∂k 0 ∂N = Z N (z, z ) (101) δZ L (z) δk(z ) ∂k 0 ∂P = Z P (z, z )(102)Z L0 (z) is the stationary longitudinal mode with injection. Using[START_REF] Kobayashi | Injection locking in AlGaAs semiconductor laser[END_REF] and Fourier transform, we obtain:|δP =M N |δN + M P |δP are defined:M P = 2 e{Z L0 (z)} |Z LO (z)| 2 |P 0 Z P (z)| (104) M N = 2 e{Z L0 (z)} |Z LO (z)| 2 |S 0 Z N (z)|(105)Finally, supposing 1 -M P inversible and noting:M = (1 -M P ) -1 M N (106)we obtain[START_REF] Tromborg | Theory of linewidth for multielectrode laser diodes with spatially distributed noise sources[END_REF]:|δP =(1 -M P ) -1 |P 0 φ (z) and H I (z) come from the ω dependance of Z L : H φ (z) = 2 e I 0 Z * L0 (z) ∂Z L ∂ω (z, ω s )(108)H I (z) = 2 m I 0 Z * L0 (z)∂Z L ∂ω (z, ω s )(109)


E-L Zins

M Guinet

Delphy Rodriguez

Sébastien Payan

Absolute absorption cross section of 2-EHN in IR region

Keywords: Absorption cross section, cetane improver, air pollutant, chemometrics, adsorption process, in-situ probing Highlights

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Introduction

The chemical industry and chemical processes of the 21st century should respect the pillars of green chemistry. In particular, the concentration of entropic pollutants should be monitored in near real-time. It is important to be able to qualitatively and quantitatively identify the accidental release of different constituents, and to document individual exposure. A major source of anthropogenic air pollutants is still linked to road traffic. In addition to fuel oil, diesel contains additives to facilitate the combustion reaction: these are species whose exothermic decomposition makes it possible to lower the required temperature for starting the combustion process.

These constituents are called cetane improvers and the most common is 2-ethyl hexylnitrate . [START_REF] Ickes | Effect of 2-ethylhexyl nitrate cetane improver on NO x emissions from premixed low-temperature diesel combustion[END_REF]- [START_REF] Kuszewski | Effect of adding 2-ethylhexyl nitrate cetane improver on the autoignition properties of ethanol-diesel fuel blend-Investigation at various ambient gas temperatures[END_REF] Liquid at room temperature, it decomposes above 100°C. It then releases free radicals that contribute to reducing the engine's self-ignition time. 2-EHN is incorporated into fuels at a level of 0.3 to 1 g.L -1 , corresponding to a production of 105 tons worldwide. [START_REF] Phd-Thesis | Biodégradtion du 2-éthylhexyl nitrate par Mycobacterium austroafricanum IFP 2173 : de l'élucidation de la voie de dégradation aux enzymes impliquées[END_REF] Since it is used in very large quantities, it may be accidentally released into the atmosphere. This gas is involved in atmospheric chemistry [START_REF] Roberts | The atmospheric chemistry of organic nitrates[END_REF][START_REF] Atkinson | Gas-phase tropospheric chemistry of volatile organic compounds: 1. Alkanes and alkenes[END_REF], and could cause secondary reactions. As a result of its exothermic decomposition, specific studies are required to characterize it. Furthermore, since 2-EHN is not easily biodegradable [START_REF] Solano-Serena | Biodegradability of 2-ethylhexyl nitrate (2-EHN), a cetane improver of diesel oil[END_REF] and not soluble in water, it can cause long term pollution. [START_REF] Bornemann | Thermal decomposition of 2-ethylhexyl nitrate (2-EHN)[END_REF]. While studies have already been carried out on its decomposition and characterization in the liquid phase, [START_REF] Insaus | Fast Determination of 2-Ethylhexyl Nitrate Diesel/Biodiesel Blends by Distillation Curves and Chemometrics[END_REF], [START_REF] Oung | Study on the Effects on Diesel LTC Combustion of 2-EHN as Cetane Improver[END_REF], [START_REF] Brito Dos Santos | Prediction of Relative Density, Distillation Temperatures, Flash Point, and Cetane Number of S500 Diesel Oil Using Multivariate Calibration of Gas Chromatographic Profiles[END_REF] only few studies have concentrated on the gas phase. FTIR is a widely used technique for atmospheric gas quantification.

Pioneering studies have been carried out on 2-EHN, [START_REF] Vrtiška | Prediction of 2-EHN content in diesel/biodiesel blends using FTIR and chemometrics[END_REF], [START_REF] Insausti | Determination of 2-ethylhexyl Nitrate in Diesel Oil Using a Single Excitation Emission Fluorescence Spectra (EEF) and Chemometrics Analysis[END_REF], [START_REF] Insausti | Single excitation-emission fluorescence spectrum (EEF) for determination of cetane improver in diesel fuel[END_REF] but no data are available for the spectral signature of the vapors in the mid-IR domain. To complete and develop the previous studies, we propose here a thorough analysis of its IR spectrum in the gaseous phase. This precise analysis could allow its rapid identification near strong sources such as fuel stations and airports.

In a second step, we propose different complementary approaches for the quantitative analysis in the gas phase based on a chemometric method. Indeed all the studies mentioned above [10-15] used the chemometric approach while atmospheric spectroscopy uses directly the absorption cross section data. Comparisons between the two complementary methods are therefore needed and useful. For an accurate quantification of a constituent by gas phase spectroscopy, it is essential to take into account the adsorption/desorption processes that can occur on the cell walls. We evaluate these effects and try to limit the biases due to exchange processes.

Material and Methods

Experimental setup

Fourier transform spectra have been recorded between room temperature and 318 K using the Bruker IFS 120 HR interferometer of MONARIS [START_REF] Jacquemart | Line parameters measurements and modeling for nu(6) band of CH3F: Generation of a complete line list for atmospheric database[END_REF]. The resolution was set to 0.5 cm -1 . A 1.57 mm aperture radius of the iris in front of the Globar source has been chosen and the focal length equals 418 mm for all spectra. A scheme of the experimental setup is presented in Figure 1.

The gas sample is obtained by evaporation of liquid 2-EHN and the pressure is recorded with a capacitive gauge. We mostly used pure 2-EHN gas but some mixtures with dry air (100-1000 mbar) have also been used. 2-EHN has a low vapor pressure of 0.27 mbar at 20°C. Moreover, it is a polar gas that undergoes a strong adsorption on the cell walls. As a consequence, a stable pressure above 0.2 mbar is difficult to maintain over 15 min. In addition, the outgassing (or micro-leaks) rate of the cell is 0.03 mbar/hour. Therefore, we have chosen pressures between 0.02 and 0.2 mbar which allowed to control the purity of the gas.

The flask of 2-EHN is first frozen in LN 2 and pumped with a turbo-molecular pump allowing to evacuate non condensable gases (N 2 , O 2 ). The gas is then slowly warmed until 20°C while keeping the pumping to evacuate condensable gases that are more volatile than the 2-EHN. Then the cell and the entrance ramp are conditioned with the chosen pressure during 10 minutes. After passivation we quickly pumped on the 2-EHN flask together with the cell and the injection ramp to not destroy the fragile passivation layer made on stainless steel surface. Therafter, the pumping is stopped and the cell is closed when the pressure of "fresh" gas reaches the passivation pressure. This procedure is adapted to evacuate the noncondensable gases, but it is difficult to separate the 2-EHN from some residual condensable gases. A time of 45 seconds is needed to record one spectrum. During this time, the total pressure of 2-EHN evolve by less than 2%.

The commercial gas sample of 2-EHN was provided by Sigma Aldrich with a stated purity of 98.0% in natural abundance. The pressures of 2-EHN (ranging from 0.02 to 0.2 mbar table 1) inside the cell have been measured with a Baratron gauge with an accuracy better than 0.25% and the temperature of the gas sample has been determined with an accuracy of ± 0.1 K using a platinum resistance thermometer inside the cell. The device was pumped over the night. Before each day of measurement, an outgassing rate was measured and taken into account to correct the purity of the sample.

The average interferograms (co-addition of 20 scans) have been Fourier transformed using the procedure included in the Bruker software OPUS package. As illustrated in Figure 2, each average spectrum is divided by a "white spectrum" recorded before each set of measurements. The interferometer was equipped with a KBr beam splitter, a MCT photovoltaic detector, and a Globar source. Two stainless steel cells of 68.8 cm and 31.8 cm path length were used. Those cells were both equipped with KBr windows. A Platinum thermometer (Pt100) was inserted into the cell to probe the gas temperature directly. The cell is surrounded by a heating cord, and 3 aluminum layers to ensure a good thermal homogeneity.

Qualitative analysis -Identification of the vibrational modes

For this molecule, different isomers may exist depending on the relative position of the carbon chains and the nitrate group. In order to allow for a reliable identification of the compound, it is first necessary to characterize its IR spectrum. In the absence of high-resolution or noble gas matrix spectra, we have made theoretical calculations. These were performed with the Gaussian09 software. [17] The aim of this part of our work was to identify the main isomers that may exist at room temperature.

For this purpose, a two-step procedure was applied, starting with a search for the different isomers in DFT, followed by a vibrational characterization. More precisely, the PBE1PBE functional was selected, with the 6-31+G(d,p) basis set, using the additional keywords scf=(Tight,MaxCycle=500) and Int=UltraFine.

Quantitative analysis -Assessment of the dataset quality

For the quantitative analysis, data have been evaluated following two approaches.

The first method takes advantage of the absorption cross sections at a given wavenumber or over a small spectral domain (area under the curve) being proportional to the partial pressure of the active molecule. This approach is very simple and allows to evaluating the limits of the data set. It is particularly adapted for sources a like laser diode, a led, or a vapor lamp. However, this approach can't be used if another gas is present and affects the IR bands in the chosen wavenumber range of the analysis. In this case a broadband instrument recording a large wavenumber area is needed and a different method which is robust against interfering gases must be used. We have employed chemometric approach, which has the advantage of being based on freely available and easy-to-use software.

The data processing was done as follows:

i. Firstly, a FORTRAN program was used to extract the absorption at a fixed wavenumber and to use it for quantitative analysis. The choice of a specific and representative wavenumber is crucial here to limit the error that may be induced by pollutants. The FORTRAN program used in this study is provided as Supplementary Material.

ii. As the presence of a molecule in a complex mixture (such as polluted air under atmospheric conditions) requires a spectrum over a wide spectral range, we have also employed a chemometric approach for the calibration. More precisely, a multivariate linear regression (MLR) study was performed using seven selected wavenumbers in the spectrum. It is worth mentioning that since 10 spectra were used for the calibration, the maximum value of independant and characteristic wavenumbers is 9. Before carrying out this study, it was checked that the quality of our spectra did not require any pre-treatment (no baseline deviation, no shift of the spectra with respect to each other, no improvement after SNV treatment). Then, a representative set of wavenumbers spread all over the spectral range and for which the nature of the vibration could be identified, was chosen. In these statistical analyses, the set of n spectra is treated as a matrix of n rows, each column corresponding to a wavenumber. This matrix is then used to handle the noise and then to perform a supervised quantitative analysis. A supervised analysis is one in which a data set is used to build a predictive model. This is the case for the calculation of a calibration curve. Following Beer-Lambert's law, it is quite natural in this case to choose a linear regression. Rather than using a single wavenumber as a basis for the model, we preferred to use a multiple linear regression model, based on the simultaneous absorbance values measured at seven given wavenumbers, to increase the robustness of the analysis.

This analysis was carried out using the ChemFlow software [START_REF] Rossard | ChemFlow, chemometrics using Galaxy[END_REF]."

Results and discussion.

Characterization of the isolated molecule a. Theoretical characterization of each isomer

Due to free rotations around simple bonds, different rotamers of 2-EHN are likely to exist at room temperature. The influence of these isomers on the vibration spectrum has been evaluated using quantum chemistry calculations. Table 2 presents some examples of structures (A,B,C,D,E) very close in energy (ΔE < 12 kJ/mol) that were identified. The IR spectrum observed at ambient pressure and temperature will therefore be the convolution of the spectra of these different isomers. All these isomers will contribute to the registered IR spectrum. The Cartesian coordinates as well as the vibration frequencies calculated for these isomers are presented as Supporting Information. For the identification in the atmosphere, the target bands must be located within the spectral range accessible for atmospheric remote sensing as provided by meteorological satellites, such as IASI [19]. It thus appears that for a qualitative analysis of 2-EHN in real samples of polluted air (determination of the presence or absence of this pollutant), it is relevant to focus on the vibrations associated with the NO 3 group, which will be more characteristic than those associated with carbon chains. This is demonstrated in the Figure 3. The calculated frequencies further show that the spectra of both isomers are too close to be experimentally separated. Indeed, the strongest bands differ by only a few cm -1 , with the notable exception of the band at 1338 cm-1, that appears to be characteristic of the B-isomer.

b. Experimental spectra and assignment of the vibration modes

The average absorption cross section (absorbance/pressure/length at a given wavenumber) is obtained by the averaging of 10 spectra (Figure 4). All these data are presented as Supplementary Information in the form of a table containing the absorption as a function of wavenumber. This formatting is directly applicable to evaluate absorption measurements at a single wavenumber or the complete data set can be integrated into a radiative transfer code. The molecular signal presents multiple absorption areas; each of them has a complex form that may be due to the coexistence of multiple isomers, as given in Table 2 and Table 3. Table 4 shows the attribution of the experimentally observed vibrations.

Study of the evolution of absorbance as a function of pressure

In order to be able to quantify traces of 2-EHN in complex samples, it is necessary to consider different calibration techniques, which can be used depending on the measurement conditions. For this study, ten spectra recorded using two different cells to cover a representative range of gas concentrations, were selected. We particularly focused on characterizing samples corresponding to low pressures of 2-EHN that are likely to be encountered in real samples of polluted air.

i. Calibration using the absorbance at 1283 cm -1

With respect to applications in atmospheric spectroscopy, it is interesting to be able to quantify species from a very restricted spectral area, or even on a single band, for which the spectra would be simple and quick to record routinely. These approaches are also relevant in the case of using spectrometers with laser diodes. To determine whether such an approach could be used in the present case, we selected the absorbance at the label 1283.7980425006 cm -1 in the spectra file, and perform a calibration. The linear regression obtained using this calibration, as well as the random distribution of the residuals (Figure 5) show that this approach is quite relevant. However, for the study of complex samples in which the 2-EHN may be present in trace amounts (polluted air), it seemed necessary to develop calibrations based on the study of a wider spectral range. The affine function obtained in

ii. Calibration using chemometric approach

For these samples, seven wavenumbers covering the entire mid-IR range considered. The intensity at each selected wavenumber for each of the samples is indicated in table 5. The Multivariate Linear Regression method allows to find a linear relation between the concentration and the intensity of all these bands. It is worth mentioning cross sections at the seven chosen wavenumbers are simultaneously used for the linear regression. In the case of the analysis of complex mixtures, this approach is more applicable than the use of Beer-Lambert's law at just a single wavenumber, if the selected wavenumbers are correctly chosen with respect to the absoption bands of polluted compounds. Indeed, in complex mixtures such as a polluted air sample, other compounds may share absorption bands with those of 2-EHN. The simultaneous recording of seven characteristic spectral bands (without any C-H or C-C vibration, common to many pollutants) after having checked the intensity ratios of the selected bands, will avoid erroneous identification of 2-EHN. The random distribution of the residuals and the linear variation of the normal QQ plot suggest that the description is correct, and can be validated.

The random distribution of the residuals (Figure 6), the high value of R 2 (0.997, Figure 7) and the linearity observed on a large central part of the Q-Q (quantile-quantile) plot (Figure 8) suggest that the description is correct, and can be validated [START_REF] Ramchani | Measurements and modeling of N-2 broadening coefficients for the nu(6) band of CH3F, comparison with CH3Cl and CH3Br molecules[END_REF]. It should be mentioned that the deviation of the Q-Q plot on the extreme values is normal, and the higher contribution of the residuals for low values of partial pressure can be explained by a lower signal to noise ratio.

- By measuring the intensity at the selected seven wavenumbers, the unknown 2-EHN concentration of a sample could be deduced (Figure 8). However, in the case of polluted air samples, the choice of the wavenumbers is crucial and should be adapted to the nature of pollutants.

Influence of resolution and pressure

After this first characterization, the effects of pressure and spectral resolution have been investigated. to help for the comparison. The difference between the two spectra shows that the shape of the spectrum is not dramatically affected due to the pressure: only a slight shift of the bands around 1250 and 1700 cm -1 is observed.

Due to the large mass of the molecule (175.23 g.mol -1 ), the rotational spectra is not resolved leading to wide band of 50 cm -1 . This spectrum is convoluted with the apparatus function and airbroadened. The resolution of the FTIR is 0.5cm -1 and the air broadening is typically around 0.1 cm -1 atm -1 . As a consequence, the data set can be used on spectrometers of different resolution.

Figure 9 illustrates this fact. The difference between the pure sampling and the mixing with air is barely observable. A slight shifting is observed, but due to the difficulties of knowing the partial pressure in the mixture, it is preferable to record the spectra with pure 2-EHN and use for evaluation even under atmospheric pressure conditions. By the same argumentation, our spectra can be used with a large set of resolutions.

Application, partial pressure in situ measurements

While capacitor gauges are accurate for the recording of total gas pressures, absolute absorption cross sections allow to perform an accurate in situ probing of the partial pressure of a target gas. Many applications monitor a trace amount of gas within a large amount of perturbing gas.

We choose to add the perturber gas following a standard protocol [START_REF] Ramchani | Measurements and modeling of N-2 broadening coefficients for the nu(6) band of CH3F, comparison with CH3Cl and CH3Br molecules[END_REF][START_REF] Raddaoui | Measurements and modeling of air broadening coefficients for the nu(6) band of CH3I[END_REF] when measuring the spectroscopic broadening by air or by N 2 . In order to provide reliable quantitative evaluations, the calibration spectroscopic studies performed in the laboratory must be as accurate as possible. However, in the case of gas phase analyses, it is known that there are important biases, which can dramatically distort the measurements, especially when the active gas is diluted in a large amount of perturber gas. For example, gas exchange with the walls interfere with the measurements whatever the precision of the spectrometeris, and the type of cell used. There are two types of phenomena that can occur:

• a desorption of residual molecules adsorbed on the walls, even in the case of the use of advanced pumping systems,

• and an adsorption on the walls of the compound to be analyzed.

This problem is well known and studied by the spectroscopists, but there is not so much detailed examples available on setups similar to those used by atmospheric spectroscopists. However this phenomenon depends both on the setup (pumping system, size and constitution of the cell ...), and on the molecule of interest (polar or non-polar ...). We wanted to estimate the order of magnitude of the possible bias that can be made in some studies when the adsorption effect is neglected. We are convinced that these adsorption/desorption phenomena on the walls lead to significant errors if they are not taken into account.

The first step is to record the outgassing rate of the residual gas, simply by closing the cell and measuring the total pressure increase. This outgassing rate is assumed to be the same with and without gas. To this end, we carried out a pumping as complete as possible, allowing the "ultimate" vacuum of the cell to be reached. Then the cell was "filled", with a precise amount of gas to be analyzed ("a" on Figure 10). After stopping the addition of gas, we follow the evolution (a priori an apparent decrease) of the partial pressure ("b" in Figure 10) of the compound of interest together with the total pressure. This step can be interpreted as the "sticking" on the walls of the gas of interest, and the desorption of residual gases (ie the degassing of the cell (hatched area in Figure 10)). After the addition of one atmosphere of dry air, the total pressure can only be measured by a 1000 mbar pressure gauge. In this case, the partial pressure of the target gas can only be probed by spectroscopy ("c" in Figure 10). Indeed, measurements of real samples, such as polluted air, will be carried out not only with the molecule of interest, but with the molecule of interest mixed with air. However, the presence of air can modify the equilibrium of the molecules adsorbed on the walls, which can modify the effective cross sections deduced from the experimental observations. More precisely, the addition of air can lead to the unsticking of molecules that were previously adsorbed on the walls of the cell (c on the Figure9), leading to an artificial increase of the effective cross section deduced for the molecule of interest. Finally the molecules readsorb on the cell walls towards a new equilibrium ("d" on Figure 10).

In the present study, 0.13 mbar of 2-EHN was introduced into the cell of the spectrometer. This pressure was determined both by the pressure gauge (total pressure) and by using the previously performed calibration curve (partial pressure). This filling took 10 min. After closing the valve, a decrease in the pressure read by the pressure gauge was noted. This decrease lasts about 50 min, and corresponds to the adsorption of 2-EHN molecules on the cell walls. According to the pressure gauge, the pressure drop reaches 0.105 mbar. But according to the calibration curve, the partial pressure of 2-EHN is lower, at 0.0955 mbar. This difference corresponds to the outgassing rate measured in the emptied cell. After the addition of 1 mbar of air, the spectroscopic monitoring shows an increase of 52% in the partial pressure of 2-EHN. Note that the addition of dry air was performed using a different inlet than the one used for addition of 2-EHN and the common parts were well pumped. The increase of 2-EHN can be attributed to desorption from the cell walls. The effective partial pressure of 2-EHN deduced from the experimental observations just after the addition of dry air is 0.145 mbar. A return to a pressure close to the first equilibrium pressure reached before the addition of dry air is observed after about 50 min and does not seem to have reached the equilibrium Table 6 shows some examples of errors that can be made depending on the time lapse after initial filling.

So depending on when the actual measurement is made, the amount of 2-EHN varies between less than 0.1 mbar and 0.145 mbar. For spectroscopic broadening measurements, for example, the average partial pressure should be adjusted, but its complicated time evolution should still lead to some residual error. For some quantitative chemistry applications the measurement of the initial amount is also not accurate enough and in situ monitoring is an advantage.

Conclusion

In this article, we have accurately (9%) measured the room temperature absorption cross section of 2-EHN, the most used cetane improver, between 850 and 3000 cm -1 , using a FT-IR spectrometer. The results are presented in a 2 columns file containing the absorption cross section as a function of wavenumber. This formatting is directly applicable for measurements at a single wavenumber or the complete set can be integrated into a radiative transfer code.

The main bands have been attributed, thanks to DFT calculations. Geometry optimizations also show the complexity of this flexible molecule, leading to almost iso-energetic conformers with slightly different vibrational spectra.

The co-existence of all these isomers experimentally leads to particularly broad and complex absorption bands with shoulders.

Two approaches based on chemometrics and spectral analysis were used to analyze the cross-section in order to set up a procedure for the accurate quantification of this molecule. The chemometric approach allows for an analysis based on several characteristic bands, which should provide a more robust quantification of 2-EHN in a complex mixture.

We use the absorption cross section to estimate the adsorption processes of 2-EHN molecules on the walls of a stainless steel cell. The perturbation induced by the addition of 1 bar of dry air in the sample was further studied. This approach allows to follow the on-line evolution of the

Figure 1 :

 1 Figure 1: Scheme of the experimental setup used for the present study. "MCT Det" and "LN2" stand for "MCT Detector", "Liquid Nitrogen", respectively.

Figure 2 :

 2 Figure 2: Generation of the absorbance spectrum from the background (I0) and the raw spectrum (I).

A ΔE = 4

 4 .5 kJ/mol 2EHN_1_juillet2020 B ΔE = 0 kJ/mol 2EHN_3_juillet2020 Sum of electronic and ZPE : -594.49497 Hartree Sum of electronic and ZPE : -594.49669 Hartree C ΔE = 11.7 kJ/mol 2EHN_5_juillet2020 D ΔE = 10.9 kJ/mol 2EHN_4suite_juillet2020.out Sum of electronic and ZPE : -594.492244 Hartree Sum of electronic and ZPE : -594.492557 Hartree E ΔE = 6.0 kJ/mol 2EHN_6suite_juillet2020.out Sum of electronic and ZPE : -594.494373 Hartree Table 2: Some stable structures obtained for the 2-EHN with their energetics.

Figure 3 :

 3 Figure 3: Radiance atmospheric spectrum (top) and radiative transfer transmittance simulations of the main absorbing gas listed in terms of their contribution (below) from [19], the arrow show the position of the main absorption bands of 2-EHN.

1 )Figure 4 :

 14 Figure 4: average absorption cross section of 2-EHN (obtained as average of 10 spectra)

Figure 5

 5 Figure 5 reflects the Beer-Lambert's law and the dispersion around this law is a good approximation of the reproducibility of our measurements. The final uncertainty can be calculated as the square root of the quadratic sum of the different systematic errors (temperature, pressure, commercial purity, pressure variation during the recording, path length…) and of the measurement reproducibility. All those contributions lead to an accuracy of 9 %.

Figure 5 :

 5 Figure 5: Y(1283.8) as a function of the pressure x length (left) and difference between the affine function and Y(1283) (right)

Figure 7 :

 7 Figure 7: Determination of the R 2 value associated with the model. Comparison of predicted and measured pressures based on the chemometric analysis of the spectra presented in Table 1.

Figure 8 :

 8 Figure 8: Q-Q (quantile-quantile) plot

Figure 9 :Figure 9

 99 Figure 9: Comparison between a spectrum of pure 2-EHN and the same pressure of 2-EHN with 1013 mbar of air added Figure 9 shows a pure spectrum of 2-EHN and a mixing of 2-EHN and air. As previously discussed, 2-EHN is a polar molecule and the introduction of air will affect the fragile layer of adsorbed molecules leading to an increase of 2-EHN in the cell. The spectrum of the mixing is arbitrarily re-scaled

Figure 10 :

 10 Figure 10: evolution of pressure both with pressure gauge and in situ spectroscopic probing. During the filling (a), the adsorption (b), the desorption due to the dry air (c) and the final adsorption (d)

  adsorption-desorption equilibrium of the active gas due to the large amount of the collision of the added air. The new steady state is almost reached

Table 1 : Experimental conditions for the IR spectra (20 scans for each spectrum) of gaseous 2-EHN that were considered in the present study

 1

	Spectrum	Pressure (mbar)	Temperature	Path length of
	number		(°C)	the cell (cm)
	1	0.025	24.4	31.8
	2	0.027	24.4	31.8
	3	0.033	25.9	68.8
	4	0.053	25.6	68.8
	5	0.098	24.3	68.8
	6	0.101	24.6	68.8
	7	0.102	24.8	68.8
	8	0.103	24.8	68.8
	9	0.133	24.1	31.8
	10	0.181	25.6	68.8

Table 3

 3 shows the calculated wavenumbers as well as associated theoretical intensities and their attributions for the most stable isomers, A and B.

	Isomer A		Attribution	Isomer B
	Wavenumber (cm -1 )	Theoretical intensity		Wavenumber (cm -1 )	Theoretical intensity
	1785	428	N-O twisting	1785	429.0552
	926	173
			N-O stretching
	931	132	linked to the carbon chain	934	240
	1368	126	H methyl wagging,	1366	111
			N-O sym stretching
	1434	108		1432	110
	1385	61		1385	63
	1050	13	C-O asymm
			stretching
	1036	57		1043	58
	1068	21	C-O asymm	1061	41
			stretching
	1068	17

Table 3 : Theoretical IR characterization of the two most stable isomers of the 2-EHN (isomers A and B in Table 2)

 3

Table 4 : Tentative attribution of the experimentally observed vibrations based on the comparison with the calculated wavenumbers for the most stable isomer of the 2-EHN (isomer A in Table 2)

 4

	Experimental Wavenumber (cm--1 )	Tentative attribution	Calculated wavenumber of the main isomer (cm -1 )
		NO 3 umbrella mode	792
	855	N-O stretching linked to the
		carbon chain	926; 931; 934
		N-O stretching linked to the	926; 931; 934
	990	carbon chain
		C-O asymm stretching	1036; 1043; 1048
	1280 1385	H methyl wagging, N-O sym stretching	1338; 1366; 1368; 1385
	1470	H methyl wagging, N-O sym stretching	1432; 1433
	1630
		N-O twisting	1784
	1655
	2870
	2890	C-H symm stretching of
		the CH 2 linked to the NO 3 group	3048; 3077
	2940
	2975

Table 5 : Presentation of the spectroscopic features of the samples selected for the chemometric analysis: absorbances (in a.u.) at the seven selected wavenumbers for ten spectra. For each registered spectrum, the corresponding partial pressure x lenght (P*L) is indicated in the first column.

 5

	λ (cm -1 )
	P × L of 2-EHN	850.0531	865.0791	1000.072	1290.065	1460.011	1638.073	1660.01
	(mbar.cm)
	0.80	0.014	0.012	0.001	0.010	0.001	0.002	0.028
	0.86	0.013	0.016	0.005	0.014	0.004	0.005	0.033
	2.27	0.021	0.023	0.006	0.025	0.004	0.005	0.067
	3.65	0.043	0.040	0.008	0.041	0.005	0.006	0.113
	4.23	0.047	0.047	0.010	0.051	0.010	0.020	0.136
	6.74	0.100	0.095	0.020	0.088	0.015	0.021	0.240
	6.95	0.071	0.074	0.017	0.080	0.013	0.016	0.219
	7.02	0.070	0.065	0.014	0.081	0.011	0.014	0.225
	7.09	0.081	0.077	0.017	0.079	0.012	0.014	0.216
	12.45	0.141	0.135	0.025	0.136	0.016	0.028	0.380

The random distribution of the residuals obtained after the chemometric treatment of the spectra presented in Table 5.

		6.E-03
		4.E-03							P × L of 2-EHN (mbar.cm)	Residuals
										0.795	5.22E-04
		2.E-03
										0.859	4.65E-03
	Residuals	0.E+00	0	3	6	9	12	15	2.270 3.646	-6.83E-03 1.11E-03
		-2.E-03							6.742	-2.60E-04
										6.949	-2.08E-03
		-4.E-03							7.018	1.71E-03
										7.086	-1.61E-04
		-6.E-03							4.229	9.12E-04
										12.453	4.25E-04
		8.E-03		Experimental value for P × L of 2-EHN (mbar.cm)
	15 Figure 6: y = 0.9969x + 0.0398 2-EHN		Experimental value	Predicted value for P × L of 2-
	of		12				R² = 0.9974			for P × L of 2-EHN
	value for P × L	(mbar.cm)	6 9							(mbar.cm) 0.795 0.859 2.27	EHN(mbar.cm) 0.763 0.700 2.752
	Theoretical		3							3.646 6.742 6.949	3.577 6.793 7.091
										7.018	6.879
			0
										7.086	7.086
				0	3	6	9	12	15
										4.229	4.188
				Experimental value for P × L of 2-EHN (mbar.cm)
										12.453	12.453

Table 6 : Possible Relative error (%) due to the adsorption/ desorption equilibrium, GM means gauge measurement, ISM stands for "in situ measurements".

 6

  2-EHN partial pressure resulting from different adsorption-desorption equilibrium. Indeed, a measurement performed without taking these adsorption/desorption processes into account could easily lead to a 52 % error in the determination of the partial pressure. 11.6812) 1511.2588 (8.9049) 1787.0253 (404.131) 3029.2753 (12.2392) 3037.8036 (1.7421) 3042.1236 (3.4322) 3047.7143 (66.8106) 3054.9309 (40.2853) 3059.612 (19.5037) 3063.566 (30.1643) 3066.9852 (10.4039) 3078.529 (3.6332) 3096.7726 (10.8215) 3099.2275 (47.3738) 3101.4951 (40.3031) 3133.3142 (49.5831) 3138.7886 (31.5489) 3141.0867 (33.8971) 3159.6343 (22.2767) 3167.8719 (3.8632)

	A 2EHN_1_juillet2020freq.out N-O twisting N-O stretching linked to the carbon chain N-O stretching linked to the carbon chain H methyl wagging, N-O sym stretching H methyl wagging, N-O sym stretching H methyl wagging, N-O sym stretching C-H stretching asymm C-O stretching asymm C-O stretching symm C-H stretching asymm C-O stretching asymm C-O stretching asymm C-H stretching symm C-H stretching symm C-H stretching symm H methyl twisting and wagging, N-O sym stretching C-H stretching asymm C-H stretching symm C-H stretching asymm C-H stretching asymm C-H stretching asymm C-H stretching symm 18.753 H methyl twisting and wagging 428.0375 172.9887 131.8025 125.9247 108.1485 60.6929 56.2806 13.1103 56.5542 47.9049 21.0466 17.0478 46.2258 41.9273 41.5464 41.1347 33.9922 31.8938 29.7327 26.8459 25.7462 16.3973 18.8254 11.8722 NO3 umbrella mode 10.9603 CH2-CH3 twisting 10.1029 C-H twisting B 2EHN_3_juillet2020freq.out 429.0552 N-O twisting 239.5948 N-O stretching linked to the carbon chain 60.5901 overall vibration 110.8232 H methyl wagging, N-O sym stretching 109.8515 H methyl wagging 63.0674 H methyl wagging, N-O sym stretching 62.1802 C-H mouvement d'ensemble 57.8258 C-O stretching symm 52.3671 C-H stretching asymm 33.8518 C-H stretching asymm 41.4644 C-O stretching asymm 41.1771 C-H stretching symm 38.416 C-H stretching asymm 35.9236 C-H stretching asym 33.6815 H methyl wagging, N-O sym stretching 31.9234 stretching symm 31.7857 C-H stretching asym des CH2 31.2232 stretching symm 20.5613 C-H stretching asym 18.9443 C-H stretching asym 15.0981 methyls vibration, overall vibration 14.2712 C-H vibration d'ensemble 12.4733 13.7066 C-H stretching symm 11.5562 CH3 rotation 11.5526 NO3 umbrella mode 8.2278 10.994 C-H stretching symm 8.6898 -0.007276 3.292981 -1.555575 1785.44 925.7941 930.886 1368.317 1433.583 1385.218 3101.448 1049.763 1036.122 3133.563 1068.163 1068.045 3051.905 3048.018 3055.14 1348.622 3141.323 3062.863 3135.978 3141.583 3095.73 3169.423 3077.843 1325.716 792.9368 776.523 1501.048 1784.551 934.154 949.1944 1366.278 1432.367 1385.447 3054.488 1043.444 3132.099 3139.785 1061.449 3048.427 3140.837 3105.188 1338.402 3060.944 3143.93 3053.644 3108.678 3154.796 1416.116 1328.826 1395.945 3085.161 1500.955 792.6936 778.6432 1509.208 1502.091 -0.636187 2.304802 0.521825 Sum of electronic and zero-point Energies= H C -594.492557 Energetic, geometric, and vibrational characteristics of all identified isomers A 2EHN_1_juillet2020 Sum of electronic and zero-point Energies= -594.494973 N -3.376696 -0.616721 0.135028 O -3.328060 -1.722019 -0.352101 O -4.332322 0.001798 0.520064 O -2.184392 0.071497 0.289701 C -1.020826 -0.648306 -0.162272 C 0.197779 0.233248 0.071717 C 1.437666 -0.584012 -0.329646 C 0.127915 1.570265 -0.686925 C 2.744407 -0.063444 0.262803 C -0.603533 2.696051 0.038689 C 3.965827 -0.857320 -0.191654 C 5.267018 -0.349303 0.417904 H -1.140328 -0.885664 -1.225577 H -0.948972 -1.585373 0.398571 H 0.259051 0.439149 1.150846 H 1.511929 -0.603428 -1.426991 H 1.310889 -1.629733 -0.015306 H -0.334053 1.394428 -1.668915 H 1.151809 1.902100 -0.896921 H 2.678761 -0.094981 1.359844 H 2.887148 0.992999 -0.002182 H -0.577744 3.617025 -0.552880 H -0.127575 2.910064 1.002589 H -1.649599 2.447662 0.232041 H 4.033549 -0.819648 -1.287442 H 3.826961 -1.915817 0.067639 H 6.126045 -0.934670 0.074942 H 5.240856 -0.407478 1.512004 H 5.448842 0.697121 0.147529 31.33 (1.0797) 47.7728 (0.1329) 53.617 (0.4399) 62.147 (0.2531) 637.717 (2.6956) 83.5157 (0.2122) 410.42 (0.1336) 490.1574 (0.4658) 527.8026 (0.6946) 729.0058 (3.8554) 732.8997 (3.2043) 776.523 (10.9603) 786.1348 (4.9281) 792.9368 (11.8722) 846.179 (0.6286) 914.285 (1.722) 925.7941 (172.9887) 93.3645 (0.4894) 930.886 (131.8025) 977.5163 (5.4331) 1011.0463 (2.2146) 1036.122 (56.5542) 1049.763 (13.1103) 1068.0458 (17.0478) 1068.163 (21.0466) 1095.1983 (6.1271) 1109.2929 B 2EHN_3_juillet2020 Sum of electronic and zero-point Energies= -594.496687 N 2.978445 -1.130103 -0.070085 O 3.256329 -2.160979 -0.621547 O 3.652394 -0.457034 0.674777 O 1.698933 -0.687221 -0.355924 C 1.335307 0.549061 0.290923 C -0.099577 0.846422 -0.119868 C -1.062436 -0.199475 0.458568 C H 1.304404 3.377995 -0.373205 H -2.888106 -2.269368 0.304071 H -3.478995 -1.100619 1.472641 H -4.654405 -1.530487 -1.337518 H -5.250654 -0.353854 -0.160115 H -5.378008 -2.084872 0.177802 37.7606 (0.3517) 43.2831 (0.2473) 43.7817 (0.7851) 63.9145 (0.0869) 68.6074 (0.3064) 381.932 (0.5234) 395.0889 (0.3713) 491.1555 (0.8895) 507.5438 (0.2058) 632.8833 (4.4388) 731.2835 (6.0996) 732.8982 (1.2854) 778.6432 (8.2278) 784.1002 (2.8513) 792.6936 (11.5526) 867.1488 (5.0238) 911.5005 (0.7955) 917.0669 (3.4569) 934.154 (239.5948) 949.1944 (60.5901) 1005.0813 (2.153) 1043.4446 (57.8258) 1051.1523 (4.5453) 1061.449 (41.4644) 1068.1006 (3.5624) 1094.2955 (0.9502) 1103.6392 (1.7617) 1164.9688 (1.5619) 1180.8117 (2.5383) 1186.6172 (0.9909) 119.5392 (0.2396) 1235.5098 (0.882) 1253.6714 (2.2024) 1282.6476 (3.4634) 1290.5735 (1.0372) 1318.5809 (0.4956) 1328.8269 (14.2712) 133.721 (0.0932) 1338.4026 (33.6815) 1343.6752 (7.5567) 1366.2782 (110.8232) 1385.447 (63.0674) 1395.9456 (12.4733) 1416.1165 (15.0981) 1417.1518 (2.5641) 1423.3152 (3.876) 1432.3674 (109.8515) 1483.5371 (2.0965) 1488.1224 (1.0922) 1491.4684 (0.5519) 1498.5179 (2.0973) 1500.955 (11.5562) 1502.091 (8.6898) 1505.8227 (5.9652) 1509.2087 (10.994) 1511.1225 (6.0251) 156.5173 (0.2551) 165.7132 (1.0003) 1784.5519 (429.0552) 216.9998 (0.1958) 236.2038 (0.2186) 249.8445 (0.1117) 264.0217 (0.9608) 3032.3062 (2.6214) 3037.3877 (6.3018) 304.3471 (0.0238) 3043.0359 (0.9983) 3048.427 (41.1771) 3053.6443 (31.2232) 3054.4882 (62.1802) 3060.9446 (31.9234) 3071.1351 (0.9) 3085.1619 (13.7066) 3089.7812 (4.3355) 3105.1886 (35.9236) 3108.6784 C -2.015167 -0.434612 -0.131481 C N 3.205677 -0.397868 0.074472 -1.718783 3.012677 -0.287660 C O 3.022177 -0.751132 1.214809 -2.501285 -1.849914 0.169271 C O 4.189721 0.088950 -0.413613 -3.985894 -2.043356 -0.115218 H O 2.179955 -0.552038 -0.848826 1.760887 1.071038 1.123319 H C 0.927329 -1.068771 -0.353069 1.952370 2.151250 -0.280439 H -0.099300 1.268448 -1.296381 H -0.350030 -0.418591 1.237561 H 0.027977 -1.020247 -0.371389 H 0.123021 3.049816 0.799150 H -1.051603 1.943273 1.471743 H -2.210335 -0.202866 -1.188778 H -2.612399 0.276785 0.454976 H -2.170470 3.827647 0.287221 H -2.519887 2.328989 -0.583240 H -1.300450 3.445147 -1.203469 H -2.293928 -2.086746 1.221749 H -1.916094 -2.566045 -0.423313 H -4.307097 -3.066042 0.106690 H -4.216354 -1.844081 -1.168105 H -4.597291 -1.364570 0.490534 C -0.164048 -0.005111 -0.312558 C E 2EHN_6suite_juillet2020.out -1.462079 -0.701412 0.127043 C Sum of electronic and zero-point Energies= -594.494373 0.188296 1.186268 0.595581 C N -3.082320 -0.693639 0.278399 -2.732743 0.069182 -0.220685 C O -3.632966 -0.900423 -0.777952 0.970036 2.306262 -0.087254 C O -3.488761 -0.877206 1.394552 -4.003889 -0.616458 0.272038 C O -1.801761 -0.170136 0.245005 -5.272136 0.145042 -0.094529 H C -1.286216 0.059690 -1.083156 1.091139 -1.511243 0.632390 H C 0.137937 0.584171 -0.954730 0.673890 -1.862716 -1.062393 H C 1.045161 -0.421205 -0.234841 -0.307095 0.365945 -1.338638 H C 0.220637 2.021998 -0.410865 -1.421473 -0.877664 1.212175 H C 2.529430 -0.071532 -0.284146 -1.525223 -1.695701 -0.338955 H C 0.004385 2.201413 1.089544 0.737709 0.819224 1.472837 H C 3.416009 -1.148532 0.334950 -0.743385 1.611441 0.987576 H C 4.898334 -0.796703 0.296705 -2.788310 0.196346 -1.311374 H H -1.308270 -0.887272 -1.632205 -2.687934 1.082682 0.200845 H H -1.935959 0.777260 -1.594820 1.187441 3.114457 0.618335 H H 0.475562 0.633912 -2.002555 0.389597 2.734652 -0.912322 H H 0.899719 -1.409366 -0.694790 1.920718 1.964230 -0.505220 H H 0.725240 -0.527564 0.809043 -3.949403 -0.737097 1.362570 H H -0.505446 2.637087 -0.959587 -4.051903 -1.632328 -0.143671 H H 1.205188 2.425875 -0.676872 -6.167651 -0.367293 0.271259 H H 2.709731 0.879133 0.235960 -5.370301 0.250910 -1.180954 H H 2.833655 0.090672 -1.329114 -5.266970 1.152972 0.335867 H 0.088047 3.259602 1.357470 -0.465176 2.275324 0.310514 C (20.5613) 3132.0992 (52.3671) 3139.7853 (33.8518) 3140.8379 (38.416) 3143.9307 (31.7857) H 0.754008 1.658013 1.674362 -2.440012 -0.224091 -0.198166 C 0.215415 3.384227 -0.486558 C H -0.979701 1.849393 1.408897 3154.7966 (18.9443) H 3.249602 -2.099284 -0.189719 -3.364202 -1.283778 0.395500 C H 3.104608 -1.317749 1.374718 -4.736804 -1.316372 -0.265781 H 1.422260 0.425596 1.376332 H 2.031413 1.327897 -0.030580 H H H H H H H H -0.135032 4.367943 -0.158834 C -0.531096 -0.243328 0.167102 -2.926068 0.757297 -0.111410 C 0.038133 1.127442 -0.213829 -2.320994 -0.411622 -1.275068 C 1.536707 1.198549 0.061489 -1.550232 2.392952 0.216162 O 2.266886 0.214614 -0.698925 -0.245028 2.393287 1.381388 O 3.297085 -1.678846 -0.715410 -0.612412 -1.194599 0.354251 O 2.606465 -0.918386 1.199370 -1.171197 -0.020524 1.539182 N 2.756988 -0.880832 0.001843 -0.145117 0.794691 -1.218171 C H 5.510092 -1.586456 0.744393 2EHN_5_juillet2020 Sum of electronic and zero-point Energies= H 5.099484 0.130445 0.845586 -594.492244 H 5.244522 -0.651903 -0.733174

(

Acknowledgements

The authors would like to thank Dr. Christof Janssen of the LERMA Laboratory, for final checking and comments.

Supplementary Information

(The spectrum will be added as a .dat file. The fortran file will also be added)