# 2.1.5: Nonlinear optical processes: getting the colors necessary for optical spectroscopy

Amplified pulses from Ti:Sapphire lasers can be readily used for time-resolved spectroscopy experiments. One question, however, remains unsolved: what do we do if the system we want to investigate absorbs the light outside the spectral range accessible by Ti:Sapphire lasers (around 800 nm). How do we get light, at, say 670 nm? It is clearly impractical to construct a new laser each time you get a new sample in your lab. Wavelength tuning and getting colors other than 800 nm wavelength is done employing the phenomena of nonlinear optics. Nonlinear optics is the entire branch of science investigating optical phenomena occurring when the electric field of light is comparable to that holding the electrons of atoms at the nuclei (roughly 1010 V/m). There are entire books on nonlinear optics [5], here we just depict several phenomena employed in getting the right colors for time-resolved spectroscopy.

##### Dispersion

Sorry for starting the discussion of nonlinear optical phenomena from the phenomenon that is acutally linear. Light dispersion is the phenomenon of linear optics, but it is important for the further discussion, therefore we will say a few words about it. In optics, dispersion is the dependence of refractive index on the frequency (or wavelength) of light. Transparent materials exhibit so-called normal dispersion, i.e. refractive index increases with frequency (or decreases with wavelength). The speed of light in the material is reversely proportional to its refractive index; therefore the blue photons (shorter wavelength) travel more slowly than the red ones. This means that if a transform limited pulse described by eq. passes through a slab of transparent medium (e.g. a piece of glass), its duration will increase because the photons of different colors spread out and do not reach the observer all at the same time. If the medium exhibits normal dispersion, the red photons will arrive earlier, and the blue ones – later (see the pulse in the inset of fig. 7). Such pulse is called chirped and resembles the pulses in the cavity of the regenerative amplifier. Dispersion broadening of the ultrashort pulses is usually an undesirable effect in the time-resolves spectroscopic experiments. It makes the pulses longer and decreases the time resolution of the experiments. To prevent it as much as possible, optical components of femtosecond beam lines (like lenses, waveplates or polarizers) are made as thin as possible. This also prevents transporting femtosecond pulses using optical fibers: in several meters of fiber a femtosecond pulse will stretch to tens of picoseconds, and different frequency components will scatter in time killing any dreams of good time resolution.

##### Harmonic generation

The easiest way of getting access to different colors than those produced by the laser is harmonic generation. This is truly a nonlinear optical phenomenon. Using harmonic generation from light with frequency w, we can obtain light with frequencies 2w, 3w, etc. The description of harmonic generation (and most of other nonlinear optical phenomena) starts with writing down the expression for nonlinear polarization. From Maxwell’s equations of electrodynamics we know that polarization is one source of electric fields (another one is charges). In a general case, material polarization can be written down as power series of electric field strength:

.

In this equation, denotes dielectric susceptibility of the i-th order. When electric field is weak, the only significant term in eq. is the first one. This is the linear term describing linear optical phenomena (dispersion, refraction, reflection, etc.) – material polarization is linearly propotional to the propagating electric field. Let us assume, that the field is described by a plane electromagnetic wave:

.

Here  is the wave vector (wave number in one-dimensional case). Plugging into , and leaving just the first term, we clearly see that no new frequencies will be produced by the polarization. The only thing changing will be the speed of light propagation (in multidimensional case, the direction may also change). However, if the electric field is strong enough and we cannot neglect the second term in eq. , it will become proportional to

,

i.e. material will start producing the light waves with the frequency twice of the incident wave.  The same way, we can show that two waves with different frequencies in the medium will result in the waves with the frequencies representing the sum and the difference of the incoming frequencies. The emergence of double frequency is called second harmonic generation, and the general case of such wave interaction (when the incident waves differ in frequencies and come from different directions) – three-wave mixing (three, because there are two incoming waves and one generated wave).

From symmetry considerations, one can show that second order nonlinear dielectric susceptibility  is zero in all isotropic materials. Therefore, harmonic generation, sum frequency generation and difference frequency generation can practically be realized only in crystals without the center of inversion. In femtosecond spectroscopy b-barium borate (abbreviated BBO) crystal is often used, because it features high optical damage threshold and has large second order dielectric susceptibility, thereby allowing efficient harmonic generation at lower light intensities. In order to generate harmonics efficiently, one needs to fulfil the phase-matching condition, which means that both incident wave and the generated wave must travel in the crystal at the same phase velocity (i.e. their respective refraction indices must be equal). This is also realized using crystal anisotropy: the polarization of the generated second harmonic wave is perpendicular to that of the incident wave. Different polarizations ‘see’ different refractive indices in the crystal. By suitably orienting the crystal, one can achieve the orientation, when both w and 2w waves experience identical refractive indices and therefore travel at the same velocities. Besides phase matching, energy and momentum conservation laws must hold for the incoming and outgoing photons. Energy conservation means that two photons with frequency w  in the incoming waves produce one photon with frequency 2w in the outgoing wave. Momentum conservation is illustrated in fig. 8. It dictates that when the incoming w frequency waves propagate in different directions, the second harmonic wave will be generated in the direction represented by the vector sum of the incoming photon momenta (fig. 8B); if the incoming waved differ in both frequency and direction, the sum frequency field will be emitted in the direction that can be calculated by adding their wave vectors and taking their lengths into account (fig. 8C).

 Fig. (i) Momentum conservation in harmonic generation processes. A: collinear second harmonic generation, the direction of second harmonic wave is the same as that of the incoming waves; B: non-collinear second harmonic generation, when the second harmonic is directed in the middle between the directions of the incoming waves; C: sum frequency generation, mixing waves with frequencies w and 2w. The direction of 3w field is given by the vector sum of the incoming waves.

Using crystals one can obtain not only the second, but also higher harmonics of the laser radiation: third, fourth, etc. The third harmonic can be generated by mixing first and second harmonic waves, and the fourth – by doubling the frequency of the second harmonic (or adding the first and third, but this is technically more complicated). These processes allow easily (one only needs a suitable crystal) extending the range of accessible laser frequencies. As the efficiency of each frequency conversion process is lower than one, some energy is always lost in the process, however, with optimum conditions, harmonic generation efficiencies of tens of percent can easily be achieved. The amount of produced light is then enough for performing time-resolved experiments.

Self-phase modulation and white light generation

 Fig. (ii) White light generation in sapphire crystal. A: beam color after passing the crystal. B – the spectrum of the same beam spread out using diffraction grating and covering the entire visible range.

As mentioned above, isotropic materials have zero second order (quadratic) dielectric susceptibility. In those materials, the first nonzero nonlinear term in eq. is cubic nonlinearity. One of the phenomena pertaining to this nonlinearity we have already discussed: it is optical Kerr effect, or self-focusing used in Kerr lens mode locking. Its basis is the nonlinear refractive index described by the eq. . The same dependence of refractive index on light intensity is the cause of white light supercontinuum generation, an effect widely used in time-resolved spectroscopic experiments. The physical basis behind this phenomenon is the fact that (nonlinear) refractive index is one of the factors in the phase of the electric field (eq. ): wave vector k  is proportional to the refractive index. This implies that the wave propagating in the medium will experience a phase shift (phase modulation) dependent on the intensity of its own electric field. Phase modulation is indistinguishable from frequency modulation, because frequency can be defined as time derivative of the phase. Thus the wave propagating in the medium with cubic nonlinearity is self-focusing and self-phase modulating. These phenomena, along with a number of other non-linear optical phenomena, broaden the spectrum of a femtosecond pulse propagating in nonlinear medium: from transform-limited pulse, a pulse of white light supercontinuum is generated, the spectrum of which can cover several hundreds of nanometers. White light continuum can easily be produced by focusing high energy Ti:Sapphire laser pulses in glass or water: 800 nm light turns into white broadband radiation (fig. 9). Even though the intensity of light produced in this manner is rather low at each particular wavelength interval, white light generation is very important for time-resolved spectroscopy: supercontinuum pulses are used as probe light in pump-probe spectroscopy (see below), and they also serve as seed light in optical parametric amplifiers discussed in the following section.

##### Optical parametric generation and amplification

Optical parametric generation is another quadratic nonlinear optical phenomenon and observed in crystals without the center of inversion. It is, in essence, the reverse of sum frequency generation: in this case a single photon from the pump wave is split into two photons (fig. 10). The resulting wave of higher frequency is called signal wave, whereas the one with the lower frequency is termed idler wave. In general, the two produced photons have different frequencies and their frequencies, wavelengths and momenta follow the conservation laws:

The most important and useful feature of the parametric generation is the fact that the division of frequencies (or photon energies) between the signal and idler wave depends on the phase matching condition, or, in other words, crystal orientation. This allows converting a given pump wave into a signal wave of desired frequency, simply by rotating the crystal to an appropriate angle, which is especially convenient for producing the tunable light source for time-resolved spectroscopy.

 Fig. (iii) A: Optical parametric generation: nonlinear crystal splits a pump photon into two different energy (frequency) photons. This process is subject to energy and momentum conservation laws. B: optical parametric amplification: seed light matching the characteristics of signal wave is applied on the crystal. It is amplified as the pump wave transfers its energy to it.

Optical parametric generation in the crystals occur when the initial photons of signal or idler waves are produced from quantum noise in the nonlinear medium and is further amplified by the pump wave. Noise is unreliable stuff, therefore, the efficiency and stability of parametric generation improves significantly, when, along with the pump wave, one applies a weak wave of desired signal wavelength to the crystal – the crystal has something to amplify from the very beginning. Here, white light supercontinuum comes in useful again: it contains all the frequencies, therefore, it serves as a perfect seed source: by directing it into the crystal together with the pump wave and rotating the crystal to an appropriate angle, one can obtain stable and strong parametric light of desired wavelength (fig. 10B).

Better picture needed

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