Until now, only two classes of organic dyes have been shown which do not produce any collateral fluorescence and works purely on second-harmonic generation. Recently, using two-photon excited fluorescence and second-harmonic generation-based microscopy, a group of Oxford University researchers showed that organic porphyrin-type molecules can have different transition dipole moments for two-photon fluorescence and second-harmonic generation, which are otherwise thought to occur from the same transition dipole moment. Second-harmonic generation microscopy is also used in material science, for instance to characterize nanostructured materials. 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 formula_19 is the effective nonlinear optical coefficient which is dependent on specific components of formula_4 that are involved in this particular interaction. The wave equation at 2ω (assuming negligible loss and asserting the slowly varying envelope approximation) is  where formula_22. At low conversion efficiency ("E(2ω) ≪ E(ω)") the amplitude formula_1 remains essentially constant over the interaction length, formula_24. Then, with the boundary condition formula_25 we obtain  In terms of the optical intensity, formula_27, 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 formula_29. 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.) 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 formula_31 denotes the complex conjugate. For simplicity, assume phase matched generation (formula_32). Then, energy conservation requires that  where formula_34 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 formula_19, the relative phases for real harmonic growth must be such that formula_42.