Second-harmonic generation (SHG), also known as frequency doubling, is the lowest-order wave-wave nonlinear interaction that occurs in various systems, including optical, radio, atmospheric, and magnetohydrodynamic systems. As a prototype behavior of waves, SHG is widely used, for example, in doubling laser frequencies. SHG was initially discovered as a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy of the initial photons (equivalently, twice the frequency and half the wavelength), that conserves the coherence of the excitation. It is a special case of two-photon sum-frequency generation, and more generally of harmonic generation. The second-order nonlinear susceptibility of a medium characterizes its tendency to cause SHG. Second-harmonic generation, like other even-order nonlinear optical phenomena, is not allowed in media with inversion symmetry (within the leading electric dipole contribution). However, effects such as the Bloch–Siegert shift, which occurs when two-level systems are driven at Rabi frequencies comparable to their transition frequencies, can give rise to second-harmonic generation in centrosymmetric systems. In addition, in non-centrosymmetric crystals belonging to crystallographic point group 432, SHG is not possible, and under Kleinman's conditions, SHG in the 422 and 622 point groups should vanish, although some exceptions exist. In some cases, almost 100% of the light energy can be converted to the second-harmonic frequency. These cases typically involve intense pulsed laser beams passing through large crystals and careful alignment to obtain phase matching. In other cases, like second-harmonic imaging microscopy, only a tiny fraction of the light energy is converted to the second harmonic, but this light can nevertheless be detected with the help of optical filters.
History
Second-harmonic generation was first demonstrated by Peter Franken, A. E. Hill, C. W. Peters, and G. Weinreich at the University of Michigan, Ann Arbor, in 1961. The demonstration was made possible by the invention of the laser, which created the required high-intensity coherent light. They focused a ruby laser with a wavelength of 694 nm into a quartz sample. They sent the output light through a spectrometer, recording the spectrum on photographic paper, which indicated the production of light at 347 nm. Famously, when published in the journal Physical Review Letters, the copy editor mistook the dim spot (at 347 nm) on the photographic paper as a speck of dirt and removed it from the publication. The formulation of SHG was initially described by N. Bloembergen and P. S. Pershan at Harvard in 1962. In their extensive evaluation of Maxwell's equations at the planar interface between a linear and nonlinear medium, several rules for the interaction of light in non-linear media were elucidated.
Types in crystals
Critical phase-matching
Second-harmonic generation occurs in three types for critical phase-matching, denoted 0, I and II. In Type 0 SHG, two photons having extraordinary polarization with respect to the crystal will combine to form a single photon with double the frequency/energy and extraordinary polarization. In Type I SHG, two photons having ordinary polarization with respect to the crystal will combine to form one photon with double the frequency and extraordinary polarization. In Type II SHG, two photons having orthogonal polarizations will combine to form one photon with double the frequency and ordinary polarization. For a given crystal orientation, only one of these types of SHG occurs. In general, to utilize Type 0 interactions, a quasi-phase-matching crystal type is required, such as periodically poled lithium niobate (PPLN).
Non-critical phase-matching Since phase-matching processes simply match the optical indices at ω and 2ω, they can also be done by controlling temperature in birefringent crystals where the optical index is temperature-dependent. For instance, LBO presents a perfect phase-matching at 25 °C for a SHG excited at 1200 or 1400 nm, but needs to be elevated at 200 °C for SHG with the usual laser line of 1064 nm. It is called "non-critical" because it does not depend on the crystal orientation, unlike usual phase-matching.
Surface second-harmonic generation
Since media with inversion symmetry are forbidden from generating second-harmonic light via the leading-order electric dipole contribution (unlike third harmonic generation), surfaces and interfaces make interesting subjects for study with SHG. In fact, second-harmonic generation and sum frequency generation discriminate against signals from the bulk, implicitly labeling them as surface specific techniques. In 1982, T. F. Heinz and Y. R. Shen explicitly demonstrated for the first time that SHG could be used as a spectroscopic technique to probe molecular monolayers adsorbed to surfaces. Heinz and Shen adsorbed monolayers of laser dye rhodamine to a planar fused silica surface; the coated surface was then pumped by a nanosecond ultra-fast laser. SH light with characteristic spectra of the adsorbed molecule and its electronic transitions were measured as reflection from the surface and demonstrated a quadratic power dependence on the pump laser power. In SHG surface spectroscopy, one focuses on measuring twice the incident frequency 2ω given an incoming electric field E ( ω ) {\displaystyle E(\omega )} in order to reveal information about a surface. Simply (for a more in-depth derivation see below), the induced second-harmonic dipole per unit volume, P ( 2 ) ( 2 ω ) {\displaystyle P^{(2)}(2\omega )} , can be written as
E ( 2 ω ) ∝ P ( 2 ) ( 2 ω ) = χ ( 2 ) E ( ω ) E ( ω ) {\displaystyle E(2\omega )\propto P^{(2)}(2\omega )=\chi ^{(2)}E(\omega )E(\omega )}
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