using Kurtz&Perry apparatus up to one part in 10 billion by volume using a SHG microscope. Due to the high sensitivity of the technique, it can be a helpful tool in the accurate determination of phase diagram and can also be used to monitor phase transitions(polymorphic transition, dehydration, ...) when at least one of the phases is NC. Theoretical derivation (plane wave) At low conversion The simplest case for analysis of second-harmonic generation is a plane wave of amplitude E(ω) traveling in a nonlinear medium in the direction of its k vector. A polarization is generated at the second-harmonic frequency: where is the effective nonlinear optical coefficient which is dependent on specific components of that are involved in this particular interaction. The wave equation at 2ω (assuming negligible loss and asserting the slowly varying envelope approximation) is where . At low conversion efficiency (E(2ω) ≪ E(ω)) the amplitude remains essentially constant over the interaction length, . Then, with the boundary condition we obtain In terms of the optical intensity, , this is, This intensity is maximized for the phase-matched condition Δk = 0. If the process is not phase matched, the driving polarization at ω goes in and out of phase with generated wave E(2ω) and conversion oscillates as sin(Δkℓ/2). The coherence length is defined as . It does not pay to use a nonlinear crystal much longer than the coherence length. (Periodic poling and quasi-phase-matching provide another approach to this problem.) With depletion When the conversion to 2nd harmonic becomes significant it becomes necessary to include depletion of the fundamental. The energy conversion states that all the involved fields verify the Manley–Rowe relations. One then has the coupled equations: where denotes the complex conjugate. For simplicity, assume phase matched generation (). Then, energy conservation requires that where is the complex conjugate of the other term, or Now we solve the equations with the premise and obtain which leads to Using we get If we assume a real , the relative phases for real harmonic growth must be such that . Then or where . From , it also follows that Theoretical expression with Gaussian beams The excitation wave is assumed to be a Gaussian beam, of amplitude: with , the direction of propagation, the Rayleigh range, the wave vector. Each wave verifies the wave equation where . With phase-matching It can be shown that: (a Gaussian), is a solution