Tutorial: Coherent Raman light matter interaction processes

Coherent Raman scattering processes such as coherent anti-Stokes Raman scattering and stimulated Raman scattering are described in a tutorial way keeping simple physical pictures and simple derivations. The simplicity of the presentation keeps however most of the key features of these coherent and resonant processes and their intimate relation with spontaneous Raman scattering. This tutorial provides a digest of introduction to the fundamental physics at work, and it does not focus on the numerous technological implementations; rather, it provides the concepts and the physical tools to understand the extensive literature in this field. The presentation is made simple enough for under-graduate students, graduate students, and newcomers with various scientific backgrounds.Coherent Raman scattering processes such as coherent anti-Stokes Raman scattering and stimulated Raman scattering are described in a tutorial way keeping simple physical pictures and simple derivations. The simplicity of the presentation keeps however most of the key features of these coherent and resonant processes and their intimate relation with spontaneous Raman scattering. This tutorial provides a digest of introduction to the fundamental physics at work, and it does not focus on the numerous technological implementations; rather, it provides the concepts and the physical tools to understand the extensive literature in this field. The presentation is made simple enough for under-graduate students, graduate students, and newcomers with various scientific backgrounds.

The stimulated Raman effect 1 was discovered accidentally by Eckhardt in 1962 shortly after the first ruby-laser action was demonstrated 2 . This is because the weak molecular stimulated Raman cross section requires laser flux density of the order of 10 8 W/cm 2 that can only be achieved through stimulated emission of radiation. In 1965 Maker and Terhune who were studying nonlinear wave mixing, reported a four wave mixing process that can be made resonant with a molecular vibration 3 , the coherent anti-Stokes Raman scattering (CARS) effect was just discovered. Since this early time and until the early 80's both stimulated Raman scattering (SRS) and CARS light matter interaction processes were extensively discussed and used in the context of nonlinear optical spectroscopy [4][5][6][7][8][9] . The first implementation of CARS in the context of imaging was reported in 1982 by Duncan 10 using two synchronized dye lasers pumped by an Ar-Ionlaser whereas the real revival using solid state lasers was reported in 1999 by Zumbusch 11 , a seminal work that launched a very active field known nowadays as coherent Raman scattering (CRS) imaging among which the first implementation of SRS microscopy imaging has been reported in 2007 [12][13][14] . This active field has been recently reviewed 15,16 and takes advantage of the last technological developments in the fields of lasers, detectors and fast imaging modalities to improve the ability of CRS microscopes to image a variety of compounds in the fields of biology, chemistry and material sciences. The key asset of the CRS technology is being label-free contrary toïňĆuorescence. Coherent Raman processes are addressing chemical bonds that are inherently present in matter.
The basics of coherent Raman light matter interaction processes can be found in reference text books such as 15,[17][18][19][20][21] . However the information is sometime scattered or embedded in more general considerations that require significant efforts from the reader to be brought together. The scope of this tutorial is to present the fundamental basics of CRS in a concise yet rigorous manner. The content is certainly not new but is aimed to provide to newcomers in the CRS field the necessary background to grasp the physics at work that is common to the numerous technical implementations that have been reported so far.

I. MOLECULAR RESONANCES
The light matter interaction in the near infra-red (NIR) is dominated by the absorption of molecular vibrational levels, the latter giving birth to narrow or broad absorption bands for electromagnetic radiation with wavelengths ranging from 3 µm to 1000 µm 22 .
Absorption is a resonant process that can be viewed in the simplest case as the interaction of a monochromatic electromagnetic wave with a molecule being modelled as a mass-spring system. We will go through this simple picture to present the NIR molecular absorption, the spontaneous Raman process, and later on, the coherent Raman processes such as CARS and SRS.

A. Harmonic oscillator
The harmonic oscillator is an ideal physical object whose temporal oscillations is a sine wave with constant amplitude and with a frequency that is solely dependent on the system parameters. It is found in many fields of physics and it is a good approximation of physical systems that are close to a stable position. In the mechanical framework the simplest harmonic oscillator is a mass m attached to a spring with a stiffness k (Fig. 1). Being set vertical the mass feels the gravity attraction g and its center of mass is described by: It is possible to study the mass displacement x from its equilibrium position x 0 . Using the fundamental principals of dynamics (energy conservation), it can be shown 23 that the mass displacement x follows where ω 0 = k/m is the system resonant frequency.
x 0 x m FIG. 1. A mass-spring system. x 0 is the equilibrium position and x the relative displacement.

Damped harmonic oscillator
Consider now that the mass experiences a friction force: F = −2γm(dx/dt), where γ is the damping. This force is zero when the mass is at rest, but increases with the mass speed.
The equation describing the mass mouvement is now 23

B. Driven harmonic oscillator
We are now interested in the mass movement when it is driven by a periodic excitation force F (t) = F 0 cos(ωt), where F 0 is the force amplitude and ω its angular frequency. The mass movement equation becomes It is convenient to use the complex notation F (t) = F 0 e −iωt , with this notation a phase lag is noted positive.
We are now looking for a displacement solution of the form x(t) = x(ω)e −iωt . Inserting this generic solution in Eq.4, it comes where x(ω) can be expressed as Close to the resonance (ω ≈ ω 0 ), and for small damping coefficient (γ ≪ ω 0 ), the harmonic oscillator solution 6 can be approximated by a complex Lorentzian fonction C. Study at resonance The complex plane [Re(x), Im(x)] is a nice way to display this Lorentzian function ( Fig.   2(a)) as the displacement x(ω) describes a circle when the angular frequency ω varies 24 .
Im Denoting ρ and ϕ the amplitude and phase of the displacement x(ω) = ρ(ω)e iϕ(ω) . Figure   2(b) displays the displacement amplitude squared with the excitation frequency ω. Figure   2(c) shows the phase response. We note that the system response can be split into three regimes depending the value of ω (Fig. 2(d)): 1. ω << ω 0 . The displacement amplitude is weak and its phase is close to zero. The oscillation is in phase with the excitation drive.
2. ω = ω 0 . The displacement amplitude is strong at its phase is π/2. The oscillation is lagging by 90 • with respect to the excitation drive (phase quadrature).
3. ω >> ω 0 . The displacement amplitude is weak and its phase is close to π. The oscillation is lagging by 180 • with respect to the excitation drive (phase opposition).
Using a simple oscillator model we have described the phenomenon of resonance. We will now move to vibrational molecular resonances that can be probed by an electromagnetic wave.

A. Vibrational modes
A molecular structure sustains many vibrational and rotational intra-molecular vibrations that can be described in terms of normal modes. Each of these "roto-vibrational" normal mode is independent from the other normal modes, it has a center of mass that is preserved and a specific energy that is quantized as described by the quantum mechanical theory.  figure 3, other molecules may exhibit other modes of vibration such as scissoring modes, twisting modes, rocking modes, torsion modes and wagging modes 25 . A molecule having a linear geometry will have only 3n−5 normal vibration modes because any rotation along its molecular axis keep the molecule unchanged. Therefore diatomic molecules (n = 2) will have only one normal vibration mode.

B. Modelling a diatomic molecule
For the sake of clarity we will consider now a single and isolated diatomic molecule. We suppose that this molecule is made with two nuclei, separated by x 0 , that are considered as two points with masses m 1 et m 2 . We consider now that the elongation vibrational mode can be described by a simple harmonic oscillator with a resonant frequency Ω R . Within this simple picture the two nuclei are connected with a spring as displayed in figure 4(a).

C. Infrared (IR) absorption
We consider now a polar diatomic molecule where an asymmetric distribution of positive and negative charges induces a dipole moment p. We suppose that the first atom holds the charge q whereas the second atom has the charge −q. The dipole moment of this We consider now the molecule embedded in an electromagnetic field − → E (t) with angular frequency ω that is linearly polarized along the molecule axis. This electric field generates a force on the molecule F lorentz (t) = qE 0 e −iωt and the elongation response of the molecule x follows a driven harmonic oscillation (Eq. 4) with µ the reduced mass: µ = m 1 m 2 /(m 1 +m 2 ). The damping term γ expresses the radiation loss of this oscillating dipole.
Let's consider now a macroscopic medium constituted with N of these molecules. Under the electric field influence, the electronic cloud displacement of each molecule induces a polarization P (ω) = Nqx(ω) .
The linear electronic susceptibility χ (1) is defined as Comparing equation 10 with 9, and introducing the solution x(ω) of equation 8, we get which gives the vibrational contribution to susceptibility χ (1) of the molecular assembly 26 .
For diluted media, the real part of χ (1) is related to the refractive index dispersion (figure where n 0 is the mean refractive index of the medium. whereas the imaginary part of χ (1) is related to the absorption coefficient of the medium where α is often expressed in cm −1 , and is the key parameter involved in the Beer-Lambert law that describes the intensity loss of a light beam that propagates in a medium over the distance L IR absorption is a powerful spectroscopic method to identify and quantify the absorption bands corresponding to molecular species in a sample 22 . : To understand the Raman effect let's consider again the diatomic molecule described earlier. The molecule is not necessary polar but its polarizability (mostly from electronic origin) depends on the intra-molecular distance x. This distance fluctuates at the resonant molecular bond frequency

D. Spontaneous Raman scattering
, where x f is the amplitude fluctuation. For tiny displacements, it is possible to perform a Taylor expansion of the polarizability α(t) near its initial value α 0 The exciting field E(t) = E 0 cos(ω p t) induices a dipole moment into the molecule Introducing the polarizability expression 15 into 16, we get The first term with frequency ω P describes the Rayleigh scattering (Fig. 6). The second term with frequency ω S = ω P − Ω R describes a red shifted scattering known as the Raman Stokes scattering. In this case the molecule moves from its ground state |f to an excited vibrational state |v (phonon creation) whereas the incoming field loses energy. The third term with frequency ω AS = ω P + Ω R describes a blue shifted scattering known as the anti-Stoke Raman scattering where the molecule goes from the excited vibrationnel level |v toward its ground state |f giving energy to the incoming photon (and absorbing a phonon).
Experimentally, the anti-Stokes scattered intensity is less than the Stokes scattered intensity ( Fig. 7). This is because at thermal equilibrium atomic level populations are described by the Boltzman statistic telling that the excited level |v is less populated than the ground state. Stokes and anti-Stokes scattered intensity become equally intense only for infinite temperature.
The selection rules for a vibration to be Raman active tells that the vibration must affect the polarizability: ∂α ∂x 0 = 0. Similar to IR absorption spectroscopy, Raman spectroscopy enables to identify and quantify the chemical composition of a medium looking at the inelastic scattered Raman spectrum. However a normal mode of a molecule with inversion symmetry is Raman active and not IR active or vice versa. Vibrations symmetric with respect of the inversion symmetry are Raman active, anti-symmetric are IR active.
The Raman scattered field can be depolarized as compared to the incident electromagnetic field. This is described by the Raman depolarization ratio ρ R (Fig. 8) This coefficient ρ R differs for various roto-vibrational modes and varies between 0 and Contrary to IR absorption, confocal Raman microscopy can perform imaging with a sub-micron resolution, however the Raman scattering efficiency process is extremely weak.
Raman scattering cross sections are of the order of 10 −30 cm 2 (as compared to a 1 photon absorption fluorescence cross section that reaches 10 −16 cm 2 , 29 ).

Nonlinear optics
Nonlinear optics encompasses optical processes that result from the nonlinear response of a medium to the incoming electric field 30 . These processes appear when the amplitude of the electric field becomes large as compared to the intra-atomic field (typically, E Coulomb = 1 4πε 0 e a 0 2 ≈ 3.10 11 V/m, where a 0 ≈ 5 · 10 −11 m is the Bohr radius). Nonlinear optical phenomena were observed when the laser was invented in 1960. One year after this discovery, the group from P.A. Frenkel could observe the first second harmonic generation (SHG) 31 .
The same year W. Kaiser and C.G.B. Garret could generate two photons excited fluorescence (TPEF) 32 . In 1965, most of the nonlinear processes were already discovered, among them are the coherent Raman processes. Coherent Raman scattering (CRS) is a resonant and coherent process that allows to gain a factor of 10 7 in efficiency as compared to spontaneous Raman scattering 30 . Let's consider two incoming plane waves, that we denote pump and Stokes, with frequencies ω P and ω S , respectively. These waves interact with a medium having a roto-vibrational vibrations Ω R .

Coherent Raman scattering processes
The total field can be written where "c.c." is the complex conjugate.
The interference between these two fields generates a beating ( Fig. 9) with the frequency where is the time average over one optical period, and K = k P − k S . If the frequency difference Ω = ω P − ω S is set to Ω = Ω R , the roto-vibrational mode enters in resonance with the wave beating.

Excitation force calculation
Let's consider again our driven harmonic oscillator describing a diatomic molecule The energy necessary to create a dipolar moment p(t) = ε 0 α(t) E(t) is given by 30 Similar to spontaneous Raman scattering, we assume that the polarizability is related to the intra-molecular distance following equation 15. Inserting 23 in 25, gives the excitation force produced by the two incoming fields on the molecule.

Harmonic oscillator solution
To solve Eq.24 considering the force 26, we look for a solution as Near the resonance ( Eq. 7), the molecular vibration amplitude is given by If the beating frequency is such that Ω = Ω R , the molecular vibration amplitude becomes large and the excitation fields will now also induce nonlinear polarizations that will be specific to the molecular resonance.

Induced nonlinear polarization
If N denotes the molecular density. The induced polarization in the medium is given by which is the sum of a linear polarization P L (z, t) = Nε 0 α 0 E(z, t) and a nonlinear polar- Expanding this expression, it appears that the nonlinear polarization radiates at 4 different frequencies: ω AS = 2ω P − ω S , ω CS = 2ω S − ω P , ω P and ω S .
where the complex amplitudes P (ω AS ), P (ω CS ), P (ω P ), P (ω S ) are Figure 10 shows the energy diagrams associated to these 4 different processes (following the polarization order above): • Coherent anti-Stokes Raman Scattering (CARS), • Coherent Stokes Raman Scattering (CSRS), • Stimulated Raman Loss (SRL), • Stimulated Raman Gain (SRG). In the following we will not consider the CSRS process whose frequency ω CS is in the IR domain where measurements are noisy and hampered by non ideal detectors. We will concentrate first on the CARS process and latter on the SRL and SRG processes.

A. Resonant and non-resonant CARS processes
The CARS process was first observed by Maker and Terhune en 1965 3 while Levenson and Bloembergen presented a detailed investigation 1972 33 . One of the discovery was to find that the CARS signal at frequency ω AS was always present, even if the frequency difference between the pump and the Stokes beams didn't match a molecular vibrational resonance (Ω = Ω R ).
To understand this, let's consider the Jablonski diagram in Fig. 11(a). As described earlier, the resonant CARS process involves the beating between the pump and the Stokes beam frequency Ω that is resonant with the molecular vibration at frequency Ω R . This molecular vibration is further probed by the pump pulse to generate the anti-Stokes signal. However there is another route, using the frequencies ω P and ω S , to generate a scattered light with frequency ω AS (Fig. 11(b)). This other four wave mixing (FWM) process is known as the non-resonant CARS as the exciting beams do not interact with the molecular vibrational state, rather it originates from the instantaneous electronic response of the medium.

B. Nonlinear polarization and susceptibility
Formaly, the nonlinear CARS polarization at location r reads where χ (3) is the third order nonlinear susceptibility, a third rank tensor that describes the possible interaction between the pump and Stokes exciting fields with the medium.
Avoiding for a short time the pump and Stokes field spectral dependance, the i -th Cartesian component (i =x,y,z ) of the nonlinear polarization P (3) generated in point r, reads 34 where indices j, k and l permute over spatial coordinates x, y and z.
It can be shown that in the case of isotropic media, where one photon transition at ω P and ω S are not occuring, the nonlinear polarization can be simply expressed in terms of the tensor element χ xxyy and the Raman depolarization ratio ρ R 35,36 .
from which we note that 1. When the pump and Stokes beam are linearly polarized and with the same polarization state, the induced nonlinear polarization is collinear with the excitation fields. Some more information on polarization resolved CARS can be found in 37 .
For the sake of simplicity, we will consider from now that the Raman line is totally polarized (ρ R = 0). Furthermore, we suppose that the pump and Stokes beams propagate along the z direction, with the same and linear polarization state. In these conditions where xxyy . We point out that tackling the problem using 1-D plane wave propagation is useful to capture basic properties of nonlinear optical microscopy. However, a full consideration of focused fields in 3D is necessary to provide a more accurate picture.

C. Anti-Stokes field generation and propagation
We concentrate in this section on the generation and propagation of the anti-Stokes field. Under the slowly varying enveloppe approximation, the anti-Stokes E AS (z, t) = A AS e i(k AS z−ω AS t) + c.c. field propagation writes 38 With the expression of the polarization P (ω AS ) defined in (39), it comes where ∆k = ∆ k · e z = (2 k P − k S − k AS ) · e z .
Suppose that the anti-Stokes signal is generated in a medium with length L. The anti- with sinc(x) = sin(x)/x.
From which we get the expression of the anti-Stokes intensity It is interesting to note that the anti-Stokes intensity in z = L is proportional to sinc 2 (∆kL/2) ( Fig. 12(a)). The anti-Stokes signal is efficiently generated only if ∆kL/2π ≪ 1 because the total anti-Stokes field is the result of the coherent summation of the anti-Stokes fields emitted from each points along z, and this interference is constructive only if all fields are in phase. For this, the anti-Stokes field must be in phase with the induced nonlinear polarization for every points along the z axis. This is what is known as the 'phase matching condition' that requires the wave vectors being In the considered collinear situation ( e P = e S = e AS = e z ), the phase matching condition is rarely fulfilled (∆k = 0) because of the medium dispersion (n P = n S = n AS ). It is useful to define the "nonlinear coherence length" L c as the length over which the maximum of the anti-Stokes signal develops When the interaction length of the exciting fields L is larger than the nonlinear coherence length, the anti-Stokes signal generation becomes less efficient (Fig. 12(c)) because of destructive interferences. It is therefore important to minimize ∆k to get the longest possible coherence length. In the context of microscopy, CARS signal is efficiently generated for two reasons. First, the phase matching condition is relaxed when the pump and Stokes beams are strongly focused into the nonlinear medium. The broad angular spectrum of the wave vectors k allows many combinations for the pump, Stokes and anti-Stokes wave vectors to satisfy the relation 44. Second, a tight focusing of the exciting fields leads to a small nonlinear excitation volume, of the order of 10µm, such that the interaction length L is too short for destructive interference.

D. χ (3) in the spectral domain
We focus in this section on the anti-Stokes intensity spectrum. We have seen previously that the CARS signal has two contributions.
• A resonant contribution described by the susceptibility χ with a = − (2Nε 0 /6µΩ R ) (∂α/∂x) 2 0 , a negative number that represents the oscillator strength of the molecular vibration.
• A non-resonant contribution described by the susceptibility χ The total CARS susceptibility writes Introducing this expression in 43, we get Where it appears that the anti-Stokes intensity is the sum of three contributions that we describe now following their order.
• A resonant contribution that contains all the vibrational mode spectral information.
This is the one that is most wanted in spectroscopy applications.
• A non-resonant contribution that is constant in the spectral domain.
• A resonant and non-resonant mixing contribution know as the heterodyne contribution, proportional to the real part of χ R .
Therefore, one can see the anti-Stokes spectral signal as the output of a two-waves interferometer, the interference being the heterodyne term. Figure 13 shows the spectral behavior of the three contributions, together with the total resulting anti-Stokes signal. Because of the heterodyne term the CARS spectrum is distorted as compared to the Raman spectrum. The anti-Stokes maximum signal is blue shifted from Ω R and an intensity minimum appears on the red side of the intensity maximum. Furthermore the CARS spectrum is not symmetric on both sides of Ω R . We will now focus on the spectral evolution of the nonlinear susceptibility χ (3) in the complex plane. As every resonance, it can be described as a circle, as already presented in the case of the spring resonance. Figure 14(a) shows the resonant contribution χ N R (in red). This shift has strong consequences on the χ (3) phase and on the CARS intensity 40 (Fig. 14(b)).
• As we have seen previously, the CARS maximum intensity, or equivalently χ (3) , (point 2) is shifted with respect to the Raman resonance (point 3).
• The χ (3) phase is lower than π/2 at the Raman resonance (point 3). Its spectrum shows a maximum (point 4) that is red shifted with respect to the Raman resonance.
• The CARS minimum intensity is in point 5, red shifted with respect to the Raman peak.
In practise the coherent nature of the CARS process makes the resonant and non-resonant contributions difficult to separate. These two contributions manifest themselves as: • At resonance, the resonant anti-Stokes field is dephased with respect to the anti-Stokes non-resonant field 41,42 .   • In the case of the resonant CARS process, the vibrational level is populated in a coherent way. This coherence is conserved during a time T 2 (known as the "coherence time"), of the order of the picosecond. The CARS signal generation is efficient only if the second photon with frequency ω P probes the vibration within a time shorter than T 2 as demonstrated in 43 . In the case of the non-resonant CARS, T 2 is much shorter (few hundreds of femtoseconds).
Even with this non-resonant background the CARS process is a powerful spectroscopic tool. CARS can bring similar information than spontaneous Raman but its resonant and coherent character makes it much more efficient and therefore, compatible with imaging.

IV. THE SRS PROCESS
In this section we will describe in detail the SRL and SRG processes. Those two processes and intimately linked and are usually described under the same stimulated Raman scattering (SRS) process. They were discovered in 1962 by Woodbury and Ng 45 as they were studying Q-switching processes in a Ruby laser containing a nitrobenzene Kerr cell. They could detect a strong IR radiation coming from the Kerr cell which remained mysterious for some times.
Some months later, Woodbury and Eckhardt hypothesized that this was coming from a stimulated Raman process that could be confirmed experimentally 1 . A detailed SRS process description can be found in review articles from Hellwarth 4 , Bloembergen 6 , and Shen and Bloembergen 5 .

A. Coherence and interferometry
As in the CARS process, we define the third order nonlinear suceptibilities χ R (ω S ) associated to the SRL and SRG processes, respectively, through their respective induced polarizations Contrary to the CARS process, the SRS polarizations are induced at the same frequencies than the incoming pump and Stokes beams. The pump and Stokes field generate P (ω P ) that, itself, generates a weak "nonlinear" field E P e i(k P z−ω P t) + c.c.. Having the same frequency as the incoming pump beam, and being coherent, this weak "nonlinear" field interferes with the incoming excitation pump field E P . As a result the SRS signal can be viewed as the result of an interference between a weak "nonlinear" field, with the strong incoming field. To understand better this phenomenon let's consider the nonlinear medium as a single dipole located in z = 0. The resulting intensity of this interference can be written as In a similar way the weak "nonlinear" field E S e i(k S z−ω S t) + c.c. interferes with the strong incoming Stokes field E S .
The second term A (3) 2 is negligible in comparison to |A| 2 . However, the quantity AA (3) can be significant and depends on the phase difference between the generated weak field and the incoming strong field. Let's have a closer look at this phase difference term.
The generated fields are related to the nonlinear polarizations by the expressions 46 : Note that the above mentioned expressions between the generated fields and the induced polarizations are strictly true in the molecule far field but a more complex approach leads to the same final result. From the induced polarizations 52 and 53, we get Comparing 34 and 35 with 52, 53, and inserting the expression of x(Ω) previously computed (Eq. 28), we get the nonlinear susceptibility involved into the SRL and SRG processes (60) Where a is a negative constant that describes the oscillator strength of the molecular vibration (a = − (2Nε 0 /6µΩ R ) (∂α/∂x) 2 0 ). It appears the SRL susceptibility is intimately linked to the resonant CARS susceptibility (Eq. 47) as χ R (ω P ) = 1 2 χ R (ω AS ). Furthermore, the SRL and SRG processes are symmetrical χ R (ω S ) = χ R (ω P ) * .
At resonance, we note that ϕ χ  (61) Where it clearly appears that, at resonance (Ω = Ω R ), the pump intensity experiences a depletion whereas the Stokes field experiences a gain, this is the origin of the terminologies "Stimulated Raman Loss" and "Stimulated Raman Gain". We focus now on the interaction of the pump and Stokes fields with a nonlinear medium of length L (Fig. 16. Within the slowly varying envelope approximation, the propagation equations for the pump and Stokes fields write

B. Field propagation
Inserting the nonlinear polarization expressions 52 and 53, it comes One of the feature of the SRS process is that phase matching is always fulfilled as ∆ k = ( k P − k P + k S − k S ) = 0. Therefore the nonlinear field generated at distance z is always in phase with the nonlinear fields generated in any other distances z' in the medium. Let's compute now the total field resulting from the interference between the nonlinear fields and the incoming excitation field. For the sake of simplicity we consider that the incoming Stokes beam (respectively the incoming pump beam) doesn't feel the result of its possible weak intensity variation when it generates the weak nonlinear pump field (respectively the weak nonlinear Stokes field) in equation 63 (respectively 64). This often valid hypothesis is usually refereed as "non depleted pump" approximation, where the "pump" terminology refers to the excitation field that induces the nonlinear parametric process 20 . Within this hypothesis it comes Where it is useful to separate χ R into its real and imaginary parts. Furthermore, we have seen previously that, following equation 60, χ R (ω P ) * . In the following we set χ Where it appears clearly that the pump amplitude loss and the Stokes amplitude gain along propagation depend on the imaginary part of the susceptibility Im(χ R ). The real part of the susceptibility induces a variation of the refractive index for the pump (respectively Stokes), depending on the Stokes (respectively pump) intensity. Using the pump intensity I P (z) = 2n P ε 0 c|A P (z)| 2 and the Stokes intensity I S (z) = 2n S ε 0 c|A S (z)| 2 , this refractive index variation writes This is known as "crossed phase modulation" (XPM) and can be viewed as a crossed Kerr effect, the refractive index at one frequency is changed by a wave at another frequency.
Therefore XPM can induce some spurious effects of focusing or defocusing, by one beam on the other. This is the origin of small non-resonant background signals in SRS described in 47 .
Let's focus now on the pump and Stokes intensity after the interaction with a nonlinear medium of length L.
It appears that the pump gain and the Stokes loss have an exponential dependance with the length L of the medium. A property interesting for applications that use long propagation distances such as in fiber optics. In the microscopy context the lenght L where the SRS process is efficient, and that corresponds to the volume where pump and Stokes beam are in focus, is sufficiently small for the quantity within the exponential to be much smaller than 1. Therefore it is possible to perform a first order development I S (L) ≈ I S (0) + 3ω S L 2n P n S ε 0 c 2 Im(χ Where the second terms of expressions 73 and 74 are the pump intensity loss ∆I P and the Stokes intensity gain ∆I S , respectively. We note that these quantities are linked by ∆I P = (ω S /ω P )∆I S . And the SRS process can be viewed as an energy transfer from the pump beam towards the Stokes beam. Another feature of expressions 73 and 74 is that both loss and gain are proportional to Im(χ R ) which is the key property that links directly SRS to spontaneous Raman as we shall see now.

Spontaneous Raman viewed as a stimulated Raman process
In quantum optics, spontaneous emission is described as a stimulated emission from the vacuum field fluctuations. The vacuum fluctuations are present in any allowed electromagnetic modes, in homogeneous media all the k vectors are allowed but in an electromagnetic cavity the local density of state can be different and the vacuum spectrum follows the electromagnetic modes that can exist within the cavity. As a consequence the spontaneous emission of a resonant atom located into such a cavity concentrates in the cavity mode only. Conversely if the cavity has no available mode, the spontaneous emission is hampered.
The possible enhancement or inhibition of the spontaneous emission rate is known as the Purcell effect 48 , it is a direct consequence of the Fermi golden rule that links the spontaneous emission rate and the local density of electromagnetic states 49 .
In the spontaneous Raman scattering process a beam with frequency ω P interact with a medium having molecular species. Because the vacuum fluctuations are present in the Stokes modes with frequency ω S such that ω P − ω S = Ω R , the vacuum fluctuation will be amplified through SRG as described in equation 74 δI S (L) ≈ δI S (0) + 3ω S L 2n P n S ε 0 c 2 Im(χ Where the first term can be neglected as compared to the second one that takes advantage of the heterodyne amplification through I P (0). Because they are coming from fluctuations, the amplified Stokes fields with intensities δI S have a random phases and propagate in every directions. The SRG process can operate between the pump field and any Stokes field directions because in SRS the phase matching is always fulfilled. As a result SRG processes between the incoming pump beam and the random vacuum fluctuations generate incoherent Stokes fields, in every directions, satisfying ω P − ω S = Ω R , it is nothing but the spontaneous Raman scattering process 50 In 1993 Cairo et al put a Raman medium (C 6 H 6 ) in a cavity to study the spontaneous Raman process 51 . They observed that the fluctuation spectrum of δI S (0) was related to the cavity modes. They demonstrated that, as spontaneous emission, it is possible to enhance or inhibit spontaneous Raman scattering for a specific Raman line, just by spectral tuning of the cavity.

C. CARS, SRS and Raman spectral responses
We have seen that the non-resonant background distorts the CARS intensity spectrum.
This is why the CARS spectrum (Eq. 76) is fundamentally different from the Raman spectrum.