Semi-classical theory of two-photon resonant third-harmonic generation

A semi-classical density matrix theory based on the Method of Averages is used to derive an expression for the third-harmonic generation intensity I^3ω as a function of the fundamental intensity I^ω in the neighbourhood of a two-photon resonance. The result takes account of optical Stark shifts, two-photon saturation and both homogeneous and inhomogeneous broadening. At high intensities, I^3ω∝I^ω, in contrast to the case where optical Stark shifts are ignored for which I^3ω∝(I^ω)^-1.     

Nonresonant third order hyperpolarizability of rare gases and N2 determined by third harmonic generation

The third order hyperpolarizabikity γ⁽³⁾₁₁₁₁(-ω₃;ω₁,ω₁,ω₁) of the rare gases He, Ne, Ar, Kr, Xe and of N₂ are determined by third harmonic generation involving picosecond light pulses of a Nd-glass laser. The results are compared with reported experimental and theoretical values.     

High harmonic generation in H2 + and HD + by two-colour femtosecond laser pulses

We have theoretically investigated the high harmonic generation (HHG) spectra of H₂ ^?+? and HD^?+? using a time-dependent wave packet approach for the nuclear motion with combined two-colour (1ω _L –3ω _L ) pulsed lasers for ω _L corresponding to wavelengths 1064 nm and 800 nm. The 1ω _L and 3ω _L lasers have peak intensities of I ₁ ⁰?=?5.0×10¹³ W/cm² and I ₂ ⁰?=?2.0×10¹⁴ W/cm², respectively. We have taken the pulse duration of T = 50 fs for both the fields, and the molecular initial vibrational level v ₀?=?0. We have argued that for these combinations, the harmonic generation due to transitions in the electronic continuum by tunnelling or multiphoton ionization may be neglected and only the electronic transitions within the two lowest electronic states would be important. Thus, the characteristic features of HHG spectra in the two-colour field are determined, in our model, by the nuclear motions on the two lowest field-coupled electronic states between which interelectronic and intraelectronic (due to the intrinsic dipole moments in case of HD^?+?) radiative transitions can take place. We have studied the role of relative phase (φ) of the two fields on the HHG spectra of the molecular ions. In case of HD^?+?, the effect of nonadiabatic (NA) nonradiative interaction between the two lowest Born-Oppenheimer (BO) electronic states (1sσ _g , 2pσ _u ) has been taken into account. Our calculations give realistic HHG spectra which are reasonably efficient and extended for both H₂ ^?+? and HD^?+? in the mixed two-colour field without involving the electronic continuum. The use of two-colour (1ω _L –3ω _L ) field enables us to generate high harmonics beyond that achievable with a single 1ω _L or 3ω _L field of the corresponding intensity, frequency and pulse time.     

Theory for the nonlinear optical response at noble-metal surfaces with nonequilibrium electrons

We present a theory for the electron-temperature dependence T _el of optical second harmonic generation (SHG). Such an analysis is required to study the dynamics of metallic systems with many hot electrons not at equilibrium with the lattice. Using a tight-binding theory for the nonlinear susceptibility χ ⁽²⁾(ω,T _e1) and the Fresnel coefficients we present results for the SHG intensity I ⁽²⁾(ω,T _e1) and its dependence on T _el for Cu. Note, χ ⁽²⁾(ω,T _e1) rather than the Fresnel coefficients determines essentially this temperature dependence. Most interestingly we find frequency ranges where I ⁽²⁾(ω,T _e1) increases for small light intensities, while it decreases for large light intensities. Our theory yields also that SHG probes effects due to hot electrons more sensitively than linear optics. The results of our calculations are compared with recent experiments on Cu and Au.     

Nonlinear susceptibilities of vapors and solutions of organic dyes

Results of theoretical studies of laser and Kerr nonlinear susceptibilities of vapors and solutions of organic dyes using a series of polycyclic arenes as an example are presented. Nonlinear susceptibilities of the third χ⁽³⁾ (?3ω; ω, ω, ω) and the fifth χ⁽⁵⁾ (?3ω; ω, ω, ω, ω, ?ω) orders of a series of organic dyes responsible for third harmonic generation of Nd:YAG laser radiation are calculated within the context of the free electron model. Results of calculations of their Kerr third-order nonlinear susceptibilities χ⁽³⁾ (?ω; ω, ?ω, ω) and non-linear refractive indices n ₂ are presented. The calculation results are compared with experimental data on third harmonic generation in naphthalene vapors and with χ⁽³⁾ (?ω; ω, ?ω, ω) as well as n ₂ of paraterphenyl and naphthalene solutions.     

Third-order optical harmonic coefficients of atomic hydrogen at low frequencies

The analytical expression is derived for the third-order optical harmonic coefficients χ⁽³⁾₁₁₁₁ (ω, ω, ω) = χ⁽³⁾₂₂₂₂ (ω, ω, ω) = χ⁽³⁾₃₃₃₃ (ω, ω, ω) of the atomic hydrogen at very low frequencies ω → 0.     

Suppression of FM-to-AM conversion in phase modulated third-harmonic generation process

Phase modulation to amplitude modulation (FM-to-AM) conversion occurring in phase modulated third-harmonic generation (THG, process 1ω + 2ω → 3ω) process is investigated. To suppress the FM-to-AM conversion, a group-velocity-matching relationship is derived, which appears to be dependent on the ratio of modulation depth of 2ω–1ω pulses. Numerical simulations indicate that the FM-to-AM conversion in phase-modulated THG process can be suppressed effectively by satisfying that group-velocity-matching relationship.     

Noncollinear phase matched four photon frequency mixing in water

An intense ultraviolet picosecond light pulse at ω_S = 2γ_P ? ω_L is generated in water by noncollinear phase matched nonresonant four photon frequency mixing of two input picosecond light pulses at frequencies ν~_P = 18 960 cm^-1 and ν~_L = 9480 cm^-1. An energy conversion of up to W_S/W_L = 0.07 was achieved. The nonlinear susceptibility components were determined to be χ⁽³⁾_yyyy(?ω_S; ω_P, ω_P, ? ω_L) = 7.5 × 10^-34 Cm/V³ and χ⁽³⁾_yxxy(?ω_S; ω_P, ω_P, ?ω_L) = 2.4 × 10^-34 Cm/V³.     

THG in dielectrics using low-energy tightly-focused ir femtosecond laser: third-order nonlinearity measurements and the evolution of laser-induced plasma

The third-order nonlinearity x ⁽³⁾ and other optical properties of LiF, CaF₂, fused silica, and free standing polymeric thin film are studied using a diagnostic technique based on the third-harmonic generation of tightly-focused femtosecond Cr:forsterite laser radiation. Time-resolved measurements of the plasma evolution in the bulk of a fused silica sample are performed by a pump-probe technique using the fundamental and third-harmonic radiation as a probe. With the use of the pump-probe technique, two types of the relaxation processes are registered.     

Spin resonant optical mixing in InSb — CW-measurements and transient effects

Optical four-wave mixing due to the resonant spin nonlinearity in n-InSb was observed in magnetic fields up to 1.4 T. The mixing of two CO-laser frequencies ω₁ and ω₂ = ω₁ - Δω yields radiation with frequencies ω₁ + 2Δω, ω₁ + Δω, ω₁ - 2Δω, and ω₁ - 3Δω. We found the resonant contribution to the third order nonlinear susceptibility arising from the spin-flip interaction to increase by more than two orders of magnitude when ω₁ is varied from 1760 cm^-1 to 1896 cm^-1. With the lasers working simultaneously Q-switched we observed radiation at ω₁ + Δω and ω₁ - 2Δω due to the mixing caused by the nonparabolicity of the conduction electrons even without an external magnetic field. At the resonant magnetic field the intensity of the mixed radiation is increased by more than one order of magnitude and is also observed when the laser pulses do not hit the sample simultaneously as long as the time difference between the two pulses does not exceed 450 ns. This points to a spin dephasing time T₂ of about 100 ns.     

Generation of the third harmonic in fullerene-containing polyimide films by picosecond radiation of the Nd: YAG laser

The results of the studies on the transformation of the frequency of laser radiation in polyimide films doped by C₇₀ are presented. The nonlinear susceptibility (χ⁽³⁾) of polyimide 6B films with various concentrations of C₇₀ is studied by third-harmonic generation. At a wavelength of 1064 nm, the value of χ⁽³⁾(−3ω; ω, ω, ω) is 9×10⁻¹³ esu. The effect of the nonlinear variation in the refractive index of the fullerene-containing films on the phase matching between the pumping and third-harmonic waves is analyzed. The third harmonic of the Nd: YAG laser radiation was generated with the efficiencies 6×10⁻⁶ and 10⁻⁶ in the films with the C₇₀ concentration of 0.5 and 0.2%, respectively.     

Four-wave mixing and hyper-Raman scattering in CuCl

Non-degenerate four-wave mixing using two non-collinear laser beams with frequencies (wavevectors) ω_p, ω_t (k_p, k_t) respectively is studied in CuCl. Two emission lines at frequencies ω⁽¹⁾=2ω_t-ω_p, and ω⁽²⁾=2ω_p-ω_t are observed. Their excitation spectrum is sharply peaked if the phase-match condition k⁽¹⁾=2k_t-k_p is fulfilled. This is the case, if ω_p coincides with the hyper-Raman lines (R⁺_T, R^-_T) of the laser labelled (t) in a well-defined geometrical configuration.     

Multiphoton SRS-Lasing in a Monoclinic β-BaYb<Subscript>2</Subscript>F<Subscript>8</Subscript> Crystal

The discovery of stimulated Raman scattering (SRS) in monoclinic fluoride β-BaYb₂F₈ known as a host matrix for Ln³⁺ lasant ions was reported. All the recorded spectral components of Stokes and anti-Stokes χ⁽³⁾-nonlinear picosecond generation were assigned to the three SRS-active photon A_g- and B_g-modes of a crystal (ω_SRS1 ~ 362 cm^–1, ω_SRS2 ~ 295 cm^–1, and ω_SRS3 ~ 230 cm^–1).     

Double resonance interference in third-order light mixing

The first observation of interference effects between two sharp resonant contributions to the third-order nonlinear susceptibility χ₍₃₎ (ω₃=ω₀+ω₁?ω₂) is reported. Using two tunable near ir frequencies, ω₀=ω₁ and ω₂, the was seen in crystalline CuCl when 2ω₁ and ω₁?ω₂ were close to the sharp Z₃ exciton at 3.21 eV and the 210 cm^-1 Raman line, respectively. Three different tunable visible frequencies were used to observe similar interference effects in a benzene-cyclohexane mixture when ω₀?ω₂ and ω₁?ω₂ were close to the 992 cm^-1 Raman line of benzene and the 801 cm^-1 Raman line of cyclohexane, respectively.     

Effects on spectroscopic properties for several low-lying electronic states of CS molecule by core-valence correlation and relativistic corrections

The potential energy curves (PECs) of six low-lying electronic states (X¹Σ⁺, a³Π, a′³Σ⁺, d³Δ, e³Σ⁻ and A¹Π) of CS molecule have been investigated using the full valence complete active space self-consistent field (CASSCF) method followed by the highly accurate valence internally contracted multireference configuration interaction (MRCI) approach with large correlation-consistent basis sets. Effects on the PECs by the core-valence correlation and relativistic corrections have been taken into account. And the two corrections are performed at the level of cc-pV5Z basis set. The way to consider the relativistic corrections is to use the second-order Douglas-Kroll Hamiltonian approximation. Using the CCSD(T), MRCI and MRCI with the Davidson modification (MRCI + Q), the PECs of electronic states involved are extrapolated to the complete basis set (CBS) limit. With the PECs, the spectroscopic parameters (T_e, R_e, ω_e, ω_ex_e, ω_ey_e, α_e, β_e, γ_e and B_e) of the six low-lying electronic states are determined. These parameters are in excellent agreement with the experimental data. The complete vibrational states are computed for the six low-lying electronic states when the rotational quantum number J equals zero, and the inertial rotation constants of the first 23 vibrational states are reported, which agree favorably with the RKR data. Comparison with the measurements shows that the two-point total-energy extrapolation scheme can obviously improve the quality of spectroscopic parameters and molecular constants.     

What is measured in a time-resolved stimulated Raman experiment?

We have performed four-beam time-resolved stimulated Raman measurements on liquid CS₂. To interpret our results we have applied the response formalism for the third-order polarization to such an experiment. It turns out that the measured quantity is proportional to (a convolution of) Abs[χ⁽³⁾(-ω_s, ω_s, ω_l, -ω_l)]. As a consequence in this four-beam e xperiment one picks up electronic contributions in contrast to a two-beam experiment where Im[χ⁽³⁾] is observed.     

Spin-orbit effects in the ground X2Π state and two low-lying excited valence states B2Π and L′2Φ for NO

The potential energy curves and spectroscopic constants B _e , ω _e , ω _e χ _e , α _e , D _e of the six Ω states (X ²Π_1/2, ? 3/2, B ²Π_1/2, ? 3/2 and L ^′2Φ_{5/2,   7/2}) of the NO radical molecule were calculated using spin-orbit multi-configuration quasi-degenerate perturbation theory (SO-MCQDPT). The spin-orbit coupling effect was considered via the state interaction approach with the full Breit-Pauli Hamiltonian. The spin-orbit splitting energy between the X ²Π_1/2 and X ²Π_3/2 states of the NO radical is 129.61 cm^-1, which agrees reasonably well with the experimental value of 123.13 cm^-1. For the B ²Π_{1/2,   3/2} states, the spin-orbit coupling (SOC) splitting energy is 35.99 cm^-1, the corresponding experimental value is 31.7 cm^-1. The SOC splitting value of the L ^′2Φ_{5/2,   7/2} states was calculated to be 103.2 cm^-1. The spectroscopic constants R _e , ω _e , ω _e χ _e , B _e , α _e , D _e are in reasonable agreement with available experimental and theoretical data for the six Ω states.     

Electron localization in 1D conductors due to the phonon scattering

Electron localization in 1D conductors due to scattering by dispersionless optical phonons at low temperatures has been studied by applying the Berezinsky diagram technique. The large localization length l_loc, much longer than the mean free path l, has been found. The low frequency optical absorption is described by the law: Re σ(ω) ～ ω² |ln³ω|.     

Stability of bipolarons in the presence of a magnetic field

The stability of large Fröhlich bipolarons in the presence of a static magnetic field is investigated with the path integral formalism. We find that the application of a magnetic field (characterized by the cyclotron frequence ω _c) favors bipolaron formation: (i) the critical electronphonon coupling parameter α _c (above which the bipolaron is stable) decreases with increasing ω _c and (ii) the critical Coulomb repulsion strength U _c (below which the bipolaron is stable) increases with increasing ω _c. The binding energy and the corresponding variational parameters are calculated as a function of α, U and ω _c. Analytical results are obtained in various limiting cases. In the limit of strong electron-phonon coupling (α ? 1) we obtain for ω _c ? 1 that E _estim ? E _estim(ω _c = 0) + c(u)ω _c/α ⁴ with c(u) an explicitly calculated constant, dependent on the ratio u = U/α where U is the strength of the Coulomb repulsion. This relation applies both in 2D and in 3D, but with a different expression for c(u). For ω _c ? α ²? 1 we find in 3D E _estim ? ω _c - α ² A(u) ln²(ω _c/α ²), (also with an explicit analytical expression for A(u)) whereas in 2D E _estim ^2D ? ω _c - α√ω _cπ(u-2-√2)/2. The validity region of the Feynman-Jensen inequality for the present problem, bipolarons in a magnetic field, remains to be examined.     

Feldgleichungen der TREDERschen Gravitationstheorie,die aus einem Variationsprinzip ableitbar sind. I

In the frame work of TREDER 's gravitational theory we consider two classes of field equations which are derivable from two classes of LAGRANGE ian densities Ω⁽¹⁾(ω₁, ω₂), Ω⁽²⁾(s?₁, s?₂). ω₁, ω₂; s?₁, s?₂ are parameters. Ω⁽²⁾(ω₁, ω₂) gives us field equations which are up to the post-NEWTON ian approximation in the sense of NORDTVEDT , THORNE and WILL equivalent to the field equations given by BRANS and DICKE . For ω₂ = ?1 ?2ω₁ field equations follow from Ω⁽¹⁾(ω₁, ?1 ?2ω₁) which are in the above mentioned sense of post-NEWTON ian approximation equivalent to EINSTEIN 's equations. The field equations following from Ω⁽¹⁾(ω₁, ω₂) have a cosmological model with the well known cosmological singularities for T → ± ∞ in case that ω₁/(1 +3ω₁ +ω₂) ? γ > 0. For ω₁/(1 +3ω₁ +ω₂) ≤ 0 cosmological models with no cosmological singularities exist. From Ω⁽²⁾(s?₁, s?₂) we obtain field equations which at the best give us perihelion rotation 7% above EINSTEIN 's value and light deflection 7% below the corresponding EINSTEIN 's value. But in that case we are able to show the existence of a cosmological model without any cosmological singularity.