Second-harmonic generation

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.^[1] 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^[2] 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 sum-frequency generation (2 photons), and more generally of harmonic generation.

The second-order

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.^[9] 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,^[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.^[10] The formulation of SHG was initially described by N. Bloembergen and P. S. Pershan at Harvard in 1962.^[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.

Types in crystals

Critical phase-matching

Second-harmonic generation occurs in three types for critical phase-matching,

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.

Optical second-harmonic generation

Since media with

In SHG spectroscopy, one focuses on measuring twice the incident frequency 2ω given an incoming electric field ${\displaystyle E(\omega )}$ 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, ${\displaystyle P^{(2)}(2\omega )}$, can be written as

${\displaystyle E(2\omega )\propto P^{(2)}(2\omega )=\chi ^{(2)}E(\omega )E(\omega )}$

where ${\displaystyle \chi ^{(2)}}$ is known as the nonlinear susceptibility tensor and is a characteristic to the materials at the interface of study.^[15] The generated ${\displaystyle E(2\omega )}$ and corresponding ${\displaystyle \chi ^{(2)}}$ 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.

From planar surfaces

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.

From non-planar surfaces

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]

Radiation pattern

The SHG radiation pattern generated by an exciting Gaussian beam also has a (homogeneous) 2D Gaussian profile if the nonlinear medium being excited is homogeneous (A). However, if the exciting beam is positioned at an interface between opposite polarities (± boundary, B) that is parallel to the beam propagation (see figure), the SHG will be split into two lobes whose amplitudes have opposite sign, i.e. are ${\displaystyle \pi }$ phase-shifted. ^[24]

These boundaries can be found in the sarcomeres of muscles (protein = myosin), for instance. Note that we have considered here only the forward generation.

Moreover the SHG phase-matching can also result in ${\displaystyle {\vec {k}}_{2\omega }=-2{\vec {k}}_{\omega }}$: some SHG is also emitted in backward (epi direction). When the

The forward (F) to backward (B) ratio is dependent on the arrangement of the different dipoles (green in figure) that are being excited. With only one dipole ((a) in the figure), F = B, but F becomes higher than B when more dipoles are stacked along the propagation direction (b and c). However, the Gouy phase-shift of the Gaussian beam will imply a ${\displaystyle \pi }$ phase-shift between the SHGs generated at the edges of the focal volume, and can thus result in destructive interferences (zero signal) if there are dipoles at these edges having the same orientation (case (d) in the figure).

Commercial uses

Second-harmonic generation is used by the laser industry to make green 532 nm lasers from a 1064 nm source. The 1064 nm light is fed through a bulk

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]

Second-harmonic generation microscopy

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

Second-harmonic generation microscopy is also used in material science, for instance to characterize nanostructured materials.[36]

Characterization of crystalline materials

Second harmonic generation is also relevant to characterize organic or inorganic crystals

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

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.)

With depletion

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

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

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

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

Theoretical expression with Gaussian beams

The excitation wave is assumed to be a Gaussian beam, of amplitude: ${\displaystyle A_{1}=A_{0}{\sqrt {\frac {2}{\pi }}}{\frac {z_{R}}{iq(z)}}\exp \left(ik_{1}{\frac {x^{2}+y^{2}}{2q(z)}}\right)}$

with ${\displaystyle q(z)=z-iz_{R}}$, ${\displaystyle z}$ the direction of propagation, ${\displaystyle z_{R}}$ the Rayleigh range, ${\displaystyle {k}_{1}}$ the wave vector.

Each wave verifies the wave equation

${\displaystyle \left[{\frac {\partial }{\partial x^{2}}}+{\frac {\partial }{\partial y^{2}}}+2ik_{1}{\frac {\partial }{\partial z}}\right]A(x,y,z;k_{1})={\begin{cases}0&{\text{for the fundamental}},\\{\frac {\omega _{n}^{2}c^{2}}{\chi ^{(n)}}}A(x,y,z;k_{1})e^{i\,\Delta k\,z}&{\text{for }}n{\text{-th harmonic}}.\end{cases}}}$

where ${\displaystyle \Delta k=k_{n}-k_{1}}$.

With phase-matching

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

No phase-matching

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.

Materials used

Materials capable of generating a second harmonic are crystals without inversion symmetry, except crystals with point group 432. This eliminates water and glass.^[50]

Here are some crystals used with certain types of laser for SHG conversion:

• Fundamental excitation at 600–1500 nm:^[53] BiBO (BiB₃O₆)
• Fundamental excitation at 570–4000 nm:^[54] lithium iodate LiIO₃.
• Fundamental excitation at 800–1100 nm, often 860 or 980 nm:^[55] potassium niobate KNbO₃.
• Fundamental excitation at 410–2000 nm: BBO (β-BaB₂O₄).^[56]
• Fundamental excitation at 984–3400 nm: KTP (KTiOPO₄) or KTA.^[57]
• Fundamental excitation at 1064 nm: monopotassium phosphate KDP (KH₂PO₄), lithium triborate (LiB₃O₅), CsLiB₆O₁₀ and barium borate BBO(β-BaB₂O₄).
• Fundamental excitation at 1319 nm: KNbO₃, BBO (β-BaB₂O₄), monopotassium phosphate KDP (KH₂PO₄), LiIO₃, LiNbO₃, and potassium titanyl phosphate KTP (KTiOPO₄).
• Fundamental excitation at ~1000–2000 nm: periodically poled crystals, like PPLN.^[58]

Notably, filamentous biological proteins with a cylindrical symmetric such as

• Half-harmonic generation
• Harmonic generation
• Nonlinear optics
• Optical frequency multiplier
• Second-harmonic imaging microscopy
• Spontaneous parametric down-conversion
• Surface second harmonic generation

References