Phase-matching

The phase-matching condition (6.11) is illustrated by Fig. 6.5. If the angles between the three wave vectors are too large, the overlap region between focused beams becomes too small and the efficiency of the sum- or difference-frequency generation decreases. Maximum overlap is achieved for collinear propagation of all three waves. In this case, k A 2 A 3 and we obtain with cjn = oyjk and = (0 zb (02 the condition 

This condition can be fulfilled in unaxial birefringent crystals that have two different refractive indices Uq and n for the ordinary and the extraordinary waves. The ordinary wave is polarized in the x-y-plane perpendicular to the optical axis, while the extraordinary wave has its E -vector in a plane defined by the optical axis and 

One distinguishes between type-I and type-II phase-matching depending on which of the three waves with 0)1,0)2,002 = o) ib 0)2 propagates as an ordinary or as an extraordinary wave. Type I corresponds to (1 e, 2 e, 3 o) in positive uniaxial crystals and to (1 o, 2 o, 3 e) in negative uniaxial crystals, whereas type II is characterized by (1 o, 2 e, 3 - o) for positive and (1 e, 2 0, 3 e) for negative uniaxial crystals [533]. Let us now illustrate these general considerations with some specific examples. 

The uniaxial crystal is called positively birefringent if and negatively 

Since the SFG process is coherent, the emitted photons are phase matched with the incident beams in a manner determined by the wavelength of the input hght and the geometry of the set-up. The usual situation in SFG spectroscopy is a system with the three beams in the same plane, as shown in Fig. 5.4. The emitted photon direction is given by [36]  

The nonlinear polarization induced in an atom or molecule acts as a source of new waves at frequencies co= co =bct)2, which propagate through the nonlinear medium with the phase velocity a)/k — c/n co). However, the microscopic contributions generated by atoms at different positions x, y, z) in the nonlinear medium can only add up to a macroscopic wave with appreciable intensity if the vectors of the phase velocities of incident inducing waves and the polarization waves are properly matched. This means that the phases of the contributions Pi (o .co2,ri) to the polarization wave generated by all atoms at different locations r/ within the pump beam must be equal at a given point within the pump beam. In this case, the amplitudes Ei(co 002) add up in phase in the direction of the pump beam and the intensity increases with the 

This condition can be fulfilled in unaxial birefringent crystals that have two different refractive indices no and n for the ordinary and the extraordinary waves. The ordinary wave is polarized in the x-y-plane perpendicular to the optical axis, while the extraordinary wave has its -vector in a plane defined by the optical axis and the incident beam. While the ordinary index no does not depend on the propagation direction, the extraordinary index n depends on the directions of both E and k. The refractive indices Uo, and their dependence on the propagation direction in uniaxial birefringent crystals can be illustrated by the index ellipsoid defined by the three principal axes of the dielectric tensor. If these axes are aligned with the jc-, y-, z-axes, we obtain with n = the index ellipsoid. 

For uniaxial crystals ni = 112 and the index ellipsoid becomes symmetric with respect to the optical axis, which we choose to be the z-axis (Fig. 5.101a). If 

---

Using now the phase matching condition, it can be seen that besides the quasi shear wave (qSV) which is obtained as usual, a second quasi shear wave (qSV(2)) results from the upper quasi shear wave part. Since the direction of the group velocity vector points downwards this wave is able to propagate and can be seen in the snapshot (see Fig. 10) if a is properly adjusted, i.e. is pointing upwards as in Fig. 2. 

The oriented overgrowth of a crystalline phase on the surface of a substrate that is also crystalline is called epitaxial growth [104]. Usually it is required that the lattices of the two crystalline phases match, and it can be a rather complicated process [105]. Some new applications enlist amorphous substrates or grow new phases on a surface with a rather poor lattice match. 

The latter condition corresponds to the phase matching requirement already mentioned—the wavelength and direction of the material polarization wave must match those of the new EM wave as closely as possible. However, for all Class I spectroscopies, this condition is automatically achieved because of quadrature. In fact, this is tnie for all quadrature spectroscopies—the Class I spectroscopies being the principal such, but, as noted, it is a nontrivial requirement in the nonquadrature Class II spectroscopies, particularly in optically dispersive media. 

Mastuura S, Blake G A, Wyss R, Pearson J, Kadow C, Jackson A and Gossard A C 1999 A travelling-wave photomixer based on angle-tuned phase matching Appl. Rhys. Lett. 74 2872-4... 

The linear and nonlinear optical responses for this problem are defined by e, 2, e and respectively, as indicated in figure Bl.5.5. In order to detemiine the nonlinear radiation, we need to introduce appropriate pump radiation fields E(m ) and (co2)- If these pump beams are well-collimated, they will give rise to well-collimated radiation emitted tlirough the surface nonlmear response. Because the nonlinear response is present only in a thin layer, phase matching [37] considerations are unimportant and nonlinear emission will be present in both transmitted and reflected directions. 

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). 

Fig. 1. (a) Phase matched second harmonic generation (2cJ = 0.49 fiTo) at cj = 0.98 where = refractive index by ordinary rays and = by extraordinary rays, (b) Hypothetical anomalous dispersion phase matching at 850 nm in similar a crystal having a Lorent2ian absorption centered at 650... 

Biaxial crystals offer the possibiUty of coincidence of the phase matching direction with one of the optic axes. This highly desirable situation, called non-critical phase matching, is quite tolerant of divergence of the incident beam from the most efficient phase matching direction. 

Fig. 2. Optical nonlinearity and phase matching windows for commonly used NLO materials. KDP = KH2PO4 BBO = j3 — BaB2 KTP = KTiOPO. ...

Cuello was excavated by Hammond and co-workers between 1975 and 1993. It is the earliest known Preclassic Maya site, with a Preclassic occupation from ca. 1200 BC to AD 300 as well as later Classic period (AD 300-900) remains the earliest pottery-using phase (Swasey, 1200-900 BC) has not yet been found at other Preclassic sites, but the Bladen (900-600 BC) and subsequent phases match occupations elsewhere in date and material culture. The Cuello excavations have been extensively described in the report edited by Hammond (1991). Of particular relevance here are the chapters on the ecology and subsistence economy (Ch. 4) by Miksicek and by Wing and Scudder, and on the human burials (Ch. 7), by Frank and Julie Saul. More recent publications have focused on the subsistence economy (Crane and Carr 1994) and on the human skeletal remains (Saul and Saul 1997). 

Hirose et al. [26] proposed a homodyne scheme to achieve the background-free detection of the fourth-order field. With pump irradiation in a transient grating configuration, the fourth-order field propagates in a direction different from that of the second-order field because of different phase match conditions. The fourth-order field is homodyned to make ffourth(td. 2 D) and spatially filtered from the second-order response hecond td, 2 D). 

---