Second-harmonic generation

Second-harmonic generation, 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, that conserves the coherence of the excitation. It is a special case of 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. However, effects such as the Bloch–Siegert shift, found when two-level systems are driven at Rabi frequencies comparable to their transition frequencies, will give rise to second-harmonic generation in centro-symmetric systems. In addition, in non-centrosymmetric crystals belonging to crystallographic point group 432, SHG is not possible and under Kleinman's conditions SHG in 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.
Generating the second harmonic, often called frequency doubling, is also a process in radio communication; it was developed early in the 20th century and has been used with frequencies in the megahertz range. It is a special case of frequency multiplication.

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 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 will be required, for example periodically poled lithium niobate.

Non-critical phase-matching

Since phase-matching process basically means to match the optical indices at ω and 2ω, it can also be done by a temperature control in some birefringent crystals, because n changes with the temperature. 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 as 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, 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 in order to reveal information about a surface. Simply, the induced second-harmonic dipole per unit volume,, can be written as
where is known as the nonlinear susceptibility tensor and is a characteristic to the materials at the interface of study. The generated and corresponding have been shown to reveal information about the orientation of molecules at a surface/interface, the interfacial analytical chemistry of surfaces, and chemical reactions at interfaces. SHG surface spectroscopy is also used extensively in the electrochemical characterization of materials, since the electric field at the interface between the electrode and electrolyte introduces an additional term that affects the interfacial inversion symmetry.

From planar surfaces

Early experiments in the field demonstrated second-harmonic generation from metal surfaces. Eventually, SHG was used to probe the air-water interface, allowing for detailed information about molecular orientation and ordering at one of the most ubiquitous of surfaces. It can be shown that the specific elements of :
where N_s is the adsorbate density, θ is the angle that the molecular axis z makes with the surface normal Z, and is the dominating element of the nonlinear polarizability of a molecule at an interface, allow one to determine θ, given laboratory coordinates. Using an interference SHG method to determine these elements of χ, the first molecular orientation measurement showed that the hydroxyl group of phenol pointed downwards into the water at the air-water interface. Additionally SHG at planar surfaces has revealed differences in pK^a and rotational motions of molecules at interfaces.

From non-planar surfaces

Second-harmonic light can also be generated from surfaces that are "locally" planar, but may have inversion symmetry on a larger scale. Specifically, recent theory has demonstrated that SHG from small spherical particles is allowed by proper treatment of Rayleigh scattering. At the surface of a small sphere, inversion symmetry is broken, allowing for SHG and other even order harmonics to occur.
For a colloidal system of microparticles at relatively low concentrations, the total SH signal, is given by:
where is the SH electric field generated by the jth particle, and n the density of particles. The SH light generated from each particle is coherent, but adds incoherently to the SH light generated by others. Thus, SH light is only generated from the interfaces of the spheres and their environment and is independent of particle-particle interactions. It has also been shown that the second-harmonic electric field scales with the radius of the particle cubed, a³.
Besides spheres, other small particles like rods have been studied similarly by SHG. Both immobilized and colloidal systems of small particles can be investigated. Recent experiments using second-harmonic generation of non-planar systems include transport kinetics across living cell membranes and demonstrations of SHG in complex nanomaterials.

Radiation pattern

The SHG radiation pattern generated by an exciting Gaussian beam also has a 2D Gaussian profile if the nonlinear medium being excited is homogeneous. However, if the exciting beam is positioned at an interface between opposite polarities that is parallel to the beam propagation, the SHG will be split into two lobes whose amplitudes have opposite sign, i.e. are phase-shifted.
These boundaries can be found in the sarcomeres of muscles, for instance. Note that we have considered here only the forward generation.
Moreover the SHG phase-matching can also result in : some SHG is also emitted in backward. When the phase-matching is not fulfilled, as in biological tissues, the backward signal comes from a sufficiently high phase-mismatch which allow a small backward contribution to compensate for it. Unlike fluorescence, the spatial coherence of the process constrain it to emit only in those two directions, where the coherence length in the backwards direction is always much smaller than in the forwards, meaning there is always more forward than backward SHG signal.
The forward to backward ratio is dependent on the arrangement of the different dipoles that are being excited. With only one dipole, F = B, but F becomes higher than B when more dipoles are stacked along the propagation direction. However, the Gouy phase-shift of the Gaussian beam will imply a phase-shift between the SHGs generated at the edges of the focal volume, and can thus result in destructive interferences if there are dipoles at these edges having the same orientation.