Non-Linear Optical Activity of Chiral Bipyrimidine-Based Thin Films.

An original series of bipyrimidine-based chromophores featuring alkoxystyryl donor groups bearing short chiral (S)-2-methylbutyl chains in positions 4, 3,4 and 3,5, connected to electron-accepting 2,2-bipyrimidine rings, has been developed. Their linear and non-linear optical properties were studied using a variety of techniques, including one- and two-photon absorption spectroscopy, fluorescence measurements, as well as Hyper-Rayleigh scattering to determine the first hyperpolarizabilities. Their electronic and geometrical properties were rationalized by TD-DFT calculations. The thermal properties of the compounds were also investigated by a combination of polarized light optical microscopy, differential scanning calorimetry measurements and small-angle X-ray scattering experiments. The derivatives were found not to have mesomorphic properties, but to exhibit melting temperatures or cold crystallization behavior that enabled the isolation of well-organized thin films. The nonlinear optical properties of amorphous or crystalline thin films were studied by wide-field second harmonic generation and multiphoton fluorescence imaging, confirming that non-centrosymmetric crystal organization enables strong second and third harmonic generation. This new series confirms that our strategy of functionalizing 3D organic octupoles with short chiral chains to generate non-centrosymmetric organized thin films enables the development of highly second order nonlinear optical active materials without the use of corona-poling or tedious deposition techniques.

Bridge improvement work: maximising non-linear optical performance in polyoxometalate derivatives.

New phenyl and stilbene-bridged polyoxometalate (POM) charge-transfer chromophores with diphenylamino donor groups produce, respectively, the highest intrinsic and absolute quadratic hyperpolarisabilities measured for such species. The ß0,zzz obtained for the phenyl bridge - at 180 × 10-30 esu - is remarkable for a short conjugated system while changing to the stilbene (260 × 10-30 esu) produces a substantial increase in non-linearity for a minimal red-shift in the absorption profile. Together with TD-DFT calculations, the results show that maximising conjugation in the π-bridge is vital to high performance in such "POMophores".

New Multifunctional Bipyrimidine-Based Chromophores for NLO-Active Thin-Film Preparation.

New synthesized bipyrimidine-based chromophores presenting alkoxystyryl donor groups carrying aliphatic achiral and chiral chains in the 4 position, connected to electron-accepting 2,2-bipyrimidine cores have been synthesized. Their linear and nonlinear optical (NLO) properties were investigated as well as their mesomorphic properties by various techniques (light-transmission measurements, polarized-light optical microscopy, differential scanning calorimetry measurements and two-photon excited fluorescence). The derivatives with achiral linear carbon chains were found to exhibit liquid-crystal properties with the formation smectic phases over large temperature ranges, which were confirmed by small-angle X-ray scattering analysis via stacking models. The nonlinear optical properties in the solid state for derivatives with C14 and the citronellol chains have been studied by wide-field second-harmonic generation and multi-photon fluorescence imaging, confirming centrosymmetry for these achiral mesogens and their excellent third-order nonlinearity whereas the chiral compound exhibits non-centrosymmetric organization resulting in a strong Second Harmonic Generation at the crystal state.

Size economy and first hyperpolarizability: synthesis and nonlinear optical behavior of ferrocene-based donor-acceptor chromophores lacking π-linkers.

The question of size economy in the design of chromophores for nonlinear optics is addressed in this investigation. We have synthesized directly linked donor-acceptor dyads, which lack a π-conjugated linker, the presence of which is usually considered obligatory in materials designed for nonlinear optics. Correlating linear optical data, electrochemical data, computational data and hyper Rayleigh scattering (HRS) data on ferrocene (Fc) based dyads, we demonstrate that the first hyperpolarizability of such size economical chromophores is significantly better compared to that of Fc based, traditional, larger, donor-π-acceptor chromophores. Arguably, a larger π-conjugated linker decreases the electronic communication between the donor and the acceptor and weakens the intramolecular charge transfer in such chromophores.

Outstanding Quadratic to Septic Optical Nonlinearity at Dipolar Alkynylmetal-Porphyrin Hybrids.

Porphyrins are important macrocycles with applications in several areas including therapy, catalysis, and sensing. Strong nonlinear optical (NLO) responses are the key to fully exploiting the potential of these biocompatible molecules. We herein report that certain metal-alkynyl donor/nitro acceptor-functionalized porphyrins are attractive candidates for NLO applications. We show that specific examples exhibit record quadratic optical nonlinearity, exceptional two-photon absorption, and outstanding three-photon absorption, and we report the first porphyrins that exhibit four-photon absorption. The two-, three-, and four-photon absorption maxima are found at the corresponding multiples of linear absorption bands that time-dependent density functional theory assigns as admixtures of porphyrin-localized π*←π and donor-porphyrin to porphyrin-acceptor charge-transfer transitions.

Electrochemically-Switched 2nd Order Non-Linear Optical Response in an Arylimido-Polyoxometalate with High Contrast and Cyclability.

Electrochemically switched 2nd order non-linear optical responses have been demonstrated for the first time in polyoxometalates (POMs), with an arylimido-derivative showing a leading combination of high on/off contrast (94 %), high visible transparency, and cyclability. Spectro-electrochemical and TD-DFT studies indicate that the switch-off results from weakened charge transfer (CT) character of the electronic transitions in the reduced state. This represents the first study of an imido-POM reduced state, and demonstrates the potential of POM hybrids as electrochemically activated molecular switches.

Label-Free Imaging of Membrane Potentials by Intramembrane Field Modulation, Assessed by Second Harmonic Generation Microscopy.

Optical interrogation of cellular electrical activity has proven itself essential for understanding cellular function and communication in complex networks. Voltage-sensitive dyes are important tools for assessing excitability but these highly lipophilic sensors may affect cellular function. Label-free techniques offer a major advantage as they eliminate the need for these external probes. In this work, it is shown that endogenous second-harmonic generation (SHG) from live cells is highly sensitive to changes in transmembrane potential (TMP). Simultaneous electrophysiological control of a living human embryonic kidney (HEK293T) cell, through a whole-cell voltage-clamp reveals a linear relation between the SHG intensity and membrane voltage. The results suggest that due to the high ionic strengths and fast optical response of biofluids, membrane hydration is not the main contributor to the observed field sensitivity. A conceptual framework is further provided that indicates that the SHG voltage sensitivity reflects the electric field within the biological asymmetric lipid bilayer owing to a nonzero χeff(2) tensor. Changing the TMP without surface modifications such as electrolyte screening offers high optical sensitivity to membrane voltage (≈40% per 100 mV), indicating the power of SHG for label-free read-out. These results hold promise for the design of a non-invasive label-free read-out tool for electrogenic cells.

The wavelength-dependent non-linear absorption and refraction of Au25 and Au38 monolayer-protected clusters.

In the past decade, the structural and electronic properties of monolayer-protected metal clusters, which can be produced size-selected in macroscopic amounts, have received a lot of attention. Their great potential for optical applications has been identified. In the high intensity regime, monolayer-protected metal clusters show pronounced nonlinear absorption and refraction. Naturally, these phenomena are wavelength-dependent, however, such dependence is largely unexplored. Here, we quantify the wavelength-dependent non-linear optical absorption and refraction cross sections of atomically precise Au25(DDT)18 and Au38(DDT)24 clusters, using the z-scan technique in combination with a tunable nanosecond laser source. Qualitatively different non-linear optical phenomena were found to take place at different excitation wavelengths (two-photon and excited-state absorption, intensity saturation and non-linear refraction). Both clusters have high nonlinear absorption cross sections at 532 nm, and present a (local) maximum at 640 nm, together with a maximum in the absorption saturation. The nonlinear refraction is always negative for Au25(DDT)18, while it changes sign for Au38(DDT)24. Depending on the wavelength, the underlying mechanism of the nonlinear absorption effects is two-photon absorption or excited state absorption. The obtained very high nonlinear cross sections, on the order of 107-109 GM, demonstrate the great potential of those clusters as nonlinear absorption or refraction materials in optical applications.

Design and synthesis of chromophores with enhanced electro-optic activities in both bulk and plasmonic-organic hybrid devices.

This study demonstrates enhancement of in-device electro-optic activity via a series of theory-inspired organic electro-optic (OEO) chromophores based on strong (diarylamino)phenyl electron donating moieties. These chromophores are tuned to minimize trade-offs between molecular hyperpolarizability and optical loss. Hyper-Rayleigh scattering (HRS) measurements demonstrate that these chromophores, herein described as BAH, show >2-fold improvement in ß versus standard chromophores such as JRD1, and approach that of the recent BTP and BAY chromophore families. Electric field poled bulk devices of neat and binary BAH chromophores exhibited significantly enhanced EO coefficients (r33) and poling efficiencies (r33/Ep) compared with state-of-the-art chromophores such as JRD1. The neat BAH13 devices with charge blocking layers produced very large poling efficiencies of 11.6 ± 0.7 nm2 V-2 and maximum r33 value of 1100 ± 100 pm V-1 at 1310 nm on hafnium dioxide (HfO2). These results were comparable to that of our recently reported BAY1 but with much lower loss (extinction coefficient, k), and greatly exceeding that of other previously reported OEO compounds. 3 : 1 BAH-FD : BAH13 blends showed a poling efficiency of 6.7 ± 0.3 nm2 V-2 and an even greater reduction in k. 1 : 1 BAH-BB : BAH13 showed a higher poling efficiency of 8.4 ± 0.3 nm2 V-2, which is approximately a 2.5-fold enhancement in poling efficiency vs. JRD1. Neat BAH13 was evaluated in plasmonic-organic hybrid (POH) Mach-Zehnder modulators with a phase shifter length of 10 µm and slot widths of 80 and 105 nm. In-device BAH13 achieved a maximum r33 of 208 pm V-1 at 1550 nm, which is ∼1.7 times higher than JRD1 under equivalent conditions.

Excited-State Dynamics and Nonlinear Optical Properties of Hyperpolarizable Chromophores Based on Conjugated Bis(terpyridyl)Ru(II) and Palladium and Platinum Porphyrinic Components: Impact of Heavy Metals upon Supermolecular Electro-Optic Properties.

A new series of strongly coupled oscillators based upon (porphinato)Pd, (porphinato)Pt, and bis(terpyridyl)ruthenium(II) building blocks is described. These RuPPd, RuPPt, RuPPdRu, and RuPPtRu chromophores feature bis(terpyridyl)Ru(II) moieties connected to the (porphinato)metal unit via an ethyne linker that bridges the 4'-terpyridyl and porphyrin macrocycle meso-carbon positions. Pump-probe transient optical data demonstrate sub-picosecond excited singlet-to-triplet-state relaxation. The relaxed lowest-energy triplet (T1) excited states of these chromophores feature absorption manifolds that span the 800-1200 nm spectral region, microsecond triplet-state lifetimes, and large absorptive extinction coefficients [ε(T1 → Tn) > 4 × 104 M-1 cm-1]. Dynamic hyperpolarizability (ßλ) values were determined from hyper-Rayleigh light scattering (HRS) measurements carried out at several incident irradiation wavelengths over the 800-1500 nm spectral region. Relative to benchmark RuPZn and RuPZnRu chromophores which showed large ßHRS values over the 1200-1600 nm range, RuPPd, RuPPt, RuPPdRu, and RuPPtRu displayed large ßHRS values over the 850-1200 nm region. Generalized Thomas-Kuhn sum (TKS) rules and experimental hyperpolarizability values were utilized to determine excited state-to-excited state transition dipole terms from experimental electronic absorption data and thus assessed frequency-dependent ßλ values, including two- and three-level contributions for both ßzzz and ßxzx tensor components to the RuPPd, RuPPt, RuPPdRu, and RuPPtRu hyperpolarizability spectra. These analyses qualitatively rationalize how the ßzzz and ßxzx tensor elements influence the observed irradiation wavelength-dependent hyperpolarizability magnitudes. The TKS analysis suggests that supermolecules related to RuPPd, RuPPt, RuPPdRu, and RuPPtRu will likely feature intricate dependences of experimentally determined ßHRS values as a function of irradiation wavelength that derive from substantial singlet-triplet mixing, and complex interactions among multiple different ß tensor components that modulate the long wavelength regime of the nonlinear optical response.

Dual photonic bandgap hollow sphere colloidal photonic crystals for real-time fluorescence enhancement in living cells.

To overcome the problems of refractive index matching and increased disorder when working with traditional heterostructure colloidal photonic crystals (CPCs) with dual or multiple photonic bandgaps (PBGs) for fluorescence enhancement in water, we propose the use of a chemical heterostructure in hollow sphere CPCs (HSCPCs). A partial chemical modification of the HSCPC creates a large contrast in wettability to induce the heterostructure, while the hollow spheres increase the refractive index difference when used in aqueous environment. With the platform, fluorescence enhancement reaches around 160 times in solution, and 72 times (signal-to-background ratio ~7 times) in cells during proof-of-concept live cardiomyocyte contractility experiments. Such photonic platform can be further exploited for chemical sensing, bioassays, and environmental monitoring. Moreover, the introduction of chemical heterostructures provides new design principles for functionalized photonic devices.

Electro-Optic Activity in Excess of 1000 pm V-1 Achieved via Theory-Guided Organic Chromophore Design.

High performance organic electro-optic (OEO) materials enable ultrahigh bandwidth, small footprint, and extremely low drive voltage in silicon-organic hybrid and plasmonic-organic hybrid photonic devices. However, practical OEO materials under device-relevant conditions are generally limited to performance of ≈300 pm V-1 (10× the EO response of lithium niobate). By means of theory-guided design, a new series of OEO chromophores is demonstrated, based on strong bis(4-dialkylaminophenyl)phenylamino electron donating groups, capable of EO coefficients (r33 ) in excess of 1000 pm V-1 . Density functional theory modeling and hyper-Rayleigh scattering measurements are performed and confirm the large improvement in hyperpolarizability due to the stronger donor. The EO performance of the exemplar chromophore in the series, BAY1, is evaluated neat and at various concentrations in polymer host and shows a nearly linear increase in r33 and poling efficiency (r33 /Ep , Ep is poling field) with increasing chromophore concentration. 25 wt% BAY1/polymer composite shows a higher poling efficiency (3.9 ± 0.1 nm2 V-2 ) than state-of-the-art neat chromophores. Using a high-ε charge blocking layer with BAY1, a record-high r33 (1100 ± 100 pm V-1 ) and poling efficiency (17.8 ± 0.8 nm2 V-2 ) at 1310 nm are achieved. This is the first reported OEO material with electro-optic response larger than thin-film barium titanate.

Enhancement of Nonlinear Optical Scattering by Gold Nanoparticles through Aggregation-Induced Plasmon Coupling in the Near-Infrared.

Gold nanoparticles (AuNPs) are regarded as promising building blocks in functional nanomaterials for sensing, drug delivery and catalysis. One remarkable property of these particles is the localized surface plasmon resonance (LSPR), which gives rise to augmented optical properties through local field enhancement. LSPR also influences the nonlinear optical properties of metal NPs (MNPs) making them potentially interesting candidates for fast, high resolution nonlinear optical imaging. In this work we characterize and discuss the wavelength dependence of the hyper-Rayleigh scattering (HRS) behavior of spherical gold nanoparticles (GNP) and gold nanorods (GNR) in solution, from 850 nm up to 1300 nm, covering the near-infrared (NIR) window relevant for deep tissue imaging. The high-resolution spectral data allows discriminating between HRS and two photon photoluminescence contributions. Upon particle aggregation, we measured very large enhancements (ca. 104 ) of the HRS intensity in the NIR, which is explained by considering aggregation-induced plasmon coupling effects and local field enhancement. These results indicate that purposely designed coupled nanostructures could prove advantageous for nonlinear optical imaging and biosensing applications.

DANPY (dimethylaminonaphthylpyridinium): an economical and biocompatible fluorophore.

Dyes with nonlinear optical (NLO) properties enable new imaging techniques and photonic systems. We have developed a dye (DANPY-1) for photonics applications in biological substrates such as nucleic acids; however, the design specification also enables it to be used for visualizing biomolecules. It is a prototype dye demonstrating a water-soluble, NLO-active fluorophore with high photostability, a large Stokes shift, and a favorable toxicity profile. A practical and scalable synthetic route to DANPY salts has been optimized featuring: (1) convergent Pd-catalyzed Suzuki coupling with pyridine 4-boronic acid, (2) site-selective pyridyl N-methylation, and (3) direct recovery of crystalline intermediates without chromatography. We characterize the optical properties, biocompatibility, and biological staining behavior of DANPY-1. In addition to stability and solubility across a range of polar media, the DANPY-1 chromophore shows a first hyperpolarizability similar to common NLO dyes such as Disperse Red 1 and DAST, a large two-photon absorption cross section for its size, substantial affinity to nucleic acids in vitro, an ability to stain a variety of cellular components, and strong sensitivity of its fluorescence properties to its dielectric environment.

Quadratic and Cubic Optical Nonlinearities of Y-Shaped and Distorted-H-Shaped Arylalkynylruthenium Complexes.

Straightforward syntheses of bis[bis{1,2-bis(diphenylphosphino)ethane}ruthenium]-functionalized 1,3,5-triethynylbenzene-cored complexes via a methodology employing "steric control" permit facile formation of Y-shaped Sonogashira coupling products and distorted-H-shaped homo-coupled quadrupolar products. Cyclic voltammetric data from these products reveal two reversible metal alkynyl-localized oxidation processes for all complexes. The wavelengths of the linear optical absorption maxima are dominated by the nature of the peripheral alkynyl ligand rather than the substituent at the unique arm of the "Y" or at the quadrupolar complex "core". The quadratic optical nonlinearities of the Y-shaped complexes were assessed by the hyper-Rayleigh scattering technique at 800 nm and employing 100 fs light pulses; introduction of donor NEt2 and/or acceptor NO2 to the wedge periphery resulted in non-zero nonlinearities, with the largest ßHRS,800 values being observed for the complexes containing the 4-nitrophenylalkynyl ligands. Depolarization ratios are consistent with substantial off-diagonal first hyperpolarizability tensor components and 2D nonlinear character. Computational studies employing time-dependent density functional theory have been employed to assign the key low-energy transitions in the linear optical spectra and to compute the quadratic nonlinear optical tensorial components. Cubic optical nonlinearities of the quadrupolar complexes were assessed by the Z-scan technique over the range 500-1600 nm and employing 130 fs light pulses; two-photon absorption cross-sections for these distorted-H-shaped complexes are moderate to large in value (up to 5500 GM at 880 nm), while one example displays significant three-photon absorption (1300×10-80  cm6 s2 at 1200 nm).

Unexpected High Second-Order Nonlinear Optical Activity of Metal Complexes with Three-Branched Hexadentate 2,2'-Bipyridine Ligands.

Two hexadentate bipyridine ligands and their RuII and NiII complexes were prepared. The helical alignment of the three electron-donor-π-bridge-electron-acceptor (d-π-A) single-strands with bundle architecture in cooperation with the metal center can strongly enhance the nonlinear optical (NLO) properties. The complexation of the novel cage-type hexadentate ligands with a paramagnetic NiII -core almost doubles the ßHRS values compared with the corresponding diamagnetic RuII complexes. The hyper-Rayleigh scattering (HRS) was performed with a highly sensitive setup for simultaneous discrimination between multi-photon fluorescence and the molecular first hyperpolarizability.

Fine-tuning polyoxometalate non-linear optical chromophores: a molecular electronic "Goldilocks" effect.

A new aryl-imido polyoxometalate non-linear optical chromophore (POMophore) with a diphenylamino donor group attains the highest ßzzz, 0 value (196 × 10-30 esu by Hyper-Rayleigh Scattering, HRS), and best transparency/non-linearity trade off yet for such materials. Stark spectroscopic and DFT investigation of this compound, plus NMe2 and carbazole analogues, show that its high performance results from a combination of strongly dipolar electronic transitions, and strong electronic communication across the π-system.

Instantaneous, Simple, and Reversible Revealing of Invisible Patterns Encrypted in Robust Hollow Sphere Colloidal Photonic Crystals.

The colors of photonic crystals are based on their periodic crystalline structure. They show clear advantages over conventional chromophores for many applications, mainly due to their anti-photobleaching and responsiveness to stimuli. More specifically, combining colloidal photonic crystals and invisible patterns is important in steganography and watermarking for anticounterfeiting applications. Here a convenient way to imprint robust invisible patterns in colloidal crystals of hollow silica spheres is presented. While these patterns remain invisible under static environmental humidity, even up to near 100% relative humidity, they are unveiled immediately (≈100 ms) and fully reversibly by dynamic humid flow, e.g., human breath. They reveal themselves due to the extreme wettability of the patterned (etched) regions, as confirmed by contact angle measurements. The liquid surface tension threshold to induce wetting (revealing the imprinted invisible images) is evaluated by thermodynamic predictions and subsequently verified by exposure to various vapors with different surface tension. The color of the patterned regions is furthermore independently tuned by vapors with different refractive indices. Such a system can play a key role in applications such as anticounterfeiting, identification, and vapor sensing.

ONIOM Investigation of the Second-Order Nonlinear Optical Responses of Fluorescent Proteins.

The first hyperpolarizability (ß) of six fluorescent proteins (FPs), namely, enhanced green fluorescent protein, enhanced yellow fluorescent protein, SHardonnay, ZsYellow, DsRed, and mCherry, has been calculated to unravel the structure-property relationships on their second-order nonlinear optical properties, owing to their potential for multidimensional biomedical imaging. The ONIOM scheme has been employed and several of its refinements have been addressed to incorporate efficiently the effects of the microenvironment on the nonlinear optical responses of the FP chromophore that is embedded in a protective ß-barrel protein cage. In the ONIOM scheme, the system is decomposed into several layers (here two) treated at different levels of approximation (method1/method2), from the most elaborated method (method1) for its core (called the high layer) to the most approximate one (method2) for the outer surrounding (called the low layer). We observe that a small high layer can already account for the variations of ß as a function of the nature of the FP, provided the low layer is treated at an ab initio level to describe properly the effects of key H-bonds. Then, for semiquantitative reproduction of the experimental values obtained from hyper-Rayleigh scattering experiments, it is necessary to incorporate electron correlation as described at the second-order Møller-Plesset perturbation theory (MP2) level as well as implicit solvent effects accounted for using the polarizable continuum model (PCM). This led us to define the MP2/6-31+G(d):HF/6-31+G(d)/IEFPCM scheme as an efficient ONIOM approach and the MP2/6-31+G(d):HF/6-31G(d)/IEFPCM as a better compromise between accuracy and computational needs. Using these methods, we demonstrate that many parameters play a role on the ß response of FPs, including the length of the π-conjugated segment, the variation of the bond length alternation, and the presence of π-stacking interactions. Then, noticing the small diversity of the FP chromophores, these results highlight the key role of the ß-barrel and surrounding residues on ß, not only because they can locally break the noncentrosymmetry vital to a ß response but also because it can impose geometrical constraints on the chromophore.

Linear Optical, Quadratic and Cubic Nonlinear Optical, Electrochemical, and Theoretical Studies of "Rigid-Rod" Bis-Alkynyl Ruthenium Complexes.

The syntheses of oligo(p-phenylene ethynylene)s (OPEs) end-functionalized by a nitro acceptor group and with a ligated ruthenium unit at varying locations in the OPE chain, namely, trans-[Ru{(C≡C-1,4-C6 H4 )n NO2 }(C≡CR)(dppe)2 ] (dppe=1,2-bis(diphenylphosphino)ethane; n=1, R=1,4-C6 H4 C≡C-1,4-C6 H4 C≡CPh, 1,4-C6 H4 NEt2 ; n=2, R=Ph, 1,4-C6 H4 C≡CPh, 1,4-C6 H4 C≡C-1,4-C6 H4 C≡CPh, 1,4-C6 H4 NO2 , 1,4-C6 H4 NEt2 ; n=3, R=Ph, 1,4-C6 H4 C≡CPh), are reported. Their electrochemical properties were assessed by cyclic voltammetry, their linear optical properties and quadratic and cubic nonlinear optical properties were assayed by UV/Vis/NIR spectroscopy, hyper-Rayleigh scattering studies employing nanosecond pulses at 1064 nm, and broad spectral range Z-scan studies employing femtosecond pulses, respectively, and their linear optical properties and vibrational spectroscopic behavior in the formally RuIII state was examined by UV/Vis/NIR and IR spectroelectrochemistry, respectively. The potentials of the metal-localized oxidation processes are sensitive to alkynyl-ligand modification, but this effect is attenuated on π-bridge lengthening. Computational studies employing time-dependent density functional theory were undertaken on model complexes, with a 2D scan revealing a soft potential-energy surface for intra-alkynyl-ligand aryl-ring rotation; this is consistent with the experimentally observed blueshift in optical absorption maxima. Quadratic optical nonlinearities are significant and cubic NLO coefficients for these small complexes are small. The optimum length of the alkynyl ligands and the ideal metal location in the OPE to maximize the key coefficients have been defined.