Kleinman Symmetry Research Articles

Optical second harmonic generation in anisotropic multilayers with complete multireflection of linear and nonlinear waves using ♯SHAARP.ml package

Optical second harmonic generation (SHG) is a nonlinear optical effect widely used for nonlinear optical microscopy and laser frequency conversion. Closed-form analytical solution of the nonlinear optical responses is essential for evaluating materials whose optical properties are unknown a priori. A recent open-source code, ♯SHAARP.si, can provide such closed form solutions for crystals with arbitrary symmetries, orientations, and anisotropic properties at a single interface. However, optical components are often in the form of slabs, thin films on substrates, and multilayer heterostructures with multiple reflections of both the fundamental and up to ten different SHG waves at each interface, adding significant complexity. Many approximations have therefore been employed in the existing analytical approaches, such as slowly varying approximation, weak reflection of the nonlinear polarization, transparent medium, high crystallographic symmetry, Kleinman symmetry, easy crystal orientation along a high-symmetry direction, phase matching conditions and negligible interference among nonlinear waves, which may lead to large errors in the reported material properties. To avoid these approximations, we have developed an open-source package named Second Harmonic Analysis of Anisotropic Rotational Polarimetry in Multilayers (♯SHAARP.ml). The reliability and accuracy are established by experimentally benchmarking with both the SHG polarimetry and Maker fringes using standard and commonly used nonlinear optical materials as well as twisted 2-dimensional heterostructures.

New group of inorganic salts of 1,3-diaminoguanidinium(1+) cation – crystal structures, vibrational spectra, linear and nonlinear optical properties

In the search for new molecular crystals for nonlinear optics (NLO), nine inorganic salts containing the 1,3-diaminoguanidinium(1+) cation as a promising organic “Y-aromatic” carrier of NLO properties were prepared and studied. Crystal structures of eight new salts, namely 1,3-diaminoguanidinium(1+) chloride (P21/c), hypophosphite (P21/c), hydrogen phosphite (P-1), dihydrogen phosphate (P21/c), hydrogen phosphate (P21/m), nitrite (P-1), sulfite (Pbcm) and chlorate (P21/c) were solved by the single crystal X-ray diffraction. The crystal structures contain centrosymmetric arrangements of pairs of counterions assembled into hydrogen-bonded undulated sheets. The charge-assisted hydrogen bonds in the resulting hydrogen bond networks are accommodated by the conformation of the 1,3-diaminoguanidinium(1+) cation as a hydrogen bond donor. All salts were also characterized by powder X-ray diffraction. The vibrational spectra (FTIR and Raman) were studied for all newly prepared salts and the previously published non-centrosymmetric sulfate (P6122). The assignment of the vibrational bands of these hydrogen-bonded crystals is derived from quantum-chemical computations (B3LYP/6-311G+(d,p) along with the Potential Energy Distribution analysis) concerning isolated 1,3-diaminoguanidinium(1+) cation and the results of correlation analysis involving present inorganic anions. The linear and nonlinear optical properties of the only non-centrosymmetric salt – 1,3-diaminoguanidinium sulfate – were studied. This uniaxial positive crystal is optically transparent down to 230 nm, and refractive indices (ne = 1.65 and no = 1.56) have been determined at 532 nm. The dispersion curves of the refractive indices were also calculated by the Q(LFT) procedure in the 532–1907 nm region. The experimentally determined (using 800 and 1000 nm laser irradiation) very low efficiency of second harmonic generation (i.e., ∼1% of KDP), which is consistent with the Kleinman symmetry assumption for the P6122 space group, limits further use of this crystal for second-order NLO applications.

Im{χ(3)} spectra of 110-cut GaAs, GaP, and Si near the two-photon absorption band edge

Spectra of the degenerate two-photon absorption coefficient β(ω), anisotropy parameter σ(ω), and dichroism parameter δ(ω)=[σ(ω)+2η(ω)]/2 of crystalline 110-cut GaAs, GaP, and Si, at 300 K were measured using femtosecond pump–probe modulation spectroscopy over an excitation range in the vicinity of each material’s half-bandgap Eg/2 (overall 0.62<ℏω<1.91 eV or 2000>λ>650 nm). Together, these three parameters completely characterize the three independent components of the imaginary part of the degenerate third-order nonlinear optical susceptibility tensor Im{χabcd(3)(ω)}. In direct-gap GaAs, these components peak at ℏω≈0.78Eg, which is close to the peak at ℏω=0.71Eg predicted by the Jones–Reiss phenomenological model. The dispersion is comparable with ab initio calculations. In indirect-gap GaP and Si, these components tend to increase with ℏω over our tuning range. In Si, the dispersion differs significantly from predictions of semi-empirical models, and ab initio calculations do not account for transitions below the two-photon direct bandgap, motivating further investigation. Kleinman symmetry was observed to be broken in all three materials. We also note anomalies observed and their possible origins, emphasizing the advantages of a 2-beam experiment in identifying the contribution of various nonlinear effects.

Second harmonic generation responses of KH2PO4: importance of K and breaking down of Kleinman symmetry.

The second harmonic generation (SHG) responses of the paraelectric and ferroelectric phases of KH2PO4 (KDP) were calculated by first-principles density functional theory (DFT) calculations, and the individual atom contributions to the SHG responses were analyzed by the atom response theory (ART). We show that the occurrence of static polarization does not enhance the SHG responses of the ferroelectric KDP, and that the Kleinman symmetry is reasonably well obeyed for the paraelectric phase, but not for the ferroelectric phase despite that the latter has a larger bandgap. This is caused most likely by the fact that the ferroelectric phase has lower-symmetry local structures than does the paraelectric phase. The contribution to the SHG response of an individual K+ ion is comparable to that of an individual O2− ion. The contributions of the O2− and K+ ions arise overwhelmingly from the polarizable parts of the electronic structure, namely, from the valence bands of the O-2p nonbonding states and from the conduction bands of the K-3d states.

Group theory description of third harmonic generation in diamond and zincblende lattice

We provide a description that shows the importance in finding a susceptibility tensor in nonlinear optics and apply group theory to derive a fourth rank tensor of Oh point group related to the bulk symmetry of monoatomic diamond semiconductor such as Silicon. It is well known that the fourth rank tensor is closely related to the third order nonlinear susceptibility tensor in nonlinear optics through Neumann’s principle and is crucial to understand third harmonic generation (THG) and electric field induced second harmonic generation (EFISH). Using a general Rodriguez rotation, we were able to obtain a three-dimmensional rotation matrix required to perform the symmetry operations, whereas their reflection matrixes are derived using a Householder fromula. We also show how a Td point group corresponding to zincblende can be obtained by breaking the symmetry of the Oh tensor rather than repeating the tedious procedure when obtaining the Oh tensor. Assuming a driven frequency far from the resonance frequency, we can apply Kleinman symmetry and demonstrate that the number of independent components in the Oh and Td fourth rank tensor is reduced from four to just two.

Measuring third-order susceptibility tensor elements of monolayer MoS2 using the optical Kerr effect method

We present a femtosecond optical heterodyne detection of the optical Kerr effect study on Chemical Vapor Deposition-grown monolayer MoS2 films at 800 nm. The third-order nonlinear optical susceptibility (i.e., χxxyy(3)+χxyyx(3)) of monolayer MoS2 is determined to be 1.4 × 10−9 esu, and the ultrafast temporal response process indicates that the susceptibility origins from nonresonant electronic polarization. Based on Kleinman symmetry, susceptibility tensor elements are determined, and further, the nonlinear refractive indexes of any elliptically polarized light could be calculated for MoS2. These results will benefit the application of MoS2 in nonlinear photonic devices.

Molecular Tensor Analysis of Third-Harmonic Scattering in Liquids.

Third-harmonic scattering is a nonlinear optical process that involves the molecular second-hyperpolarizability, γ. This work presents a rigorous quantum electrodynamical analysis of the scattering process, involving a partially index-symmetric construction of the fourth-rank γ tensor-dispensing with the Kleinman symmetry condition. To account for stochastic molecular rotation in fluids, methods of isotropic averaging must be employed to relate the molecular properties to accessible experimental quantities such as depolarization ratio. A complete eighth-rank tensor rotational average yields results for observable third-harmonic scattering rates, cast as a function of the natural-invariant γ components, and the polarization geometry of the experiment. Decomposing the tensor γ into irreducible weights allows specific predictions to be made for each molecular point group, allowing greater discrimination between the results for different molecular symmetries.

Changes of collagen ultrastructure in breast cancer tissue determined by second-harmonic generation double Stokes-Mueller polarimetric microscopy.

Second-harmonic generation (SHG) double Stokes-Mueller polarimetric microscopy is applied to study the alteration of collagen ultrastructure in a tissue microarray containing three pathological human breast cancer types with differently overexpressed estrogen receptor (ER), progesterone receptor (PgR), and human epidermal growth factor receptor 2 (HER2). Kleinman symmetry is experimentally validated in breast tissue for 1028 nm laser wavelength and it has been shown that measurements with only linearly polarized incoming and outgoing states can determine molecular nonlinear susceptibility tensor component ratio, average in-plane orientation of collagen fibers and degree of linear polarization of SHG. Increase in the susceptibility ratio for ER, PgR, HER2 positive cases, reveals ultrastructural changes in the collagen fibers while the susceptibility ratio increase and decrease in degree of linear polarization for ER and PgR positive cases indicate alteration of the ultrastructure and increased disorder of the collagen fibers within each focal volume. The study demonstrates a potential use of polarimetric SHG microscopy for collagen characterization and cancer diagnostics.

The role of Kleinman Symmetry in the Simplified Bond Hyperpolarizability Model

We have shown previously in [J. Opt. Soc. Am. 31, 526-533 (2014)] that Kleinman symmetry plays an important role when comparing the third rank tensor associated with Second Harmonic Generation (SHG) obtained from the Simplified Bond Hyperpolarizability Model (SBHM) and from group theory (GT) for diamond and zincblende lattice. In this work we show that the third rank tensor derived from the SBHM has inherently the assumption of Kleinman symmetry.

Evaluation of the molecular static and dynamic first hyperpolarizabilities

By considering two prototypical π‐conjugated compounds, several technical aspects associated with the evaluation of the first hyperpolarizabilities have been addressed in this article, that is, (i) the automatization of the Romberg's scheme to improve the numerical accuracy in the finite field method, (ii) the evaluation of the frequency dispersion at correlated levels using approximate schemes, and (iii) the deviations from Kleinman's symmetry conditions. It results from this study that accurate numerical derivatives can be obtained by resorting to the Romberg's method and by analyzing the Romberg's table in terms of two quantities, the field error and the iteration error. Indeed, the resulting first hyperpolarizability values are in close agreement with those obtained using an analytical differentiation procedure. The reliability of the multiplicative and additive approximate schemes to describe the frequency dispersion at correlated levels from using HF (Hartree‐Fock) frequency dispersion has been confirmed to be limited to large wavelengths or far‐from‐resonance wavelength regions. Kleinman's symmetry conditions have been assessed, showing that for off‐diagonal components of these two π‐conjugated compounds, the deviations could be substantial. Nevertheless, good accuracy can be achieved for experimentally related quantities like βHRS because the diagonal tensor components are dominant. © 2014 Wiley Periodicals, Inc.

Simplified bond-hyperpolarizability model of second harmonic generation, group theory, and Neumann’s principle

We discuss the susceptibility third-rank tensor for second harmonic and sum-frequency generation, associated with low index surfaces of silicon (Si(001), Si(011), and Si(111)), from two different approaches: the simplified bond-hyperpolarizability model (SBHM) and group theory (GT). We show that the SBHM agrees very well with the experimental results for simple surfaces because the definitions of the bond vectors implicitly include the geometry of the crystal and therefore the symmetry group. However, for more complex surfaces it is shown that one can derive from GT the SBHM tensor, if Kleinman symmetry is allowed.

Investigation of symmetries of second-order nonlinear susceptibility tensor of GaSe crystals in THz domain

Following our measurements and analysis made on several GaSe crystals, we demonstrated that terahertz (THz) generation from ultrafast laser pulses can be developed into a sensitive technique for investigating symmetries of second-order nonlinear susceptibility tensor of a nonlinear crystal. Indeed, for GaSe crystals, both Kleinman's symmetry condition and spatial symmetry were violated due to the contribution of ionic displacement to nonlinear polarization and deviation of GaSe lattice from hexagonal symmetry. When the pump photon energy was increased from that below the bandgap of GaSe to that above it, the mechanism for the THz generation was switched from optical rectification to photocurrent surge.

Determination of the second-order nonlinear susceptibility elements of a single nanoparticle using coherent optical microscopy

We present a technique based on the use of coherent confocal microscopy that can be used to estimate, to within a scale factor, all of the elements of the second-order susceptibility tensor of a single pointlike nanoparticle under permutation and Kleinman symmetry. An estimate of the three-dimensional position of a nanoparticle is also obtained. A forward model for the problem is presented and a method for solving the inverse problem is demonstrated. The approach is tested through simulations which show that the position and the elements of the susceptibility tensor can be robustly retrieved using this method.

High Order Symmetry Structural Properties of Vibrational Resonances Using Multiple-Field Polarization Coherent Anti-Stokes Raman Spectroscopy Microscopy

Polarization-resolved coherent anti-Stokes Raman scattering (CARS) is usually applied to measure the depolarization ratio in solutions or evidence orientation effects in anisotropic media. We present an extensive approach based on multiple-field polarization-resolved CARS, in order to unravel the complexity of vibrational resonances up to the fourth-order symmetry, at the microscopic scale in nonisotropic media. The CARS polarized signals measured under a continuous variation of the incident pump and/or Stokes excitation beams are analyzed using a full tensorial picture both in the nonresonant and resonant regimes. This method evidences the strong influence of vibrational symmetries on polarized CARS, and more specifically the occurrence of Kleinman symmetry deviations at the vicinity of the Raman lines frequencies. This technique, illustrated on a cubic symmetry crystal, is general and can be applied to other medium symmetries.

Mapping the nonlinear optical susceptibility by noncollinear second-harmonic generation

We present a method, based on noncollinear second-harmonic generation, to evaluate the nonzero elements of the nonlinear optical susceptibility. At a fixed incidence angle, the generated signal is investigated by varying the polarization state of both fundamental beams. The resulting polarization charts allows us to verify if Kleinman's symmetry rules can be applied to a given material or to retrieve the absolute value of the nonlinear optical tensor terms, from a reference measurement. Experimental measurements obtained from gallium nitride layers are reported. The proposed method does not require an angular scan and thus is useful when the generated signal is strongly affected by sample rotation.

Reflective second harmonic generation near resonance in the epitaxial Al-doped ZnO thin film

The second harmonic (SH) generation from the highly epitaxial Al-doped ZnO film on sapphire was measured, using the femtosecond Ti:Sapphire laser at the near-resonant SH wavelength, in reflection geometry to avoid the sapphire's contribution in the conventional Maker fringes technique. By investigating SH intensities as a function of the azimuthal angle along the film's normal, we found that the sapphire substrate had a negligible contribution to the reflective SH signal and the film had a pure and well-aligned c-domain. We also developed a new method to calculate the component's ratios of the nonlinear susceptibility tensor by analyzing the polarization diagrams of SH intensities under the incidence with two different angles. The ratios indicate that Kleinman's symmetry is broken due to the absorption at SH wavelength and the dominant component of the nonlinear susceptibility tensor is d(33). Calibration using the Z-cut quartz shows a possible overestimate of the nonlinear response by Maker fringes technique.

Multipolar tensor analysis of second-order nonlinear optical response of surface and bulk of glass

We use two-beam second-harmonic generation to perform a quantitative tensor analysis of the effective dipolar surface nonlinearity and the separable multipolar bulk nonlinearity for BK7 glass. The most straightforward, self-consistent interpretation of the results is obtained when the effective surface response is assumed to have approximate Kleinman symmetry and the bulk contribution is dominated by magnetic, rather than quadrupole, effects.

Reconsideration of second-harmonic generation from isotropic liquid interface: Broken Kleinman symmetry of neat air/water interface from dipolar contribution

It has been generally accepted that there are significant quadrupolar and bulk contributions to the second-harmonic generation (SHG) reflected from the neat air/water interface, as well as common liquid interfaces. Because there has been no general methodology to determine the quadrupolar and bulk contributions to the SHG signal from a liquid interface, this conclusion was reached based on the following two experimental phenomena: the breaking of the macroscopic Kleinman symmetry and the significant temperature dependence of the SHG signal from the neat air/water interface. However, because the sum frequency generation vibrational spectroscopy (SFG-VS) measurement of the neat air/water interface observed no apparent temperature dependence, the temperature dependence in the SHG measurement has been reexamined and proven to be an experimental artifact. Here we present a complete microscopic analysis of the susceptibility tensors of the air/water interface, and show that dipolar contribution alone can be used to address the issue of the breaking of the macroscopic Kleinman symmetry at the neat air/water interface. Using this analysis, the orientation of the water molecules at the interface can be obtained, and it is consistent with the measurement from SFG-VS. Therefore, the key rationales to conclude significantly quadrupolar and bulk contributions to the SHG signal of the neat air/water interface can no longer be considered as valid as before. This new understanding of the air/water interface can shed light on our understanding of the nonlinear optical responses from other molecular interfaces as well.

Hyperpolarizabilities for the one-dimensional infinite single-electron periodic systems. I. Analytical solutions under dipole-dipole correlations

The analytical solutions for the general-four-wave-mixing hyperpolarizabilities chi(3)[-(omega1 + omega2 + omega3);omega1,omega2,omega3] on infinite chains under both Su-Shrieffer-Heeger and Takayama-Lin-Liu-Maki models of trans-polyacetylene are obtained through the scheme of dipole-dipole correlation. Analytical expressions of dc Kerr effect chi(3)(-omega;0,0,omega), dc-induced second-harmonic generation chi(3)(-2omega;0,omega,omega), optical Kerr effect chi(3)(-omega;omega,-omega,omega), and dc electric-field-induced optical rectification chi(3)(0;omega,-omega,0) are derived. By including or excluding nabla(k) terms in the calculations, comparisons show that the intraband contributions dominate the hyperpolarizabilities if they are included. nabla(k) term or intraband transition leads to the break of the overall permutation symmetry in chi(3) even for the low-frequency and nonresonant regions. Hence it breaks the Kleinman [Phys. Rev. 126, 1977 (1962)] symmetry that is directly based on the overall permutation symmetry. Our calculations provide a clear understanding of the Kleinman symmetry breaks that are widely observed in many experiments. We also suggest a feasible experiment on chi(3) to test the validity of overall permutation symmetry and our theoretical prediction. Finally, our calculations show the following trends for the various third-order nonlinear optical processes in the low-frequency and nonresonant region: chi(nonres)(3)(-3omega;omega,omega,omega)>chi(nonres)(3)(-2omega;0,omega,omega)>chi(nonres)(3)(-omega;omega,-omega,omega)>chi(nonres)(3)(-omega;0,0,omega) > or = chi(nonres)(3)(0;omega,-omega,0), and in the resonant region: chi(res)(3)(-omega;0,0,omega)>chi(res)(3)(-omega;omega,-omega,omega)>chi(res)(3)(-2omega;0,omega,omega)>chi(res)(3)(0;omega,-omega,0)>chi(res)(3)(-3omega;omega,omega,omega).

Sign Change of Hyperpolarizabilities of Solvated Water, Revised: Effects of Equilibrium and Nonequilibrium Solvation

The second harmonic and electro-optical responses for the water molecule in its liquid state are theoretically revised. The continuum, semicontinuum, and supermolecular solvation models are employed using quadratic response theory at the Hartree-Fock level, either in the equilibrium or nonequilibrium implementation. The experimentally observed sign change of the second harmonic response of water on liquefaction is reproduced using the supermolecular and semicontinuum model for both equilibrium and nonequilibrium solvation. The conclusions of a previous study, which rested upon linear response theory in an equilibrium implementation, are confirmed. Also, the assumption of Kleinman symmetry and dispersion is addressed.