Model Of Second-harmonic Generation Research Articles

Polarization-Resolved Extreme-Ultraviolet Second-Harmonic Generation from LiNbO3

Second harmonic generation (SHG) spectroscopy ubiquitously enables the investigation of surface chemistry, interfacial chemistry, as well as symmetry properties in solids. Polarization-resolved SHG spectroscopy in the visible to infrared regime is regularly used to investigate electronic and magnetic order through their angular anisotropies within the crystal structure. However, the increasing complexity of novel materials and emerging phenomena hampers the interpretation of experiments solely based on the investigation of hybridized valence states. Here, polarization-resolved SHG in the extreme ultraviolet (XUV-SHG) is demonstrated for the first time, enabling element-resolved angular anisotropy investigations. In noncentrosymmetric LiNbO_{3}, elemental contributions by lithium and niobium are clearly distinguished by energy dependent XUV-SHG measurements. This element-resolved and symmetry-sensitive experiment suggests that the displacement of Li ions in LiNbO_{3}, which is known to lead to ferroelectricity, is accompanied by distortions to the Nb ion environment that breaks the inversion symmetry of the NbO_{6} octahedron as well. Our simulations show that the measured second harmonic spectrum is consistent with Li ion displacements from the centrosymmetric position while the Nb─O bonds are elongated and contracted by displacements of the O atoms. In addition, the polarization-resolved measurement of XUV-SHG shows excellent agreement with numerical predictions based on dipole-induced SHG commonly used in the optical wavelengths. Our result constitutes the first verification of the dipole-based SHG model in the XUV regime. The findings of this work pave the way for future angle and time-resolved XUV-SHG studies with elemental specificity in condensed matter systems.

Analytical model of surface second-harmonic generation

The process of second-harmonic generation (SHG) in a finite one-dimensional nonlinear medium is analyzed in parallel by the Green-function technique and the Fourier-transform method. Considering the fundamental pump field propagating along a given direction and eliminating back-reflections at the boundaries the terms giving the surface second-harmonic fields in the particular solution of the wave equation are uniquely identified. Using these terms the flow of energy corresponding to the surface second-harmonic fields is analyzed in the vicinity of the boundaries. The formula giving the depth of the nonlinear medium contributing to the surface SHG is obtained. Both approaches for describing the SHG are compared considering complexity and quantization of the interacting fields. In addition, a theoretical model of surface SHG in centrosymmetric media is proposed. The model is built upon assumption that the second-order nonlinearity decays exponentially with distance from the boundary. As an important example, the generation of surface SHG from a thin dielectric nonlinear layer placed on a silicon substrate is analyzed by the proposed model.

Influence of attenuation on self-organized second-harmonic generation in a germanium-doped microstructured silica fiber.

Observation of the efficient second-harmonic generation (SHG) in silica fibers drew a lot of interest, as second-order processes in centrosymmetric materials should not occur. In this Letter, we extended a theoretical model of self-organized SHG to include an attenuation and investigated the influence of fiber loss on the self-organized SHG process. We performed calculations of energy conversion efficiency for SHG in microstructured optical fibers. Finally, we referred the simulation results to the measured SHG efficiency in a microstructured side-hole germanium-doped silica fiber. The extended model should be applied for the fibers with loss beyond 0.01dB/m. Even in such fibers, efficient SHG without an external second-harmonic beam is possible.

Second-harmonic generation in a metal-clad nonlinear optical waveguide.

Efficient optical nonlinear effects in waveguides play an important role in integrated photonic functionalities. The dispersion characteristics need to be well designed to satisfy the phase-matching condition of the interacting waves in waveguides. Here we demonstrate a novel phase-matching process of second-harmonic generation (SHG) in a symmetrical metal-cladding optical waveguide (SMCOW) with a nonlinear guiding layer. Ultrahigh order modes in SMCOWs possess small propagation constants, and can be actively tuned to satisfy the phase-matching condition via free-space coupling. We establish a model of SHG in the SMCOW and experimentally verify it as well. This mechanism could also be applied to or referenced in other nonlinear frequency conversion processes.

Direct observation of bulk second-harmonic generation inside a glass slide with tightly focused optical fields

Bulk second-harmonic generation (SHG) inside glass slides is directly detected unambiguously without interference from surface contributions. This is enabled by tightly focused and highly localized ultrashort laser pulses. The theoretical calculations based on vector diffraction theory and the phenomenological model of SHG inside centrosymmetric materials agree well with the measured far-field SHG radiation patterns for different polarization states of the fundamental beam. The results indicate that the observed bulk SHG is predominantly related to the bulk parameter ${\ensuremath{\delta}}^{\ensuremath{'}}$ and originates from the three-dimensional field gradient in the focal region.

Fully automated muscle quality assessment by Gabor filtering of second harmonic generation images

Although structural changes on the sarcomere level of skeletal muscle are known to occur due to various pathologies, rigorous studies of the reduced sarcomere quality remain scarce. This can possibly be explained by the lack of an objective tool for analyzing and comparing sarcomere images across biological conditions. Recent developments in second harmonic generation (SHG) microscopy and increasing insight into the interpretation of sarcomere SHG intensity profiles have made SHG microscopy a valuable tool to study microstructural properties of sarcomeres. Typically, sarcomere integrity is analyzed by fitting a set of manually selected, one-dimensional SHG intensity profiles with a supramolecular SHG model. To circumvent this tedious manual selection step, we developed a fully automated image analysis procedure to map the sarcomere disorder for the entire image at once. The algorithm relies on a single-frequency wavelet-based Gabor approach and includes a newly developed normalization procedure allowing for unambiguous data interpretation. The method was validated by showing the correlation between the sarcomere disorder, quantified by the M-band size obtained from manually selected profiles, and the normalized Gabor value ranging from 0 to 1 for decreasing disorder. Finally, to elucidate the applicability of our newly developed protocol, Gabor analysis was used to study the effect of experimental autoimmune encephalomyelitis on the sarcomere regularity. We believe that the technique developed in this work holds great promise for high-throughput, unbiased, and automated image analysis to study sarcomere integrity by SHG microscopy.

Back-reflection Second-harmonic Generation of (111)Si: Theory and Experiment

We consider second-harmonic generation (SHG) from a (111) surface of a tetrahedrally bonded semiconductor illuminated at normal incidence by a focused pump beam of Gaussian cross section as a model of SHG by focused beams. Calculations are done in the anisotropic bond model (ABM) and the results are applied to Si. The unit-cell configuration is simple enough for the calculations to be done analytically, so the results can be compared directly to similar calculations done for amorphous material. Although the differences in unit-cell symmetry occur on the atomic scale, they lead to large differences in the spatial distribution of the emerging radiation. Lateral focusing, which might be expected to increase the bulk contribution to SHG by increasing the lateral field gradient, has little effect; the spatial-dispersion contribution remains dominated by the phase term. Focusing does not inhibit backscattered SHG from the bulk, although our data on the oxidation of H-terminated (111)Si clearly show that in some cases the interface contribution dominates by a wide margin.

Correlation between second-order optical response and structure in thermally poled sodium niobium-germanate glass

We have carried out Raman and second harmonic generation (SHG) measurements to probe thermal poling-induced phenomena in glass 20Na2O-80[0.35Nb2O5-0.65GeO2]. A SHG response of 0.6 pm/V was measured after poling in a ∼3 μm thick layer under the anode and found to deviate from the widely used electric field-induced SHG model. This effect was associated with complex structural rearrangements in the subanode layer involving destruction of nonbridging oxygen atoms, formation of molecular oxygen, and enhancement of cross-linking in the glass network.

Second-harmonic and reflectance-anisotropy spectroscopy of vicinalSi(001)∕SiO2interfaces: Experiment and simplified microscopic model

We directly compare experimental second-harmonic generation (SHG) spectra and reflectance-anisotropy spectra (RAS) of native-oxidized vicinal Si(001) interfaces with off-cut angles $\ensuremath{\zeta}=0\ifmmode^\circ\else\textdegree\fi{}$, 4\ifmmode^\circ\else\textdegree\fi{}, 6\ifmmode^\circ\else\textdegree\fi{}, 8\ifmmode^\circ\else\textdegree\fi{}, and 10\ifmmode^\circ\else\textdegree\fi{} from (001) toward [110]. We fit the measured azimuthal rotational-dependence of $p$-in/$p$-out SHG at two-photon energies $2.8\phantom{\rule{0.3em}{0ex}}\mathrm{eV}l\ensuremath{\hbar}\ensuremath{\omega}l3.5\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ including the ${E}_{1}$ critical point resonance using a simplified bond hyperpolarizability model. The model includes dipolar contributions from tetrahedrally coordinated interfacial bonds as well as quadrupolar contributions from the Si bulk. The fit yields complex axial hyperpolarizability spectra ${\ensuremath{\beta}}_{\ensuremath{\Vert},\ensuremath{\Vert},\ensuremath{\Vert}}(2\ensuremath{\omega})$ and averaged bond angles of the interfacial back-edge, terrace-edge and step-edge bonds, and bulk quadrupolar hyperpolarizability ${\ensuremath{\delta}}_{\ensuremath{\Vert},\ensuremath{\Vert},\ensuremath{\Vert}}(2\ensuremath{\omega})$. The fitted microscopic model accurately reproduces measured $s$-in/$p$-out SHG spectra. We then used a Miller's rule approximation to generate axial linear polarizability spectra ${\ensuremath{\alpha}}_{\ensuremath{\Vert}}(\ensuremath{\omega})$ from the fitted ${\ensuremath{\beta}}_{\ensuremath{\Vert},\ensuremath{\Vert},\ensuremath{\Vert}}(2\ensuremath{\omega})$. The measured RAS was satisfactorily reproduced within the same photon energy range. Including the bulk quadrupole contribution to SHG was crucial to accurately retrieving the RAS and to achieving fitted real and imaginary $\ensuremath{\beta}(2\ensuremath{\omega})$ spectra consistent with a nonlinear Kramers-Kronig relation. The results demonstrate the possibility of formulating a common microscopic model of SHG and RAS responses of complicated interfaces.

Surface plasmon resonance enhanced second-harmonic generation in Kretschmann configuration

We report second-harmonic generation (SHG) from a thin gold film at surface plasmon resonance (SPR) in an attenuated total reflection (ATR) geometry having the Kretschmann configuration. The SHG intensity was observed to be 350–400 larger than the reflected SHG intensity in the Fresnel reflection geometry at the surface of the thin gold film. A similar enhancement was observed in the case of a thin gold film covered with a hemicyanine monolayer. These results suggest that the SHG source of the thin gold film is located outside this film. Simulation based on this model yields an enhancement factor of ∼285, in agreement with the experimental results, whereas the simulation supposing that the SHG source is located inside the gold thin film gives a factor of over 6000. This model is consistent with a hydrodynamic model of SHG in a metal, in which the nonlinear current runs outside the bulk metal and the electron density penetrates from the metal to the air at their interface.

Characteristics of a UV beam generated by a frequency doubled Ar-ion laser

Frequency doubled Ar-ion lasers have become important UV sources for many different applications in photonics and semiconductor industry. In this study, we analyze the beam characteristics of an UV beam generated by a frequency doubled Ar-ion laser by using a second harmonic generation (SHG) model. Beam profiles at different locations are computed and compared with the measured profiles. The beam propagation factor, M 2, is also estimated using the SHG model and the parameters of the laser. It has been demonstrated that in the walk-off plane the profiles of the second harmonic beam are not Gaussian, and changing along the beam propagation direction. The calculations also reveal the wavefront shapes of the second-harmonic beam at different locations. As the existence of the walk-off, the shapes of the wavefronts are asymmetrical, and changing with the distance of propagation from the near- to far-field.

Embedded solitons in Lagrangian and semi-Lagrangian systems

We develop the technique of the variational approximation (VA) for solitons in two directions. First, one may have a physical model which does not admit the usual Lagrangian representation, as some terms were discarded for various reasons. For instance, the second-harmonic-generation (SHG) model considered here, which includes the Kerr nonlinearity, lacks the usual Lagrangian representation if one ignores the Kerr nonlinearity of the second-harmonic, as compared to that of the fundamental. However, we show that, with a natural modification, one may still apply the VA to those seemingly flawed systems as efficiently as it applies to their fully Lagrangian counterparts. We call such models, that do not admit the usual Lagrangian representation, semi-Lagrangian systems. Second, we show that, upon adding an infinitesimal tail that does not vanish at infinity, to a usual soliton ansatz, one can obtain an analytical criterion which (within the framework of VA) gives a condition for finding embedded solitons (ESs), i.e., isolated truly localized solutions existing inside the continuous spectrum of the radiation modes. The criterion takes a form of orthogonality of the radiation mode in the infinite tail to the soliton core. To test the criterion, we have applied it to both the semi-Lagrangian truncated version of the SHG model and to the same model in its full form. In the former case, the criterion (combined with VA for the soliton proper) yields an exact solution for the ES. In the latter case, the criterion selects the ES with a relative error ≈1%.

Influence of phase transitions on the second harmonic generation in metal nanoparticles

In this work we use nonlinear optical properties of Ga nanoparticles monolayers of different average sizes embedded in dielectric matrices to investigate the liquid-solid phase transitions in these materials. Ga nanoparticles, formed by exploiting the partial wetting of liquid Ga over a SiOx surface are irradiated with fs laser pulses from a Ti:sapphire source. The resulting Second Harmonic (SH) generated in the reflection and transmission directions is measured along the phase transitions by cooling the sample from 320 K down to liquid nitrogen temperature. Hysteresis cycles are observed in the nonlinear transmission, which exhibit a strong amplification from the solid to liquid values as compared to the linear optical results. A simple model for SH generation, based on Mie scattering calculations which includes the effect of surface plasmon resonance provides a fair key for the interpretation of the observed effects.

Embedded solitons: solitary waves in resonance with the linear spectrum

It is commonly held that a necessary condition for the existence of solitons in nonlinear-wave systems is that the soliton’s frequency (spatial or temporal) must not fall into the continuous spectrum of radiation modes. However, this is not always true. We present a new class of codimension-one solitons (i.e., those existing at isolated frequency values) that are embedded into the continuous spectrum. This is possible if the spectrum of the linearized system has (at least) two branches, one corresponding to exponentially localized solutions, and the other to radiation modes. An embedded soliton (ES) is obtained when the latter component exactly vanishes in the solitary-wave’s tail. The paper contains both a survey of recent results obtained by the authors and some new results, the aim being to draw together several different mechanism underlying the existence of ESs. We also consider the distinctive properties of semi-stability of ESs, and moving ESs. Results are presented for four different physical models, including an extended fifth-order KdV equation describing surface waves in inviscid fluids, and three models from nonlinear optics. One of them pertains to a resonant Bragg grating in an optical fiber with a cubic nonlinearity, while two others describe second-harmonic generation (SHG) in the temporal or spatial domain (i.e., respectively, propagating pulses in nonlinear-optical fibers, or stationary patterns in nonlinear planar waveguides). Special attention is paid to the SHG model in the temporal domain for a case of competing quadratic and cubic nonlinearities. In particular, a new result is that when both harmonics have anomalous dispersion, an ES can exist which is, virtually, completely stable.

In Situ Observation of a Phase Transition in a Thin Molecular Film by Optical Second Harmonic Generation

By use of optical second harmonic generation (SHG), an irreversible phase transition from a disordered to an ordered structure is observed in thin pyridine (C5H5N) films dosed on a Ag(111) surface under ultrahigh vacuum. This phase transition occurs when the surface temperature is raised above 125 K. An analytical model of SHG from the annealed pyridine film including optical interference is presented. It is most likely that the annealed films are composed of crystallites randomly oriented on the surface plane with an average length scale comparable to or larger than half a micrometer.

Second-harmonic generation in atomic vapor with picosecond laser pulses

Picosecond laser pulses were used to study the highly forbidden resonant second-harmonic generation (SHG) in potassium vapor. The input intensity dependence, vapor density dependence, buffer-gas pressure dependence, and spatial profile of the SHG were measured. A pump–probe experiment was conducted to probe the time dependence of the SHG signal. The experimental results can be understood from an ionization-initiated dc-field-induced SHG model. A theory of a dc-field-induced SHG model is developed that takes into account the time development of the dc electric field in detail. This temporal buildup of the dc field along with transient coherent excitation between two-photon-allowed transitions can explain the experimental results quantitatively, including the previous vapor SHG results with nanosecond laser pulses.

Key trade-offs for second harmonic generation in poled polymers

We discuss the critical trade-off between the spectral dispersion in the second order nonlinearity d(2) and the linear absorption coefficient for second harmonic generation (SHG) by the mixing of copropagating or counterpropagating fundamental beams in poled polymers. The 4-dimethylamino–4′-nitrostilbene side-chain polymer is used as a specific example. The predictive power of the classical two-level model for SHG is also assessed.

Picosecond Studies of Optical Second Harmonic Generation in Atomic Vapor.

Picosecond laser pulses are used to study the forbidden resonant second harmonic generation (SHG) in atomic vapor. The results can be understood quantitatively by an ionization-initiated dc-field-induced SHG model taking into account the temporal buildup of the dc field and transient behavior of the two-photon coherent excitation of the vapor.