Third harmonic frequency

Question. Show that the third term in Equation (9.11) results in the generation of radiation of frequency 3v when radiation of frequency v is incident on a crystal capable of third harmonic generation. 

Because the second term in the brackets contains 3v in the sine function, radiation at a frequency which is three times that of the incident radiation is generated. This is referred to as third harmonic generation. The first term in brackets indicates that some radiation of unchanged frequency also results. 

Almost every unique frequency contained in the vibration signature of a machine-train can be directly attributed to a corresponding mechanical motion within the machine. For example, the constant endplay or axial movement of the rotating element in a motor-generator set generates elevated amplitude at the fundamental (lx), second harmonic (2x), and third harmonic (3x) of the shaft s true running speed. In addition, this movement increases the axial amplitude of the fundamental (lx) frequency. 

Many electrical problems, or problems associated with the quality of the incoming power and internal to the motor, can be isolated by monitoring the line frequency. Line frequency refers to the frequency of the alternating current being supplied to the motor. In the case of 60-cycle power, monitoring of the fundamental or first harmonic (60 Hertz), second harmonic (120 Hz), and third harmonic (180 Hz) should be performed. 

The index ms indicates that j s transforms according to the mixed symmetry representation of the symmetric Group 54 [33]. 7 5 is an irreducible tensor component which describes a deviation from Kleinman symmetry [34]. It vanishs in the static limit and for third harmonic generation (wi = u>2 = W3). Up to sixth order in the frequency arguments it can be expanded as [33] ... 

We note that 02 4 vanishs if two of the four the frequency arguments Wq, wi, u>2, W3 become zero. For the case that three frequency arguments are equal (third harmonic generation) both mixed-symmetry effective frequencies w 2 vanish. The coefficients Amsi Bms, etc. 

A similar convergence is found for the third harmonic generation process at the lower of the two frequencies, 671.5 nm. At the higher frequency, 476.5 nm, the Taylor approximations for the third harmonic generation hyperpolarizability converge only very slowly, even with a tenth-order Taylor approximation a one-percent accuracy is not obtained. This accuracy, however, is still achieved with a [1,2] Fade approximant calculated from the dispersion coefficients up to sixth order. 

Fig. 8. Examples of some of the donor-acceptor substituted TEEs prepared for the exploration of structure-property relationships in the second- and third-order nonlinear optical effects of fully two-dimensionally-conjugated chromophores. For all compounds, the second hyperpolarizability y [10 esu], measured by third harmonic generation experiments in CHCI3 solution at a laser frequency of either A = 1.9 or 2.1 (second value if shown) pm is given in parentheses. n.o. not obtained...

Schaller, R. D., Johnson, J. C Wilson, K R., Lee, L. F., Haber, L. H. and Saykally, R. J. (2002) Nonlinear chemical imaging nanomicroscopy from second and third harmonic generation to multiplex (broad-bandwidth) sum frequency generation near-freld scanning optical microscopy. 

If any of the frequencies (w, 2m or 3w in third harmonic generation or similar frequency combinations in three wave mixing experiments) pproach the region of intense absorption situated around 16000 cm appreciable changes in the values of are expected... 

Third-order susceptibilities of the PAV cast films were evaluated with the third-harmonic generation (THG) measurement [31,32]. The THG measurement was carried out at fundamental wavelength of 1064 nm and between 1500 nm and 2100 nm using difference-frequency generation combined with a Q-switched Nd YAG laser and a tunable dye laser. From the ratio of third-harmonic intensities I3m from the PAV films and a fused quartz plate ( 1 thick) as a standard, the value of x(3) was estimated according to the following equation derived by Kajzar et al. [33] ... 

By this nonlinear polarization different incident lightwaves, or lightwaves and phonon waves, are coupled together, resulting in the production of sum and difference frequencies and second or third harmonics. (This is to some extent analogous to rf frequency mixing in a nonlinear diode). 

Where P is the polarisation and the others the linear (1) and non-linear, second (2) and third order (3) terms. Examples of important second order effects are frequency doubling and linear electro-optic effects (Pockles effect), third order effects are third-harmonic generation, four-wave mixing and the quadratic electro-optic effect (Ken-effect). 

Hill, S. C., Leach, D. H., and Chang, R. K. 1993. Third-order sum-frequency generation in droplets—Model with numerical results for third-harmonic generation. J. Opt. Soc. Am. B 10 16-33. 

In addition to the blueshifling of the optical absorbance of Q-state semiconductor particles, a linear optical effect, there are nonlinear optical effects demonstrated by Q-state semiconductors. Two types have been observed by MCs prepared in organized films. One is the third harmonic generation or frequency tripling (44). A... 

This scheme of frequency tripling was successfully tested with fuchsin in hexafluorisopropanol (a solvent selected for its low index of refraction and relatively flat dispersion curve) to frequency-triple the output of a neodymium laser 67,68) With an input power of 10 MW/cm2 a third-harmonic output of 0.2 mW/cm2 was measured. This low value was mainly due to the relatively high absorption of fuchsin at 355 nm. An improvement of the efficiency by a factor of 80 was found with hexamethylindocarbocyanine iodide in hexafluorisopropanol because of the much lower absorption of this dye at 355 nm. Since the absorption minimum of this dye is at 383 nm, one could expect an additional efficiency increase by a factor of 70 for a fundamental laser wavelength of 1.15 / 69>. Other cyanine dyes have been used for frequency tripling a fundamental wavelength of 1.89 /mi 70>. 

In a series of experiments we have tested the type and range of entrainment of glycolytic oscillations by a periodic source of substrate realizing domains of entrainment by the fundamental frequency, one-half harmonic and one-third harmonic of a sinusoidal source of substrate. Furthermore, random variation of the substrate input was found to yield sustained oscillations of stable period. The demonstration of the subharmonic entrainment adds to the proof of the nonlinear nature of the glycolytic oscillator, since this behavior is not observed in linear systems. A comparison between the experimental results and computer simulations furthermore showed that the oscillatory dynamics of the glycolytic system can be described by the phosphofructokinase model. 

Experiments were performed using a commercial kilohertz femtosecond Ti Sapphire laser system (Spectra Physics) delivering laser pulses at 790 nm with a duration of 110 fs and an energy of 750 pj. The pump beam at 263 nm (third harmonic) was produced by frequency doubling and sum-frequency mixing in two BBO crystals. Then, the pump beam was focused on a 300 pm thick ethylene glycol jet in order to produce electrons by photoionisation of the... 

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