Tutorial: Coherent Raman light matter interaction processes Open Access

The stimulated Raman effect¹ 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 10⁸ W/cm² 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 and the coherent anti-Stokes Raman scattering (CARS) effect was just discovered. Since this early time and until the early 1980s, both stimulated Raman scattering (SRS) and CARS light matter interaction processes were extensively discussed and used in the context of nonlinear optical spectroscopy.^4–9 The first implementation of CARS in the context of imaging was reported in 1982 by Duncan¹⁰ using two synchronized dye lasers pumped by an Ar-ion laser, whereas the real revival using solid state lasers was reported in 1999 by Zumbusch,¹¹ a seminal study that launched a very active field nowadays known as coherent Raman scattering (CRS) imaging among which the first implementation of SRS microscopy imaging has been reported in 2007.^12–14 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 in contrary to fluorescence. Coherent Raman processes address 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 Refs. 15 and 17–21. However, the information is sometimes 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.

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.

It is convenient to use the complex notation F(t) = F₀e^−iωt, with this notation a phase lag is noted positive.

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 

Denoting ρ and φ as 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 on the value of ω [Fig. 2(d)]:

1. ω ≪ ω₀: The displacement amplitude is weak and its phase is close to zero. The oscillation is in phase with the excitation drive.

2. ω = ω₀: The displacement amplitude is strong at its phase which is π/2. The oscillation lags by 90° with respect to the excitation drive (phase quadrature).

3. ω ≫ ω₀: 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 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 modes is independent of the other normal modes, and it has a center of mass that is preserved and a specific energy that is quantized as described by the quantum mechanical theory. The atomic mass, the number of involved chemical bonds, the atomic species, the molecule geometry and symmetry, and the possible hydrogen bond interactions affect the stiffness of the vibrational forces at work, which at the end dictate the possible vibrational energies. All these vibrational energies belong to the infrared domain and can be probed by absorption spectroscopy. Rotational modes are only observable when molecules are in the gas phase and correspond to energies ranging from 10 to 400 cm⁻¹, which correspond to the far infrared electromagnetic domain (wavelengths between 25 and 10 000 μm). Roto-vibrational modes correspond to energies ranging from 1 to 4000 cm⁻¹ corresponding to the mid-infrared domain (wavelengths between 2.5 and 25 μm). Near infrared radiations ranging from 4000 to 14 000 cm⁻¹ (wavelengths from 0.7 to 2.5 μm) can also excite overtones of these vibrations.

A molecule having a geometry which is not linear and constituted of n atoms will have 3n − 6 normal vibration modes. For instance, water molecules (H₂O) have 3 modes of vibrations that are shown in Fig. 3, and 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 keeps the molecule unchanged. Therefore diatomic molecules (n = 2) will have only one normal vibration mode.

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 x₀, that are considered as two points with masses m₁ and m₂. 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 Fig. 4(a).

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 two punctual charge system can then be expressed in vector form $p→=qd→$⁠, where $d→$ is the displacement vector oriented from the negative to the positive charge.

IR absorption is a powerful spectroscopic method to identify and quantify the absorption bands corresponding to molecular species in a sample.²² 

At thermodynamic equilibrium, the interaction of light with matter is governed by absorption (the molecule retains the light energy for a certain time) and scattering (the molecule instantaneously scatters the incoming light in a different direction). When a molecule scatters light, most of the scattered photons keep their original frequencies, a phenomenon known as elastic scattering or “Rayleigh scattering” (Fig. 6). However, a small fraction of the incident light is scattered in an inelastic way, i.e., scattered with a frequency that is different from the original incoming light, a phenomenon known as the “Raman effect.” Inelastic Raman scattering was first observed in 1928 by Raman and Krishnan²⁷ in India and independently by Landsherg and Mandelstam²⁸ in USSR. By focusing the spectrally filtered sun light, they could observe new frequencies in the scattered light.

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 vibrational 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 Boltzmann statistic indicating 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 rule for a vibration to be Raman active is that the vibration must affect the polarizability, $∂α∂x0≠0$⁠. Similar to IR absorption spectroscopy, Raman spectroscopy enables to identify and quantify the chemical composition of a medium by 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 to the inversion symmetry are Raman active and those anti-symmetric are IR active.

This coefficient ρ_R differs for various roto-vibrational modes and varies between 0 and $34$⁠.

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 cm² (as compared to a 1 photon absorption fluorescence cross section that reaches 10⁻¹⁶ cm², Ref. 29).

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, $ECoulomb=14πε0ea02≈3.1011$ V/m, where a₀ ≈ 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 Frenkel could observe the first second harmonic generation (SHG).³¹ In the same year, Kaiser and Garrett generated two photon excited fluorescence (TPEF).³² In 1965, most of the nonlinear processes were already discovered; among them were the coherent Raman processes.

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.

Figure 10 shows the energy diagrams associated with 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 later on the SRL and SRG processes.

The CARS process was first observed by Maker and Terhune in 1965,³ while Levenson and Bloembergen presented a detailed investigation in 1972.³³ One of the discoveries 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 did not match a molecular vibrational resonance (Ω ≠ Ω_R).

To understand this, let us 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.

Some more information on polarization resolved CARS can be found in Ref. 37.

We point out that tackling the problem using 1-D plane wave propagation is useful to capture the basic properties of nonlinear optical microscopy. However, a full consideration of focused fields in 3D is necessary to provide a more accurate picture.

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 Δk to get the longest possible coherence length.

In the context of microscopy, the 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 (42). Second, a tight focusing of the exciting fields leads to a small nonlinear excitation volume, of the order of 10 μm, such that the interaction length L is too short for destructive interference.

Therefore, one can see the anti-Stokes spectral signal as the output of a two-wave 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 $χR(3)$ as a dashed black circle. As $χNR(3)$ is real, the total CARS susceptibility $χ(3)=χR(3)+χNR(3)$ also describes a circle (in blue) but is shifted on the real axis by $χNR(3)$ (in red). This shift has strong consequences on the χ⁽³⁾ phase and on the CARS intensity⁴⁰ [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 and red shifted with respect to the Raman peak.

In practice, the coherent nature of the CARS process makes the resonant and non-resonant contributions difficult to separate. These two contributions manifest themselves as follows:

• At resonance, the resonant anti-Stokes field is dephased with respect to the anti-Stokes non-resonant field.^41,42

• If the pump and Stokes field polarization states are different, the resulting resonant and non-resonant anti-Stokes fields will not have the same polarization state. This effect is related to the Raman depolarization ratio and the tensor properties of χ⁽³⁾.³⁵ 

• In the case of the resonant CARS process, the vibrational level is populated in a coherent way. This coherence is conserved during a time T₂ (known as the “coherence time”), of the order of picoseconds. The CARS signal generation is efficient only if the second photon with frequency ω_P probes the vibration within a time shorter than T₂, as demonstrated in Ref. 43. In the case of the non-resonant CARS, T₂ 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. Contrary to spontaneous Raman scattering, the CARS signal is blue shifted as compared to the pump and Stokes beams and not polluted by possible molecular unwanted single photon fluorescence. Regnier and Taran were the first to use CARS spectroscopy to measure specific molecular concentration in gas in 1973.⁴⁴ Begley et al. measured one year later specific molecular concentrations in toluene-benzene mixtures$$solutions⁸ using CARS.

In this section, we will describe in detail the SRL and SRG processes. Those two processes are intimately linked and are usually described under the same stimulated Raman scattering (SRS) process. They were discovered in 1962 by Woodbury and Ng⁴⁵ 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 time. Few months later, Woodbury and Eckhardt hypothesized that this came 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 

The second term $A(3)2$ is negligible in comparison to $A2$⁠. 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 us have a closer look at this phase difference term.