Third-order frequency conversion

G. Hilber, A. Lago, R. Wallenstein Broadly tunable VUV/XUV-adiation generated by resonant third-order frequency conversion in Kr. J. Opt. Soc. Am. B 4, 1753 (1987) ... 

Third- and higher-order frequency conversion are often done with beams that are tightly focused within the nonlinear medium to increase the peak intensity. In this situation, optimal performance can require either a positive or neg-... 

Some of the applications of third- and higher-order frequency conversion are given in Table VII. The th harmonic generation is used to produce radiation at a frequency that is q times the incident frequency. The most commonly used interaction of this type is third-harmonic conversion. It has been used to produce radiation at wavelengths ranging from the infrared to the extreme ultraviolet. Third-harmonic conversion of radiation from high power pulsed lasers such as CO2, Ndiglass, Nd YAG, ruby, and various rare-gas halide and rare-gas excimer lasers has... 

TABLE VII Selected Results for Third- and Higher-Order Frequency Conversion Processes... 

Both second-order and third-order materials have technological applications because of their ability to convert low-frequency light to high-frequency light. However, the efficiency with which they are able to accomplish this feat decreases dramatically from second order to third order. Even the second-order process is small compared to the first-order process. From the design point, one faces a dilemma avoid the symmetry constraint and live with the low efficiency of third-order materials or adhere to the symmetry constraints and reap the benefit of better conversion. In Chapter 12, more about synthetic strategies as they relate to producing nonlinear optical materials will be covered. 

Let us finally notice that, investigating the dispersion theory for the effective third-order nonlinear susceptibility of nanocomposite media, Peiponen et al. established that Kramers-Kronig relations are not valid for whereas they are valid for other nonlinear processes such as frequency conversion [95]. 

If the electron density of a crystal could be accurately described by a single cosine wave that repeats three times in the unit cell dimension, d, that is, has a periodicity of d/S, its diffraction pattern would have intensity only in the third order (only one Bragg reflection, 3 0 0). Conversely, if only one order of the diffraction spectrum is observed (the Bragg reflection h = 3, for example), then the diffracting density amplitude must correspond to a cosine wave with frequency d/h [dl3) (29, 30). This can be considered as an electron-density wave, one of the components summed to give an electron density map. Each Bragg reflection provides an electron-density wave that contributes to Equation 2, the total electron density. In this way the relationship between the order (hkl) of a Bragg reflection and its contribution to the electron density is established. 

Applications of second order nonlinear optical materials include the generation of higher (up to sixth) optical harmonics, the mixing of monochromatic waves to generate sum or difference frequencies (frequency conversion), the use of two monochromatic waves to amplify a third wave (parametric amplification) and the addition of feedback to such an amplifier to create an oscillation (parametric oscillation). 

Parametric frequency-conversion interactions can occur in any order of the perturation expansion of Eq. (2). The most commonly observed processes have involved interactions of second and third order, although interactions up to order 11 have been reported. Parametric interactions are characterized by a growth rate for the intensity of the generated wave that depends on a power, or a product of powers, of the intensities of the incident waves, and they are strongly dependent on difference in wave vectors between the nonlinear polarization and the optical fields. 

Third-order and higher odd-order processes can be observed with electric dipole interactions in materials with any symmetry. They are used most commonly in materials that have a center of symmetry, such as gases, liquids, and some solids, since in these materials they are the lowest-order nonzero nonlinearities allowed by electric dipole transitions. Fourth-order and higher even-order processes involving electric dipole interactions are allowed only in crystals with no center of symmetry, and, although they have been observed, they are relatively inefficient and are seldom used for frequency conversion. 

Let us now deduce the factors that control the rate of conversion of B and C to D and E by imagining the transformation process is portrayed well by what is known as a collision rate model. (Strictly speaking, the collision rate model applies to gas phase reactions here we use it to describe interactions in solution where we are not specifying the roles played by the solvent molecules.) First, in order to be able to react, the molecules B and C have to encounter each other and collide. Hence, the rate of reaction depends on the frequency of encounters of B and C, which is proportional to the product of their concentrations. The rate is also related to how fast B and C move in the aqueous solution. Next, the rate is proportional to the probability that B and C meet with the right orientation to be able to react, which we may refer to as the orientation probability . Third, only a fraction of collisions have a sufficient amount of energy (greater then or equal to Ea) to break the relevant bonds in B and C... 

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