Nonlinear Spectroscopies

            Nonlinear spectroscopy results when a light field perturbs the optical properties of a molecule so that the subsequent light fields see changes in the molecular states. It is important that the different light fields interact with the molecular states over time periods that are shorter than periods associated with dephasing and population relaxation so that the effects of the first interaction are not lost. By exciting a state with one light field and probing it at a later time, nonlinear experiments allow one to investigate how quantum states evolve in time and how different states are related to each other. We must first understand the evolution of coherences and populations that characterize the different nonlinear methods. We use second order spectroscopies as an example.

 

Second Harmonic Generation (SHG), Sum Frequency Generation (SFG), Difference Frequency Generation (DFG), and Optical Rectification                [back to the top]

            The second order nonlinear spectroscopies of SHG, SFG, DFG, and optical rectification are the simplest examples of nonlinear molecular spectroscopies and will serve as our example of diagrams that describe nonlinear spectroscopy. They are three wave mixing (TWM) methods. Second order spectroscopies vanish for isotropic samples and are therefore surface selective. Figure 1a shows the flow of coherences that describes all the processes. The second order nonlinear spectroscopies involve two transitions from the initial ground state population, gg, to cg for SFG, ba for DFG, or back to gg for optical rectification. SFG is a parametric process where the final emitting coherence involves the initial state, g, so the molecule returns to the ground state after the cg coherence emits. It has only a single coherence pathway, gg→bg→cg. SHG is a special case of SFG where the two excitation frequencies are identical. DFG is a nonparametric process where the final emitting coherence, ba, does not involve the ground state so the molecule is left in an aa population after the ba coherence emits. It has two coherence pathways that interfere, gg→bg→ba and gg→ga→ba. Optical rectification is a parametric process with one pathway, gg→bg→gg. If the excitation beam frequencies are labeled ω₁ and ω₂, SHG, SFG, DFG, and optical rectification have output frequencies at 2ω₁ (or 2ω₂), ω₁+ ω₂, ω₁- ω₂, and dc, respectively.

            SFG and DFG spectra result from scanning one of the excitation frequencies while monitoring the intensity of the output signal from the final coherence. The final coherence emits light at a different frequency than the excitation beams so it is simple to spectrally discriminate between the excitation beams and the output signal. The output intensity increases when the excitation frequencies match molecular resonances with vibrational and/or electronic states. Figure 1b shows the WMEL diagrams that describe the possible resonances for the three processes. Figure 1c shows Feynman diagrams that contain the same information more explicitly. Here, the two vertical lines represent the ket (left) and bra (right) states as time advances vertically from the bottom. Interactions are represented by sloped lines. The quantum states before and after an interaction are indicated by letters. Lines sloped downward from a ket or bra vertex are absorption events ( for ket side interactions and  for bra side interactions) and lines sloped upward from a vertex are emission events ( and  for ket and bra side interactions, respectively.) The resonance enhancements occur when the arrows or combination of arrows match the frequencies of the coherences. The coherences are defined by identifying the ket and bra states at a given time (eg. after the second interaction, the ket-bra  states are cg for SFG and ba for DFG in figure 1c). Resonant enhancements occur when a combination of excitation frequencies matches the coherence frequency.

            The relative intensity of the different TWM methods depends on the phase matching conditions. The polarization induced in any given molecule will have all the possible frequency components formed by linear combinations of the excitation frequencies but the observed emission at these frequencies will depend on how the phases of the emission from the polarizations of all the molecules add together. Phase matching requires , , ,  and for SHG, SFG, DFG, and optical rectification, respectively. On a surface, the perpendicular component of is unimportant and phase matching requires matching the component parallel to the surface.

           

Coherent Raman Spectroscopies- coherent anti-Stokes Raman (CARS), stimulated Raman (SRS), impulsive stimulated Raman (ISRS), inverse Raman (IRS), Raman gain and loss, coherent Stokes Raman (CSRS), multiply enhanced nonparametric (MENS), multiply enhanced parametric (MEPS)       [back to the top]

            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 ω_L and ω_S (indicating laser field at ω_o and Stokes field at ω_o–ω_v) to create an output at the anti-Stokes frequency relative to ω_L, 2ω_L-ω_S  or the Stokes frequency relative to ω_S,  2ω_S-ω_L, respectively (see figure 2a,b). Phase matching is important for optimizing the output intensity. In CARS, the phase matching is  and in CSRS, the phase matching is . 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 ω_L and ω_S, respectively. The fast ω_S pulse overlaps in time with only part of the longer ω_L pulse and selects the frequencies present in the ω_L pulse at the time of overlap. The broad band of frequencies in the ω_S pulse can achieve resonance with many vibrational states at ω_L-ω_S  so 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 ω_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 . 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, . In the steady state, the vibrational contribution is

                                                        (3)

and its importance depends on the resonant enhancement from the detuning factor . In CARS, a strong Raman transition has a peak that is ~25x larger than .

            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 and  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 ω_S can interfere (heterodyne) with the excitation beam at ω_S. The transitions involving the ω_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 ω_S intensity or decrease in the transmitted ω_L intensity, respectively. If there is only a single excitation frequency, ω_L present and its intensity is high, vacuum fluctuations in the quantized electromagnetic field can act as an ω_S beam to stimulate emission at ω_S so an ω_S beam can be created from the vacuum. The creation of a beam at ω_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, ω_S, and a broad band source at ω_L. The transmitted light at ω_L is measured with a monochromator and multiplexing detector like a CCD and absorption lines appear in the dispersed output at ω_L from vibrational transitions at ω_L-ω_S. One way to view these processes is to realize that in the absence of a sample, the fields at ω_L and ω_S would be independent. However, if a sample is present, the fields can exchange energy using the sample as an intermediary. If the ω_S beam is initially absent, the ω_S beam intensity grows and the ω_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 ω_L and ω_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.

 

Stimulated Fluorescence, Pump-Probe Spectroscopies, and Time Resolved Pump-Probe Spectroscopies      [back to the top]

            Fluorescence and resonance Raman spectroscopies are closely related. Both are FWM processes and both involve similar excitation frequencies. They differ in the time ordering of the exciting fields and they differ in whether the intermediate state is a coherence or population. Figure 2d shows an example WMEL diagram for stimulated emission of fluorescence. The initial ket and bra side excitations create an excited bb population. This population is stimulated to emit by interaction with the third field which forms the ba output coherence. In contrast, the intermediate state in stimulated Raman spectroscopy is an ag coherence.

            Pump-probe methods are also closely related. These methods almost always use FWM processes. Example WMEL diagrams of single color pump-probe methods are shown in Figure 3. Pump-probe methods involve two overlapped excitation pulses- an initial pump pulse that creates the first two interactions in each diagram in figure 3 and a probe pulse that creates the third interaction leading to the final output coherence. Since the output field has the same frequency as the probe field, the two fields interfere. The transmitted intensity of the probe field can either increase because the pump induced stimulated emission (figure 3a) or ground state bleaching (figure 3b) or decrease because the pump created an excited state population that allow excited state absorption. The net change depends on the relative strength of all these processes.

            Two color pump-probe experiments use different frequencies for the pump and probe pulses. Figure 3d-f shows the two color pump-probe pathways. The first two electric field interactions represent the pump and create a population. The third interaction is the probe that creates the output coherence. The bleaching, stimulated emission, and excited state absorption  pathways are shown in figures 3d; 3e; and 3f, respectively.

            The quantum states involved in the pump or probe steps can be electronic/vibronic or vibrational. Electronic states are excited through uv/visible absorption while vibrational states are excited by infrared or Raman (or stimulated Raman) transitions. Consequently, time resolved (TR) pump-probe methods are classified as TR-UV-UV, UV-IR, IR-IR, UV-Raman and IR-Raman, etc. depending upon whether an electronic or vibrational state is pumped or probed. Since Raman is actually a four wave mixing process, the UV-Raman and IR-Raman pump-probe experiments are actually six wave mixing processes. Note that if the two pump interactions are created instead by separate controllable beams, these pump-probe pathways become rephasing stimulated photon echo (figures 4c, e, g) or nonrephasing (figures 4d, f, h) pathways.

           

 

Photon Echo, Stimulated Photon Echo, Transient Grating, Reverse Photon Echo, and Reverse Transient Grating Spectroscopies           [back to the top]

            Photon echo was one the first examples of an optical analogue of NMR. Figure 4a,b show the two photon echo WMEL diagrams. An initial pulse creates a ga coherence after the first interaction. After a delay time, τ, a second pulse induces the second and third interactions to create an ag coherence that reemits. Phase matching is important. For photon echo, the phase matching condition is  where the subscripts indicate the time ordering of the two pulses. The usefulness of photon echo rests on having spectral transitions that are inhomogeneously broadened so molecules that have different environments or conformations have a distribution of different transition frequencies. The temporal dependence of the excited ga coherences is  which becomes  after the τ delay. The temporal dependence of the final ag coherence is , conjugate to the ga coherence. After a delay of τ, the net phase will be zero, regardless of the ω_ag frequency. At this point in time, all of the coherences will be again in phase and the re-emission becomes completely coherent (it scales as N²). The large output intensity that results at time τ is called the echo. By measuring the echo intensity as function of delay time, one can measure the dephasing rate (Γ_ag) of the ga coherence.

            Stimulated photon echo experiments are three pulse experiments so the three excitation interactions in figure 4 occur at different times. The phase matching condition is . Figure 5a-c shows there are three pathways- a) ground state bleaching, b) stimulated emission, c) excited state absorption. The sign of the polarization is (-1)ⁿ where n is the number of bra-side interactions so excited state absorption has the opposite sign from the other pathways, that is it creates intensity decreases rather than increases. After the second interaction in figure 4a,b, one has either a gg ground state population (figure 4a) or an aa excited state population (figure 4b). After a delay time T, the third pulse arrives to create the ag output coherence. Measuring the stimulated photon echo intensity as a function of the T delay time measures the population changes of the ground and excited state populations. 

            Figure 4e,f shows the WMEL diagrams for transient grating experiments. Transient grating methods use two pulses that occur at the same time but are angled relative to each other. The phase matching condition is  where the subscripts again indicate the time ordering. These pulses are represented by the first two interactions in fig. 4e,f. They create gg (figure 4e) and aa (figure 4f) populations that are spatially modulated () to form a grating. The third pulse acts as a probe to create the ag output coherence. Its direction corresponds to its reflection off the grating created by the first two pulses. By measuring the transient grating intensity as a function of the delay between the two excitation pulses, one measures the population decay of the ground and excited state populations. A second delay can be introduced by separating the first two pulses in time. The transient grating experiment now looks very similar to the stimulated photon echo experiment. The most important difference lies in how they respond to inhomogeneous broadening. Note that the coherence formed after the first pulse is an agcoherence and so is the coherence formed after the last pulse. The phase difference created during the first delay no longer cancels the phase change formed by the last coherence; they now add, and rephasing never occurs. This form of transient grating experiment is called a non-rephasing pathway.

            Methods called reverse photon echo and reverse transient grating also exist. These methods are based on reversing the time orderings of the two pulses. In photon echo, the last two interactions occurred simultaneously. In reverse photon echo, the two simultaneous interactions now occur first. The phase matching is . Similarly, the first two interactions were simultaneous in transient grating methods. In reverse transient grating experiments, the two simultaneous interactions now occur last. The phase matching is . Again, the subscripts indicate the time ordering. Typical WMEL for these methods are shown in figure 4d and h. These methods have not been used appreciably except in the developing field of coherent multidimensional spectroscopy.

 

Incoherent Fluorescence and Raman Spectroscopies and Their Relationship to Coherent Spectroscopies      [back to the top]

            Fluorescence and Raman (affectionately known as COORS or Common Old Ordinary Raman Spectroscopy) spectroscopies are also four wave mixing nonlinear spectroscopies but they are incoherent, i.e. there is no phase relationships between excited molecules in different parts of a sample. The WMEL diagrams for Raman and fluorescence spectroscopy are still represented by figures 2c and d, respectively. However, instead of the ω_S excitation being caused by a real excitation beam, it is instead caused by the vacuum fluctuations which have no long range phase relationships so the coherences they stimulate are incoherent and therefore not directional. The emission intensity then scales as the concentration N.

 

Coherent Multidimensional Spectroscopies           [back to the top]

            Ever since Feynman et al showed that the transitions between two quantum states were equivalent to the spin excitations in NMR experiments, there has been great interest in developing the optical analogues of NMR, particularly because of the power of NMR to probe complex systems with great selectivity. The invention of the laser stimulated a great deal of work to develop the optical analogues because the laser provided the required coherent source.

            Early experiments closely followed NMR methods. π/2 and π pulses were delivered to samples by carefully controlling the phase and intensity of the excitation. It was difficult to preserve the phase information of the coherences because the vibrational and electronic coherences dephase ~9 orders of magnitude more quickly than NMR coherences because of the stronger interactions with the environmental thermal bath. These dephasing rates are much larger than typical Rabi frequencies, so it was difficult to create coherences with π/2 and π pulses owing to the time required being longer than that permitted by the dephasing times. Success was achieved only by making the pulses stronger so the Rabi periods were shorter or increasing the dephasing times so the dephasing times were comparable to the Rabi period. Typically, the dephasing times were increased by cooling the sample to extremely low temperatures. Although these approaches allowed limited studies, they did not provide a viable approach for measuring typical samples.

            Coherent multidimensional spectroscopy (CMDS) has emerged as a technique with widespread use because it was discovered that phase matching and time ordering of the excitation pulses could isolate particular coherence pathways just as well as supplying a series of phased π/2 and π pulses. For example, a molecule in its ground state can interact with a field to create a coherence either through absorption on its ket- or bra-side. However, if the experiment uses a stimulated photon echo phase matching geometry  , the first interaction must be a bra-side absorption; so for a two state system, the pathways are restricted to the two shown in figure 5.

            CMDS can be based on time domain or frequency domain methods. In time domain methods, one directly measures the temporal oscillations of the coherences and then performs a Fourier transform to obtain a frequency spectrum. If excitation pulses were available that were very short compared with the period of a coherence’s oscillation, the measurement could be performed by scanning the delay of the excitation pulse relative to the coherence being measured and one would immediately obtain the temporal phase oscillations of the coherence. Unfortunately, the excitation pulses are not that short so one must use a local oscillator. A local oscillator usually is a fourth field that is almost identical to the signal beam. The electric fields of the local oscillator and the signal heterodyne so the intensity depends on . If the time delay between the local oscillator and the signal is changed, one directly measures the oscillations of their relative phases. A Fourier transform will provide the frequency domain spectrum. If the delay between the first two excitation beams changes, the signal beam will reflect the changes in phase of the coherence created by the first beam and these in turn will be measured by the local oscillator as well. Fourier transform of these variations provide a second frequency axis for the 2D-IR spectroscopy. If the first two excitation beams create a coherence, the delay between the second and third excitation beams will scan the temporal phase changes of that intermediate coherence and Fourier transformation will provide a third frequency dimension. It is very important that the phase relationships between the excitation beams remains stable over the entire measurement time. This factor requires that the multiple excitation beams be derived from a single source that acts to define the phase of each beam. It also requires that the relative path lengths within the experimental system are interferometrically stable to a small fraction of a wavelength. Finally, in a pure time domain  experiment, all the quantum states are excited impulsively and one measures all of the coherences simultaneously through the temporal oscillations. Practically, the range of quantum states that can be measured depends on the excitation pulse bandwidth. Typically, the pulse widths are ~50-100 fs and these bandwidths excite quantum states over a range of ~150-300 cm^-1. 

            In frequency domain methods, one scans the frequency of the excitation beams while monitoring the intensity of the output coherence. The intensity is enhanced by each resonance and the enhancements are multiplicative. Phase coherence is again required but it is only required during the pulse sequence. Long term phase coherence is not required. Finally, in a pure frequency domain experiment, only specific quantum states are excited at any one time so individual coherence pathways and specific states are selected. The range is limited only by the tunability of the excitation sources. There are advantages to working in a mixed frequency/time domain where the excitation pulses are long enough to excite individual quantum states and avoid the difficulties involved with maintaining long term phase coherence but they are also short enough that one can measure the temporal dynamics of the coherences and populations of the quantum states.

            More about coherent multidimensional spectroscopy...

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