Second-harmonic generation

Different types of second-harmonic generation phase-matching of a coherent light for strong conversion. The case of negative crystals (${\displaystyle n_{o}>n_{e}}$) is considered, invert indices if positive crystal (${\displaystyle n_{e}>n_{o}}$).

Second-harmonic generation occurs in three types for critical phase-matching,[12] 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 utilise Type 0 interactions a quasi-phase-matching crystal type will be required, for example periodically poled lithium niobate (PPLN).

Since phase-matching process basically means to adapt the optical indices n 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,[13] 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.

A depiction of the second-harmonic generation setup for measuring the orientation of phenol at the air-water interface.

Early experiments in the field demonstrated second-harmonic generation from metal surfaces.[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.[17] It can be shown that the specific elements of ${\displaystyle \chi ^{(2)}}$:

${\displaystyle {\begin{aligned}\chi _{zzz}^{(2)}&=N_{s}\left\langle \cos ^{3}(\theta )\right\rangle \alpha _{zzz}^{(2)}\\\chi _{xzx}^{(2)}&={\frac {1}{2}}N_{s}\left\langle \cos(\theta )\sin ^{2}(\theta )\right\rangle \alpha _{zzz}^{(2)}\end{aligned}}}$

where N_s is the adsorbate density, θ is the angle that the molecular axis z makes with the surface normal Z, and ${\displaystyle \alpha _{zzz}^{(2)}}$ 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 pK^a and rotational motions of molecules at interfaces.

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.

Second-harmonic light can also be generated from surfaces that are ‘locally’ planar, but may have inversion symmetry (centrosymmetric) on a larger scale. Specifically, recent theory has demonstrated that SHG from small spherical particles (micro- and nanometer scale) is allowed by proper treatment of Rayleigh scattering, (scattering without a change in frequency from absorbed to emitted waves).[19] 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 ${\displaystyle I_{2\omega }^{\text{total}}}$, is given by:

${\displaystyle I_{2\omega }^{\text{total}}\propto \sum \limits _{j=1}^{n}\left(E_{j}^{2\omega }\right)^{2}=n\left(E^{2\omega }\right)^{2}=nI_{2\omega }}$

where ${\displaystyle E_{j}^{2\omega }}$ is the SH electric field generated by the jth particle, and n the density of particles.[20] The SH light generated from each particle is coherent, but adds incoherently to the SH light generated by others (as long as density is low enough). 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 ${\displaystyle E(2\omega )}$ scales with the radius of the particle cubed, a³.

Besides spheres, other small particles like rods have been studied similarly by SHG.[21] 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[22] and demonstrations of SHG in complex nanomaterials.[23]

Characterizing an ultrashort pulse (like measuring its temporal width) cannot be done directly with electronics only, as the time-scale is below 1ps (${\displaystyle 10^{-12}}$sec) : it needs to use the pulse itself, that is why an autocorrelation function is often used. 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. Optical autocorrelation, in its intensity or fringe-resolved (interferometric) version use SHG,[28] unlike field autocorrelation. Also, most versions of the FROG (called SHG-FROG) use SHG to mix the delayed fields.[29]

In biological and medical science, the effect of second-harmonic generation is used for high-resolution optical microscopy. Because of the non-zero second-harmonic coefficient, only non-centrosymmetric structures are capable of emitting SHG light. One such structure is collagen, which is found in most load-bearing tissues. Using a short-pulse laser such as a femtosecond laser and a set of appropriate filters the excitation light can be easily separated from the emitted, frequency-doubled SHG signal. This allows for very high axial and lateral resolution comparable to that of confocal microscopy without having to use pinholes. SHG microscopy has been used for studies of the cornea[30] and lamina cribrosa sclerae,[31] both of which consist primarily of collagen. Second-harmonic generation can be produced by several non-centrosymmetric organic dyes; however, most of the organic dyes also generate collateral fluorescence along with second-harmonic generation signals.[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.[35]

Second-harmonic generation microscopy is also used in material science, for instance to characterize nanostructured materials.[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.[38] In addition, this technique can be used on single crystal as well as on powdered samples. One should recall that SHG is only possible (from the bulk) in non-centrosymmetric (NC) crystals. The part of non-centroysmmetric crystals in Nature is much lower than centrosymmetric crystals (circa 22% of the Cambridge structural database[39]), but the frequency of NC crystals increases by a lot in pharmaceutical, biological and electronic fields because of the particular properties of these crystals (piezoelectricity, pyroelectricity, polar phases, chirality,...).

In 1968,[40] (7 years after the first experimental evidence of SHG on single crystal[9]), Kurtz and Perry started to develop a SHG analyzer to rapidly detect the presence or not of inversion center in powdered crystalline samples. The detection of a SHG signal has been shown to be reliable and sensitive test for the detection of crystalline non-centrosymmetry with the confidence level higher than 99%. It is a relevant tool to resolve space group ambiguities that can arise from Friedel’s Law in single-crystal X-ray diffraction.[41] Furthermore, the method is referenced in the International Tables for Crystallography and is described as a “powerful method of testing crystalline materials for the absence of a symmetry center.[42]

One possible application is also to rapidly discriminate chiral phases such as conglomerate that are of particular interest for pharmaceutical industries.[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.[45]

Due to the high sensitivity of the technique, it can be a helpful tool in the accurate determination of phase diagram[46] and can also be used to monitor phase transitions(polymorphic transition, dehydration, ...) when at least one of the phases is NC.[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. A polarization is generated at the second-harmonic frequency:[50]

${\displaystyle P(2\omega )=\varepsilon _{0}\chi ^{(2)}E^{2}(\omega )=2\varepsilon _{0}d_{\text{eff}}(2\omega$ ;\omega ,\omega )E^{2}(\omega ),\,}

where ${\displaystyle d_{\text{eff}}}$ is the effective nonlinear optical coefficient which is dependent on specific components of ${\displaystyle \chi ^{(2)}}$ that are involved in this particular interaction. The wave equation at 2ω (assuming negligible loss and asserting the slowly varying envelope approximation) is

${\displaystyle {\frac {\partial E(2\omega )}{\partial z}}=-{\frac {i\omega }{n_{2\omega }c}}d_{\text{eff}}E^{2}(\omega )e^{i\,\Delta k\,z}}$

where ${\displaystyle \Delta k=k(2\omega )-2k(\omega )}$.

At low conversion efficiency (E(2ω) ≪ E(ω)) the amplitude ${\displaystyle E(\omega )}$ remains essentially constant over the interaction length, ${\displaystyle \ell }$. Then, with the boundary condition ${\displaystyle E(2\omega ,z=0)=0}$ we obtain

${\displaystyle E(2\omega ,z=\ell )=-{\frac {i\omega d_{\text{eff}}}{n_{2\omega }c}}E^{2}(\omega )\int _{0}^{\ell }e^{i\,\Delta k\,z}\,dz=-{\frac {i\omega d_{\text{eff}}}{n_{2\omega }c}}E^{2}(\omega )\ell \,{\frac {\sin \left({\frac {1}{2}}\,\Delta k\,\ell \right)}{{\frac {1}{2}}\,\Delta k\,\ell }}e^{{\frac {i}{2}}\,\Delta k\,\ell }}$

In terms of the optical intensity, ${\displaystyle I=n/2{\sqrt {\varepsilon _{0}/\mu _{0}}}|E|^{2}}$, this is,

${\displaystyle I(2\omega ,\ell )={\frac {2\omega ^{2}d_{\text{eff}}^{2}\ell ^{2}}{n_{2\omega }n_{\omega }^{2}c^{3}\varepsilon _{0}}}\left({\frac {\sin \left({\frac {1}{2}}\,\Delta k\,\ell \right)}{{\frac {1}{2}}\,\Delta k\,\ell }}\right)^{2}I^{2}(\omega )}$

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 ${\displaystyle \ell _{c}={\frac {\pi }{\Delta k}}}$. 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.)

Diagram of second-harmonic generation with perfect phase matching ${\displaystyle \Delta k=0}$.

Diagram of second-harmonic generation with an imperfect phase matching ${\displaystyle \Delta k\neq 0}$. 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.

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:[51]

${\displaystyle {\begin{aligned}{\frac {\partial E(2\omega )}{\partial z}}&=-{\frac {i\omega }{n_{2\omega }c}}d_{\text{eff}}E^{2}(\omega )e^{i\,\Delta k\,z},\\[5pt]{\frac {\partial E(\omega )}{\partial z}}&=-{\frac {i\omega }{n_{\omega }c}}d_{\text{eff}}^{*}E(2\omega )E^{*}(\omega )e^{-i\,\Delta k\,z},\end{aligned}}}$

where ${\displaystyle *}$ denotes the complex conjugate. For simplicity, assume phase matched generation (${\displaystyle \Delta k=0}$). Then, energy conservation requires that

${\displaystyle n_{2\omega }\left[E^{*}(2\omega ){\frac {\partial E(2\omega )}{\partial z}}+{\text{c.c.}}\right]=-n_{\omega }\left[E(\omega ){\frac {\partial E^{*}(\omega )}{\partial z}}+{\text{c.c.}}\right]}$

where ${\displaystyle {\text{c.c.}}}$ is the complex conjugate of the other term, or

${\displaystyle n_{2\omega }\left|E(2\omega )\right|^{2}+n_{\omega }|E(\omega )|^{2}=n_{2\omega }E_{0}^{2}.}$

Phase-matched SHG with source depletion (blue), and corresponding excitation (orange). L is the interaction length (ℓ in the text).

Now we solve the equations with the premise

${\displaystyle {\begin{aligned}E(\omega )&=\left|E(\omega )\right|e^{i\varphi (\omega )}\\E(2\omega )&=\left|E(2\omega )\right|e^{i\varphi (2\omega )}\end{aligned}}}$

and obtain

${\displaystyle {\frac {d\left|E(2\omega )\right|}{dz}}=-{\frac {i\omega d_{\text{eff}}}{n_{\omega }c}}\left[E_{0}^{2}-\left|E(2\omega )\right|^{2}\right]e^{2i\varphi (\omega )-i\varphi (2\omega )}}$

which leads to

${\displaystyle \int _{0}^{\left|E(2\omega )\right|\ell }{\frac {d\left|E(2\omega )\right|}{E_{0}^{2}-\left|E(2\omega )\right|^{2}}}=-\int _{0}^{\ell }{\frac {i\omega d_{\text{eff}}}{n_{\omega }c}}e^{2i\varphi (\omega )-i\varphi (2\omega )}\,dz.}$

Using

${\displaystyle \int {\frac {dx}{a^{2}-x^{2}}}={\frac {1}{a}}\tanh ^{-1}{\frac {x}{a}}}$

we get

${\displaystyle \left|E(2\omega )\right|_{z=\ell }=E_{0}\tanh \left({\frac {-iE_{0}\ell \omega d_{\text{eff}}}{n_{\omega }c}}e^{2i\varphi (\omega )-i\varphi (2\omega )}\right).}$

If we assume a real ${\displaystyle d_{\text{eff}}}$, the relative phases for real harmonic growth must be such that ${\displaystyle e^{2i\varphi (\omega )-i\varphi (2\omega )}=i}$. Then

${\displaystyle I(2\omega ,\ell )=I(\omega ,0)\tanh ^{2}\left({\frac {E_{0}\omega d_{\text{eff}}\ell }{n_{\omega }c}}\right)}$

or

${\displaystyle I(2\omega ,\ell )=I(\omega ,0)\tanh ^{2}(\Gamma \ell ),}$

where ${\displaystyle \Gamma =\omega d_{\text{eff}}E_{0}/nc}$. From ${\displaystyle I(2\omega ,\ell )+I(\omega ,\ell )=I(\omega ,0)}$, it also follows that

${\displaystyle I(\omega ,\ell )=I(\omega ,0)\operatorname {sech} ^{2}(\Gamma \ell ).}$

It can be shown that: ${\displaystyle A_{n}=-i{\frac {\omega _{n}}{2n_{n\omega }c}}{\left(A_{0}{\sqrt {\frac {2}{\pi }}}\right)^{n}}z_{R}^{2}\int _{-\infty }^{z}{\frac {\chi ^{(n)}(u)}{q(u)^{2}}}\,du\exp \left(ik_{n}{\frac {x^{2}+y^{2}}{2q(z)}}\right)}$

(a Gaussian), is a solution of the equation (n = 2 for SHG).

Intensity SHG, phase-matched or not. The medium width is supposed to be much higher than z, the Rayleigh range at 20µm, excitation wavelength of 0.8µm, and optical index of 2.2.

A non-perfect phase-matching is a more realistic condition in practice, especially in biological samples. The paraxial approximation is however supposed still valid: ${\displaystyle k_{n}=nk_{1}}$, and in the harmonic expression, ${\displaystyle \chi ^{(n)}(z)}$ is now ${\displaystyle \chi ^{(n)}(z)e^{i\,\Delta k\,z}}$.

In the special case of SHG (n = 2), in a medium of length L and a focus position ${\displaystyle z_{0}}$, the intensity writes:[52]

${\displaystyle I_{2\omega }={\frac {2\omega ^{2}}{\pi c^{2}\varepsilon _{0}w_{0}^{2}n_{2\omega }n_{\omega }^{2}}}I_{\omega }^{2}(\chi ^{(2)})^{2}\left(\int _{z_{0}}^{z_{0}+L}{\frac {e^{i\,\Delta k\,z}}{1+iz/z_{R}}}\right)^{2}\,dz.}$

where ${\displaystyle c}$ is the speed of light in vacuum, ${\displaystyle \varepsilon _{0}}$ the vacuum permittivity, ${\displaystyle n_{n\omega }}$ the optical index of the medium at ${\displaystyle n\omega }$ and ${\displaystyle w_{0}}$ the waist size of excitation.

Thus, the SHG intensity quickly decays in the bulk (${\displaystyle 0<z_{0}<L}$), due to the Gouy phase-shift of the Gaussian beam.

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. Interestingly, the signal also vanishes in bulk for higher orders, like THG.

• Parameswaran, K. R.; Kurz, J. R.; Roussev, M. M.; Fejer (2002). "Observation of 99% pump depletion in single-pass second-harmonic generation in a periodically poled lithium niobate waveguide". Optics Letters. 27 (1): 43–45. Bibcode:2002OptL...27...43P. doi:10.1364/ol.27.000043. PMID 18007710.
• "Frequency doubling". Encyclopedia of laser physics and technology. Retrieved 2006-11-04.