Note that the polarization density P t and electrical field E t are considered as scalar for simplicity. Central to the study of electromagnetic waves is the wave equation. Starting with Maxwell's equations in an isotropic space, containing no free charge, it can be shown that. For nonlinear medium, Gauss's law does not imply that the identity. However, even when this term is not identically 0, it is often negligibly small and thus in practice is usually ignored, giving us the standard nonlinear wave equation:.
The nonlinear wave equation is an inhomogeneous differential equation. The general solution comes from the study of ordinary differential equations and can be obtained by the use of a Green's function. Physically one gets the normal electromagnetic wave solutions to the homogeneous part of the wave equation:. One of the consequences of this is a nonlinear interaction that results in energy being mixed or coupled between different frequencies, which is often called a "wave mixing".
As an example, if we consider only a second-order nonlinearity three-wave mixing , then the polarization P takes the form. Plugging this into the expression for P gives. These three-wave mixing processes correspond to the nonlinear effects known as second-harmonic generation , sum-frequency generation , difference-frequency generation and optical rectification respectively. Note: Parametric generation and amplification is a variation of difference-frequency generation, where the lower frequency of one of the two generating fields is much weaker parametric amplification or completely absent parametric generation.
In the latter case, the fundamental quantum-mechanical uncertainty in the electric field initiates the process. The above ignores the position dependence of the electrical fields. In a typical situation, the electrical fields are traveling waves described by. The above equation is known as the phase-matching condition.
Typically, three-wave mixing is done in a birefringent crystalline material, where the refractive index depends on the polarization and direction of the light that passes through. The polarizations of the fields and the orientation of the crystal are chosen such that the phase-matching condition is fulfilled.
This phase-matching technique is called angle tuning. Typically a crystal has three axes, one or two of which have a different refractive index than the other one s. Uniaxial crystals, for example, have a single preferred axis, called the extraordinary e axis, while the other two are ordinary axes o see crystal optics. There are several schemes of choosing the polarizations for this crystal type.
Polymers and molecular assemblies for second-order nonlinear optics
If the signal and idler have the same polarization, it is called "type-I phase matching", and if their polarizations are perpendicular, it is called "type-II phase matching". However, other conventions exist that specify further which frequency has what polarization relative to the crystal axis. These types are listed below, with the convention that the signal wavelength is shorter than the idler wavelength.
Most common nonlinear crystals are negative uniaxial, which means that the e axis has a smaller refractive index than the o axes. In those crystals, type-I and -II phase matching are usually the most suitable schemes. Types II and III are essentially equivalent, except that the names of signal and idler are swapped when the signal has a longer wavelength than the idler. One undesirable effect of angle tuning is that the optical frequencies involved do not propagate collinearly with each other.
This is due to the fact that the extraordinary wave propagating through a birefringent crystal possesses a Poynting vector that is not parallel to the propagation vector. This would lead to beam walk-off, which limits the nonlinear optical conversion efficiency. These methods are called temperature tuning and quasi-phase-matching. Temperature tuning is used when the pump laser frequency polarization is orthogonal to the signal and idler frequency polarization. The birefringence in some crystals, in particular lithium niobate is highly temperature-dependent.
The crystal temperature is controlled to achieve phase-matching conditions. The other method is quasi-phase-matching. Hence, these crystals are called periodically poled. This results in the polarization response of the crystal to be shifted back in phase with the pump beam by reversing the nonlinear susceptibility. This allows net positive energy flow from the pump into the signal and idler frequencies.
Quasi-phase-matching can be expanded to chirped gratings to get more bandwidth and to shape an SHG pulse like it is done in a dazzler. SHG of a pump and self-phase modulation emulated by second-order processes of the signal and an optical parametric amplifier can be integrated monolithically. At high peak powers the Kerr effect can cause filamentation of light in air, in which the light travels without dispersion or divergence in a self-generated waveguide. When a noble gas atom is hit by an intense laser pulse, which has an electric field strength comparable to the Coulomb field of the atom, the outermost electron may be ionized from the atom.
Once freed, the electron can be accelerated by the electric field of the light, first moving away from the ion, then back toward it as the field changes direction. The electron may then recombine with the ion, releasing its energy in the form of a photon. The light is emitted at every peak of the laser light field which is intense enough, producing a series of attosecond light flashes. The photon energies generated by this process can extend past the th harmonic order up to a few K eV.
Polymers and molecular assemblies for second-order nonlinear optics — NYU Scholars
This is called high-order harmonic generation. The laser must be linearly polarized, so that the electron returns to the vicinity of the parent ion. Low power optical limiting studies on nanocrystalline benzimidazole thin films prepared by modified liquid phase growth technique P A Praveen , S P Prabhakaran , R. Synthesis reactions characterisation and supramolecular association of some organotellurium derivatives Tripurari Pujan.
Theoretical studies on nonlinear optical properties of formaldehyde oligomers by ab initio and density functional theory methods Hui-Yin Wu , Ajay Chaudhari , Shyi-Long Lee.
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Optical properties of cross-conjugated isopolydiacetylene oligomers as measured by ultraviolet-visible spectroscopy and the optical Kerr effect : Molecular photonics and electronics Aaron D. The results Tab. As a conclusion, there is an optimal NLO active moiety content to obtain the maximum second-order nonlinear coefficient.
Polymers for Second-Order Nonlinear Optics
This behaviour can be explained bearing in mind the influence of the molecular environments on the chromophoric dipole orientability. Indeed, these samples have lower Tg Tab. Of course, azo content plays a key role to increase the d33 value in this process: at higher concentration there are more oriented chromophores with consequent enhanced d33 values. As a consequence, their mobility is reduced and their orientability restricted, so the d33 value decreases [10, 24, 30].
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An idealized illustration of chromophoric orientations and dipolar aggregations before and after corona pooling of some significant copolymeric films is presented in Fig. Thus, the real d33 values of these polymers should be higher. Temporal evolution of SHG signal and temperature dependence of the poled sample of poly[ S -MAP-S] during a typical dipole reorientational dynamic experiment.
The increase of dipolar interactions between the chromophores also plays an important role in enhancing the temporal and thermal stability of the electrically oriented polymers. For an electrically poled polymeric system, the long term stability of NLO properties is a critical requirement for real applications. In this study, the temporal stability of the NLO properties was studied by monitoring the SHG signal variation as a function of time and temperature. These measurements were performed one week after the first poling process.
In general, the effective d33 coefficient of NLO polymers remains stable at low temperatures, but decays significantly at a specific temperature, thus providing information on the maximum device operating temperatures that the film can undergo, and allowing quick evaluation of the temporal and thermal stability of the material.
The results of this dynamic thermal stability assessment for the poled samples are shown in Fig. A decrease of the SHG signal at temperatures appreciably lower than Tg is observed for all the samples.
Similar behaviours were reported by other researchers . This specific temperature value is defined by some authors as the effective electrooptic relaxation temperature . It was found that dielectric measurements on a typical NLO side-chain polymer exhibited multiple relaxation processes even below Tg, which are all related to the motions of the chromophoric dipoles in the side-chain .
Such behaviour could be due to lower order alignment of chromophores as evidenced by nonlinear coefficients Tab. After an initial small decay in the first h, the NLO response of the polymer tends to be stable. All our polymeric derivatives show at room temperature similar effects that persist for at least six months and are well reproducible.
It therefore appears that the investigated samples are characterized by better thermal and long-term stability with respect to analogous side-chain polymers with similar NLO-active chromophores covalently linked to the polymer backbone [30, ].
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These effects are probably due to their different structures. In fact, the azoaromatic moiety in these last materials is linked to the backbone through a flexible spacer and their dipole orientation may relax quickly to the original isotropic state. By contrast, in poly[ S -MAP-S] and the copolymers the azoaromatic group is conformationally blocked by the rigid pyrrolidine ring interposed between the chromophore and the methacrylic main chain, with consequent reduced mobility of the macromolecular chains that improves the Tg values. In this way the electrically induced dipole orientation remains at room temperature at a higher value and is stable for a longer time.
Conclusions In conclusion, we have investigated the corona poling behaviour of the homopolymer and a series of side-chain azobenzene MMA copolymers with different azo moiety molar contents. The noncentrosymmetrical orientation of chromophores was achieved by corona poling process to impart second-order NLO properties. The samples were characterized by UV-vis spectroscopy before and after poling.
The NLO properties of the investigated polymers are significantly influenced by the chromophore molar content. Thermal dipole reorientation experiments on the poled polymeric films clearly show that chromophore dipoles are significantly mobile even below the Tg of the NLO polymers. The crude products were precipitated from solution by pouring the reaction mixture into a large excess of methanol ml and collected by filtration.
The purification was performed for three times by dissolution in DMF and precipitation from methanol and the last traces of unreacted monomer were eliminated from the product by Soxhlet extraction with methanol. Details on the identification label, initial feed monomer, copolymer composition, average molecular weight, initial decomposition and glass transition temperatures are reported in Tab.
Tetramethylsilane TMS was used as internal reference. Azoaromatic chromophore concentrations of about 3 mol L-1 were used. Calibration curves were obtained by using several monodisperse polystyrene standards. The film thickness, measured by a Tencor P profilometer, was in the range nm. The native films resulted to be optically isotropic by inspection with a Zeiss Axioscope2 polarizing microscope through crossed polarizers fitted with a Linkam THMS hot stage.
Absorption spectra of the films were carried out under the same instrumental conditions as the related solutions.
The samples were inserted in a box, mounted on the hot stage and corona-wire poled. The poled and SHG-technique used are shown schematically in Fig. Experimental set-up for poling and SHG measurements.