Second-harmonic generation

Second-harmonic generation, like other even-order nonlinear optical phenomena, is not allowed in media with inversion symmetry (in the leading electric dipole contribution).

These cases typically involve intense pulsed laser beams passing through large crystals and careful alignment to obtain phase matching.

Famously, when published in the journal Physical Review Letters,[9] 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.

[11] 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.

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.

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.

[16] 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.

: where Ns 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 (x, y, z).

[18] Using an interference SHG method to determine these elements of χ(2), the first molecular orientation measurement showed that the hydroxyl group of phenol pointed downwards into the water at the air-water interface (as expected due to the potential of hydroxyl groups to form hydrogen bonds).

Additionally SHG at planar surfaces has revealed differences in pKa and rotational motions of molecules at interfaces.

Second-harmonic light can also be generated from surfaces that are "locally" planar, but may have inversion symmetry (centrosymmetric) on a larger scale.

[19] At the surface of a small sphere, inversion symmetry is broken, allowing for SHG and other even order harmonics to occur.

Recent experiments using second-harmonic generation of non-planar systems include transport kinetics across living cell membranes[22] and demonstrations of SHG in complex nanomaterials.

[26] The forward (F) to backward (B) ratio is dependent on the arrangement of the different dipoles (green in figure) that are being excited.

The 1064 nm light is fed through a bulk nonlinear crystal (typically made of KDP or KTP).

Nevertheless, some "green laser pointer" products have become available on the market which omit the expensive infrared filter, often without warning.

SHG has the advantage of mixing two input fields to generate the harmonic one, it is thus a good candidate (but not the only one) to perform such a pulse measurement.

SHG microscopy has been used for studies of the cornea[30] and lamina cribrosa sclerae,[31] both of which consist primarily of collagen.

[32] 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.

[32][33] 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,[34] which are otherwise thought to occur from the same transition dipole moment.

[36] Second harmonic generation is also relevant to characterize organic or inorganic crystals[37] since is one of the most discriminant and rapid technique to detect non-centrosymmetry.

It is a relevant tool to resolve space group ambiguities that can arise from Friedel's law in single-crystal X-ray diffraction.

[43] It could also be used as a technique to probe the structural purity of material if one of the impurities is NC reaching a detection threshold as low as 1 ppm[44] using Kurtz–Perry apparatus up to one part in 10 billion by volume using a SHG microscope.

[47][48][49] 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.

The wave equation at 2ω (assuming negligible loss and asserting the slowly varying envelope approximation) is 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

In conformity with experiments, the SHG signal vanishes in the bulk (if the medium thickness is too large), and the SHG must be generated at the surface of the material: the conversion therefore does not strictly scales with the square of the number of scatterers, contrary to what the plane wave model indicates.

[50] Notably, filamentous biological proteins with a cylindrical symmetric such as collagen, tubulin or myosin, but also certain carbohydrates (such as starch or cellulose) are also quite good converters of SHG (fundamental in the near infrared).

[53] Examples of crystals used with for SHG conversion: For common types of diode-pumped solid state lasers with input wavelengths:

Energy level scheme of SHG process
An electron (purple) is being pushed side-to-side by a sinusoidally oscillating force, i.e. the light's electric field. But because the electron is in an anharmonic potential energy environment (black curve), the electron motion is not sinusoidal. The three arrows show the Fourier series of the motion: The blue arrow corresponds to ordinary (linear) susceptibility , the green arrow corresponds to second-harmonic generation, and the red arrow corresponds to optical rectification .
Different types of second-harmonic generation phase-matching of a coherent light for strong conversion. The case of negative crystals ( ) is considered, invert indices if positive crystal ( ).
Diagram of the second-harmonic generation process
A depiction of the second-harmonic generation setup for measuring the orientation of phenol at the air-water interface.
Cartoon depicting ordered molecules at a small spherical surface. An ultrafast pump laser pumps light with frequency ω which generates light at 2ω from the locally non-centrosymmetric media.
SHG radiation pattern excited with a Gaussian beam, in a homogeneous medium (A), or at an interface between opposite polarities that is parallel to the propagation (B). Only the forward SHG is represented.
SHG radiation pattern in forward (F) and backward (B) from different dipoles arangment: (a) single dipoles, thus F = B ; (b) a small stack of dipoles, F > B ; (c) a large stack of dipoles, F >> B ; (d) the Gouy phase-shift cancels the SHGs, F & B weak
Diagram of second-harmonic generation with perfect phase matching .
Diagram of second-harmonic generation with an imperfect phase matching . In this case energy flows forth and back from the pump to the frequency doubled signal, and having a thick crystal can lead to a smaller amount of SHG produced.
Phase-matched SHG with source depletion (blue), and corresponding excitation (orange). L is the interaction length ( ℓ in the text).
Intensity SHG, phase-matched or not. The medium width is supposed to be much higher than z , the Rayleigh range at 20 μm, excitation wavelength of 0.8 μm, and optical index of 2.2.