The coherent Raman spectroscopies are four wave mixing processes where three excitation fields create the output coherence and one of the intermediate coherences involves a vibrational state. The WMEL diagrams for the different methods

are sketched in Figure 2. States a, b, and c are typically vibrational, electronic, and a vibrational state of the electronic state, respectively. Often in Raman methods, the electronic states are so energetic that the excitation fields are far from resonance. In this case, the states are indicated by dotted lines and are called virtual states, even though they are actually simply off-resonant molecular states. If the excitation fields are resonant, the methods are now called resonance-Raman methods. The output intensities are typically 10³-10⁴ larger because of the additional resonances.

Both the coherent anti-Stokes Raman spectroscopy (CARS) and coherent Stokes Raman spectroscopy (CSRS, pronounced “scissors”) typically use two excitation beams labeled $\omega_L$ and $\omega_S$ (indicating laser field at $\omega_o$ and Stokes field at $\omega_o-\omega_v$) to create an output at the anti-Stokes frequency relative to $\omega_L$, $2 \omega_L-\omega_S$, or the Stokes frequency relative to $\omega_S$, $2 \omega_S-\omega_L$, respectively (see figure 2a,b). Phase matching is important for optimizing the output intensity. In CARS, the phase matching is $\vec{k}_{CARS} = 2\vec{k}_L - \vec{k}_S$ and in CSRS, the phase matching is $\vec{k}_{CSRS} = 2\vec{k}_S - \vec{k}_L$. CARS is used almost exclusively for coherent Raman experiments because its output is at higher frequencies than the excitation sources and is therefore less sensitive to fluorescence that might be created by the excitation beams.

Experiments are performed by changing one excitation frequency relative to the other and monitoring the output intensity. The output intensity increases by ~10³ when ω_L-ω_S is resonant with a strong vibrational transition. With monochromatic excitation sources, the frequency is changed by scanning the frequency of one relative to another. Multiplex CARS is performed by using a broadband source for one excitation frequency  and spectrally resolving the output frequencies, typically with a monochromator and a CCD camera. This allows one to acquire complete vibrational spectra on a single laser shot. For ultrafast multiplex CARS spectroscopy, a chirped pulse (pulses where the frequency changes rapidly during the pulse duration) with a several picosecond duration and a fast femtosecond pulse (~100 fs) are used for $\omega_L$ and $\omega_S$, respectively. The fast $\omega_S$ pulse overlaps in time with only part of the longer $\omega_L$ pulse and selects the frequencies present in the $\omega_L$ pulse at the time of overlap. The broad band of frequencies in the $\omega_S$ pulse can achieve resonance with many vibrational states at $\omega_L - \omega_S$ so that a series of $ag$ vibrational coherences are created (see figure 2a) whose FID occurs at specific vibrational frequencies. The last excitation is also the chirped $\omega_L$ pulse so the vibrational coherences will interact with the frequencies present in the chirped pulse during the time they overlap and create output frequencies at $\omega_{ag} + \omega_L$. Detection of the signal with a monochromator and a CCD camera then resolves the anti-Stokes frequencies from the different vibrational coherences.

CARS is attractive for spectroscopy because the output signals are large and directional, so interference from incoherent fluorescence can be discriminated against both spectrally using a monochromator and spatially by using an aperture to define the CARS output beam. Phase matching is usually important to achieve reasonable signal levels.

One of CARS most important limitations is the presence of signals that arise from nonresonant electronic states since state a in figure 2a can be a resonant vibrational state or a nonresonant electronic state. Electrons are much lighter than nuclei so the polarization induced by an electric field is usually dominated by the electronic component of the polarization rather than the nuclear component. Resonance with a vibrational state raises its contribution to the polarization so it can dominate over a nonresonant electronic polarization but the nonresonant electronic polarization must always be considered. The relative contributions are described by the nonlinear susceptibilities, $\chi^{(3)} = \chi^{(3)}_{vibrational} + \chi^{(3)}_{NR}$. In the steady state, the vibrational contribution is

$$ \chi^{(3)}_{vibrational} = \frac{\mu_{gb} \mu_{ba} \mu_{ac} \mu_{cg}} {8\hbar^3 \Delta_{bg} \Delta_{ag} \Delta_{cg}}$$

and its importance depends on the resonant enhancement from the detuning factor $\Delta_{ag} = \omega_{ag} - (\omega_L - \omega_S) - i \Gamma_{ag}$. In CARS, a strong Raman transition has a $\chi^{(3)}_{vibrational}$ peak that is ~25x larger than $\chi^{(3)}_{NR}$.

A number of methods have been developed to discriminate against the nonresonant background. Since the nonresonant electronic polarization decays almost instantaneously after the excitation field is turned off, ultrafast pulses can be used to excite a vibrational coherence. Since vibrational coherences typically live for ~1-10 ps, delaying the third pulse relative to the first two can strongly discriminate against any nonresonant electronic coherence but still excite the vibrational coherences to create the output coherence.

Polarization techniques can also be used for discrimination. The output CARS signal is polarized if the excitation beams are polarized. A polarizer can be adjusted to block the signal beam from being detected if the exciting frequencies are not resonant with a vibrational state. However, if the excitation frequencies are changed to a vibrational resonance, the output polarization can change because the $\chi^{(3)}_{vibrational}$ and  $\chi^{(3)}_{NR}$ tensors are different, their transition moments have different dependences on the polarization of the exciting electromagnetic fields. A portion of the output signal can now pass through the polarizer and be detected.

Figure 2c shows their WMEL diagram for Raman gain or stimulated Raman and Raman loss or inverse Raman spectroscopies. The most important difference between these processes is that the output frequency matches an excitation frequency, so the output beam at $\omega_S$ can interfere (heterodyne) with the excitation beam at $\omega_S$. The transitions involving the $\omega_L$ beams are absorption transitions, so the intensity of the transmitted beams decreases, while the transitions involving the ω_s beams are stimulated emission transitions, so the intensity of the transmitted beams increases. There is no need for phase matching considerations when the output frequency matches one of the excitation frequencies since the phases have to be identical. Consequently, these spectroscopies use collinear beams.

Raman gain and loss spectroscopies are performed by changing one excitation beam frequency while monitoring the increase in the transmitted $\omega_S$ intensity or decrease in the transmitted $\omega_L$ intensity, respectively. If there is only a single excitation frequency, $\omega_L$ present and its intensity is high, vacuum fluctuations in the quantized electromagnetic field can act as an $\omega_S$ beam to stimulate emission at $\omega_S$ so that an $\omega_S$ beam can be created from the vacuum. The creation of a beam at $\omega_S$ is stimulated Raman scattering (SRS). Usually, the Raman transition with the largest transition moment dominates the process. Inverse Raman spectroscopy (IRS) is performed with a monochromatic excitation frequency, $\omega_S$, and a broad band source at $\omega_L$. The transmitted light at $\omega_L$ is measured with a monochromator and multiplexing detector like a CCD and absorption lines appear in the dispersed output at $\omega_L$ from vibrational transitions at $\omega_L- \omega_S$. One way to view these processes is to realize that in the absence of a sample, the fields at $\omega_L$ and $\omega_S$ would be independent. However, if a sample is present, the fields can exchange energy using the sample as an intermediary. If the $\omega_S$ beam is initially absent, the $\omega_S$ beam intensity grows and the $\omega_L$ beam intensity decreases.

An ultrafast pulse can have a sufficiently wide range of frequencies (a 100 fs pulse has a frequency bandwidth of ~350 cm^-1) that it alone can provide both the $\omega_L$ and $\omega_S$ frequencies and therefore excite vibrational states by stimulated Raman scattering. This process is called impulsive stimulated Raman scattering (ISRS) and it represents an important way to excite many vibrational states simultaneously.