Harmonic generation second

Second-harmonic generation (SHG), the special case of sum-frequency generation where a i = CO2 = and CO3 = 2o , is an invaluable frequency upconver-sion technique in lasers [5,6]. Most near-UV lasers are frequency-doubled beams originating in visible dye lasers, and Nd YAG-pumped dye lasers are excited by the 532-nm SHG rather than the 1064-nm fundamental from the YAG laser. Autocorrelation diagnoses of pulse durations generated by mode-locked lasers also rely on frequency doubling. 

It is clear from Eq. 11.11 that for SHG (in which o represents both the initial and final state) 

Since the second-order nonlinear susceptibility is proportional to c oo), we have in the El approximation 

The practical problems associated with SHG may be appreciated by considering a classical theory for wave propagation in the medium. By combining the Maxwell equations (1.37c) and (1.37d), we obtain the homogeneous wave equation [1] 

When an electromagnetic wave propagates through a frequency-doubling medium, the homogeneous equation (11.15) becomes superseded by [1,5] 

Second-harmonic generation (SHG), or frequency doubling, is defined as the conversion of a specific wavelength of light into half its original, i.e. 2i— 1/2 2i, or with respect to frequency ca, 2u i. It was not until the 

These figures also clearly illustrate that accurate SHG efficiencies can only be determined by measuring similar particle size ranges. For example, if the SHG of a-Si02 is measured with particles 90 pm and the unknown material is measured at a smaller particle size, the SHG efficiency of the unknown material would be overestimated. Thus, it is critical that the SHG of Si02 and the unknown material be measured at the sample particle size range, i.e. 45-63 pm. 

Once the phase-matching capabilities of the material are known, the bulk nonlinear optical susceptibility, can be estimated. The value 

The SHG efficiency of the unknown compound (A) is either compared with LiNbOs - SHG efficiency of 600 x Si02 - or Si02 depending on the phase-matching behaviour of the unknown compound. The units for (ieff are picometres per volt (pmA ). 

As detailed above, a necessary condition for the generation of nonlinear polarization of even order is the lack of inversion symmetry in the nonlinear medium. In media with inversion symmetry, only nonlinear terms of odd order exist (begiiming with well as higher order bulk contributions 

6) A detailed account of the principles of surface SHG is given in Brevet (1997). 

At the surface, however, the inversion symmetry of the bulk is broken. Electric dipole contributions to the nonlinear polarization become possible due to the spatial structure of the surface and due to the discontinuity of the normal component of the electric field at the surface. In the case of a nonlocal interaction, the polarization at a given position r depends on the external field of the surroundings (i.e, spatial dispersion ). In most cases, however, one assumes for the sake of simplicity a local interaction, in which case the susceptibility is independent of the polarization in the surroimdings. If one assumes that the main reason for this nonlinear polarization is the generation of a strong (static) dipole field at the surface, then it becomes clear that it should be localized in the uppermost atomic layers down to a depth of 0.5-1.5 nm (Sipe et al. 1987). 

The additional electric quadrupole and magnetic dipole contributions from the bulk can be separated from the real surface contributions only under special conditions for example, if one modifies the surface layer in the form of an evaporated thin film and if one extracts the bulk contributions by calculating the interface contributions (Koopmans et al. 1993). The bulk contributions are proportional to the field gradient, meaning that a zone of about A/27T 5... 10 nm contributes to the total signal intensity. 

Based on Equation 3.124, when the incident fields have the frequency (0i = (O2 = (O, the second-order optical process in ZnO can be reduced to the following forms 

Let us now discuss the second-order susceptibility, diu= (X h)/2 for ZnO with a brief reference to the well-established nonlinear optical crystals. Table 3.12 collates the relevant parameters for commonly used optical crystals and semiconductors. 

The macroscopic second-harmonic susceptibilities are treated as tensorial sums of the microscopic single-bond second-order susceptibilities. Assuming that the bonds in the wurtzite (w) and the zinc blende (z) crystals are identical, the bond additivity concept gives rise to the following simple expression [189]  

Further, symmetry arguments associated with wurtzitic crystals lead 

The superscripts z and w distinguish the zinc blende and wurtzitic phases. 

For the case = 2 = w the phase-matching condition (5.118) for Second-Harmonic Generation (SHG) becomes 

In favorable cases phase-matching is achieved for 6 = 90 . This has the advantage that both the fundamental and the SH beams travel collinearly through the crystal whereas for 6 90 the power flow direction of the ex- 

This nonlinear polarization generates a wave with amplitude E(2a ) which travels with the phase velocity v(2a ) = 2o /k(2ci ) through the crystal. The effective nonlinear coefficient depends on the nonlinear crystal and 

According to the definition at the end of Sect.5.8.1, type-I phasematching is achieved in uniaxial, negatively birefringent crystals when ng(2c , ) = n (w). The polarizations of the fundamental wave and the SH 

Material Transparency range [nm] Spectral range of phase matching of type I or II Damage threshold [GW/cm2] Relative doubling efficiency Reference 

For the case coi =o 2=(o, the phase-matching condition (5.119) for second-harmonic generation (SHG) becomes 

Let us estimate how a possible slight phase mismatch = n (jo) — n 2(jo) affects the intensity of the SH wave. The nonlinear polarization P 2co) generated at the position r by the driving field Eo cos[cjot — k o)) can be deduced from (5.113) as 

Assume that the pump wave propagates in the z-direction. Over the path length z a phase difference 

For type-II phase matching the polarization of the fundamental wave does not fall into the plane defined by the optical axis and the A -vector. It therefore has one component in the plane, which travels with v = c/uo, and another component with V = c/hq perpendicular to the plane. The phase-matching condition now becomes 

Both can be done on single crystals and on oriented thin films. The former yields (-2co co,co) susceptibility, while the latter (-co co,0). Depending of the relative orientation of the incident and output fields with respect to the crystal (or thin film) symmetry axes, different components of both susceptibilities can be obtained. The SHG technique is also often used to study the kinetics of orientation and relaxation processed by in situ measurements [ 16]. 

The second harmonic generation is a coherent technique giving the fast, electronic in origin, second-order NLO susceptibility (-2co co,co) at a given, measurement frequency co. Here, we limit the discussion to poled films, with °o mm symmetry, which exhibit two nonzero x tensor components the diagonal A zzz( 2co co,co) and the off diagonal xzz( 2co co,co), where Z is the poling (preferential orientation) direction. Usually, thin films are deposited on one side of substrate only (thin film deposited on both sides is discussed in Swalen and Kaj- 

As already mentioned, the SHG measurements are done in the fundamental (/) and harmonic (h) polarization configuration, starting with the s-polarized fundamental and p-polarized harmonic beams. In that case the projection factor is given by  

By using appropriate calibration (e.g., a quartz single-crystal plate) the off diagonal susceptibility (or sp sp deduced directly from these 

We discuss in some detail the so called p-in p-out configuration, in which a / -polarized laser beam (with its polarization vector perpendicular to the interface) is used, and the signal with p-polarization is investigated. On flat metal-solution interfaces, there are three sources that give rise to frequency doubling, and the observed signal is caused 

It must be borne in mind that the signal is caused by the interference of all three terms so an increase in a(u ) does not necessarily lead to a corresponding increase in the SHG intensity. Nevertheless, a strong change in a uj) will always show up in the SHG signal. 

As an example [13] we consider the underpotential deposition of thallium on silver (Fig. 15.13). At potentials above the onset of the upd of thallium the SHG signal decreases, at first slowly, then more rapidly. The adsorption of thallium causes a strong rise in a(o ), because the region in which the electronic density decays to zero becomes more extended with an angle of incidence of 45° this shows up as a drastic increase in the signal. A similar behavior is seen in other systems, and often even fractions of a monolayer can be detected. 

At present the whole field is developing rapidly we mention a few other applications of SHG that will certainly become important in the future  

On single crystal surfaces the SHG signal depends on the polar angle of incidence. This can be used to investigate the structure and symmetry of the surface. 

Generally, ordinary light sources are too weak to allow the effect to be observed. However laser light is of a sufficient intensity that the higher 

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Optical second-harmonic generation (SHG) has recently emerged as a powerful surface probe [95, 96]. Second harmonic generation has long been used to produce frequency doublers from noncentrosymmetric crystals. As a surface probe, SHG can be caused by the break in symmetry at the interface between two centrosymmetric media. A high-powered pulsed laser is focused at an angle of incidence from 30 to 70° onto the sample at a power density of 10 to 10 W/cm. The harmonic is observed in reflection or transmission at twice the incident frequency with a photomultiplier tube. 

SHG Optical second-harmonic generation [95, 96] A high-powered pulsed laser generates frequency-doubled response due to the asymmetry of the interface Adsorption and surface coverage rapid surface changes... 

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Brevet P F 1997 Surface Second Harmonic Generation (Lausanne Presses Polyteohniques et Universitaires Romandes)... 

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Pumping is with a flashlamp, as in the case of the ruby laser, and a pulse energy of the order 1 J may be achieved. Frequency doubling (second harmonic generation) can provide tunable radiation in the 360-400 nm region. 

In the single-domain state, many ferroelectric crystals also exhibit high optical nonlinearity and this, coupled with the large standing optical anisotropies (birefringences) that are often available, makes the ferroelectrics interesting candidates for phase-matched optical second harmonic generation (SHG). 

The second term on the right-hand side, a component oscillating at frequency 2co, represents the second harmonic of the incident beam. This component of the polarization vector can radiate light at the frequency 2co. Observation of the second harmonic generation was demonstrated in the early 1960s using mby lasers (59). 

Only certain types of crystalline materials can exhibit second harmonic generation (61). Because of symmetry considerations, the coefficient must be identically equal to zero in any material having a center of symmetry. Thus the only candidates for second harmonic generation are materials that lack a center of symmetry. Some common materials which are used in nonlinear optics include barium sodium niobate [12323-03-4] Ba2NaNb O lithium niobate [12031 -63-9] LiNbO potassium titanyl phosphate [12690-20-9], KTiOPO beta-barium borate [13701 -59-2], p-BaB204 and lithium triborate... 

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