Nonlinear Dipole Research Articles

Giant Second Harmonic Generation in Supertwisted WS2 Spirals Grown in Step-Edge Particle-Induced Non-Euclidean Surfaces.

In moiré crystals resulting from the stacking of twisted two-dimensional (2D) layered materials, a subtle adjustment in the twist angle surprisingly gives rise to a wide range of correlated optical and electrical properties. Herein, we report the synthesis of supertwisted WS2 spirals and the observation of giant second harmonic generation (SHG) in these spirals. Supertwisted WS2 spirals featuring different twist angles are synthesized on a Euclidean or step-edge particle-induced non-Euclidean surface using carefully designed water-assisted chemical vapor deposition. We observed an oscillatory dependence of SHG intensity on layer number, attributed to atomically phase-matched nonlinear dipoles within layers of supertwisted spiral crystals where inversion symmetry is restored. Through an investigation into the twist angle evolution of SHG intensity, we discovered that the stacking model between layers plays a crucial role in determining the nonlinearity, and the SHG signals in supertwisted spirals exhibit enhancements by a factor of 2 to 136 when compared with the SHG of the single-layer structure. These findings provide helpful perspectives on the rational growth of 2D twisted structures and the implementation of twist angle adjustable endowing them great potential for exploring strong coupling correlation physics and applications in the field of twistronics.

Manipulation of Chiral Nonlinear Optical Effect by Light-Matter Strong Coupling.

Light-matter strong coupling (LMSC) is an intriguing state in which light and matter are hybridized inside a cavity. It is increasingly recognized as an excellent way to control material properties without any chemical modification. Here, we show that the LMSC is a powerful state for manipulating chiral nonlinear optical (NLO) effects through the investigation of second harmonic generation (SHG) circular dichroism. At the upper polariton band in LMSC, in addition to the enhancement of SHG by more than 1 order of magnitude, the responsivity to the handedness of circularly polarized light was largely modified, where sign inversion and increase of the dissymmetry factor were achieved. Quarter waveplate rotation analysis revealed that the LMSC clearly influenced the coefficients associated with chirality in the NLO process and also contributed to the enhancement of nonlinear magnetic dipole interactions. This study demonstrated that LMSC serves as a great platform for controlling chiral and magneto-optics.

Deep learning with autoencoders and LSTM for ENSO forecasting

El Niño Southern Oscillation (ENSO) is the prominent recurrent climatic pattern in the tropical Pacific Ocean with global impacts on regional climates. This study utilizes deep learning to predict the Niño 3.4 index by encoding non-linear sea surface temperature patterns in the tropical Pacific using an autoencoder neural network. The resulting encoded patterns identify crucial centers of action in the Pacific that serve as predictors of the ENSO mode. These patterns are utilized as predictors for forecasting the Niño 3.4 index with a lead time of at least 6 months using the Long Short-Term Memory (LSTM) deep learning model. The analysis uncovers multiple non-linear dipole patterns in the tropical Pacific, with anomalies that are both regionalized and latitudinally oriented that should support a single inter-tropical convergence zone for modeling efforts. Leveraging these encoded patterns as predictors, the LSTM - trained on monthly data from 1950 to 2007 and tested from 2008 to 2022 - shows fidelity in predicting the Niño 3.4 index. The encoded patterns captured the annual cycle of ENSO with a 0.94 correlation between the actual and predicted Niño 3.4 index for lag 12 and 0.91 for lags 6 and 18. Additionally, the 6-month lag predictions excel in detecting extreme ENSO events, achieving an 85% hit rate, outperforming the 70% hit rate at lag 12 and 55% hit rate at lag 18. The prediction accuracy peaks from November to March, with correlations ranging from 0.94 to 0.96. The average correlations in the boreal spring were as large as 0.84, indicating the method has the capability to decrease the spring predictability barrier.

Nonlinear effects in manybody van der Waals interactions

Van der Waals interactions are ubiquitous and they play an important role for the stability of materials. Current understanding of this type of coupling is based on linear response theory, while optical nonlinearities are rarely considered in this context. Many materials, however, exhibit strong optical nonlinear response, which prompts further evaluation of dispersive forces beyond linear response. Here we present a approach that takes into account linear and nonlinear properties of all dipolar nanoparticles in a given system. This method is based on a Hamiltonian for nonlinear dipoles, which we apply in different systems uncovering a complex interplay of distance, anisotropy, polarizabilities, and hyperpolarizabilities in the vdW energy. This investigation broadens our basic understanding of dispersive interactions, especially in the context of nonlinear materials. Published by the American Physical Society 2024

Homogeneous Dephasing in Photosynthetic Bacterial Reaction Centers: Time Correlation Function Approach.

A new tractable linear electronic transition dipole moment time correlation function (ETDMTCF) that accurately accounts for electronic dephasing, asymmetry, and width of 1-phonon profile, which the zero-phonon line (ZPL) contributes to it, in Rhodopseudomonas viridis bacterial reaction center is derived. This time correlation function proves to be superior to other frequency-domain expressions in case of strong electron-phonon coupling (which is often the case in bacterial RCs and pigment-protein complexes), many vibrational modes involved, and high temperature, whereby more vibronic and electronic (sequence) transitions would arise. The Fourier transform of this ETDMTCF leads to asymmetric multiphonon profiles composed of Lorentzian distribution and Gaussian distribution on the high- and low-energy sides, respectively, whereby the overtone widths fold themselves with that of the one-phonon profile. This ETDMTCF also features expedient computation in large systems using asymmetric phonon profiles to account correctly for dephasing and pigment-protein interaction (electron-phonon coupling). The derived ETDMTCF allows computing all nonlinear optical signals in both time and frequency domains, through the nonlinear dipole moment time correlation functions (as guided by nonlinear optical response theory) in line with the eight Liouville space pathways. The linear transition dipole moment time correlation function is of a central value as the nonlinear transition dipole moment time correlation function is expressed in terms of the linear transition dipole moment time correlation function, derived herein. One of the great advantages of presenting this ETDMTCF is its applicability to nonlinear transition dipole moment time correlation functions in line with the eight Liouville space pathways needed in computing nonlinear signals. As such, there is more to the utility and applicability of the presented ETDMTCF besides computational expediency and efficiency. Results show good agreement with the reported literature. The intimate connection between a one-phonon profile and the corresponding bath spectral density in photosynthetic complexes is discussed.

Entropy analysis of Casson nanofluid flow across a rotating porous disc with nonlinear thermal radiation and magnetic dipole

The theme of the current effort is to theoretically analyze the entropy generation and heat transfer aspects of Casson nanofluid flow triggered by rotating porous disc with the presence of magnetic dipole, nonlinear thermal radiation, viscous dissipation and Joule heating. The modeling of the nanofluid can be described with the combination of Brownian motion and thermophoresis by incorporating the passive control boundaries, and the governing PDEs are transformed into a set of highly nonlinear ODEs. The resulting equations are then solved analytically using HAM technique. The present results are compared with previously published results, which are in excellent agreement. The effect of pertinent nondimensional parameters on the entropy generation, hydrodynamic, heat and mass transport aspects is discussed via graphical illustrations. Both radial and tangential velocities are affected by accelerating the values of Hartmann number and porosity parameter. The temperature profile is upsurged by improving the radiation and thermal ratio parameter. Increasing the Casson parameter and Brinkman number leads to improved entropy generation rate. Moreover, skin friction, heat and mass transfer rates are examined with the help of the tables. It is believed that this study can be utilized as coolants by numerous automotive and engineering industries, namely the electronic devices, electrical motor, spin coating, fabrication of spacecraft, thermal insulation, nuclear reactors, etc.

From Model to Algorithms: Distributed Magnetic Sensor System for Vehicle Tracking

A novel vehicle localization and tracking methods are presented based on magnetic anomaly detection by distributed magnetic sensors. First, taking advantage of total magnetic field, in this article, we propose a total field matching (TFM) method that is immune of rotational vibrations to perform target localization. Instead of directly inverting the nonlinear magnetic dipole equations, we use the TFM approach to find the suboptimal target position, and then apply the linear Kalman filter to tail after the target. Because the relationship is linear between the target dynamics and the localization equations. A case study is performed by simulation to result in an estimated trajectory of ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ϕ</i> ) = (70.8 m, 44.9°) that agrees well with the real one of ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ϕ</i> ) = (70.5 m, 45°). For a vehicle tracking, the outdoors experiment results show good estimation accuracy based on four different sensor networking configurations.

Numerical Simulation of Coherent Extreme Ultraviolet Radiation by Considering Simple Hydrogen Atomic Potential

In this study, the ionization of a single electron exposed to an intense laser field is computed, and the nonlinear dipole response of a single electron is obtained. The ionization of a single electron exposed to an intense laser field can be computed using the strong field approximation (SFA), also known as the Keldysh theory. This method is based on the idea that the laser field is so strong that it can ionize the electron by tunneling through the Coulomb barrier. In this paper, the Xe, Ne, and H2 gas species are modeled since they have simple atomic systems. All gas species have relative or close ionization potentials. Ultra-short pulse duration (50 fs) is accepted because of the shorter time scale than the electron energy-lattice transfer. The ionization potentials of gas species result in the Keldysh parameter being smaller than one. The electron dipole oscillation spectra of these gas species are simulated by calculating the dipole spectrum considering the Lewenstein model. The electron propagation under the different wavelengths is simulated. The effects of the different driving wavelengths have noticeable effects on the enhancement and the extension of the extreme ultraviolet radiation signal.

Three-dimensional vortex dipole solitons in self-gravitating systems.

We derive the nonlinear equationsgoverning the dynamics of three-dimensional (3D) disturbances in a nonuniform rotating self-gravitating fluid under the assumption that the characteristic frequencies of disturbances are small compared to the rotation frequency. Analytical solutions of these equations are found in the form of the 3D vortex dipole solitons. The method for obtaining these solutions is based on the well-known Larichev-Reznik procedure for finding two-dimensional nonlinear dipole vortex solutions in the physics of atmospheres of rotating planets. In addition to the basic 3D x-antisymmetric part (carrier), the solution may also contain radially symmetric (monopole) or/and antisymmetric along the rotation axis (z-axis) parts with arbitrary amplitudes, but these superimposed parts cannot exist without the basic part. The 3D vortex soliton without the superimposed parts is extremely stable. It moves without distortion and retains its shape even in the presence of an initial noise disturbance. The solitons with parts that are radially symmetric or/and z antisymmetric turn out to be unstable, although, at sufficiently small amplitudes of these superimposed parts, the soliton retains its shape for a very long time.

Incomplete spectrum QSM using support information.

Reconstructing a bounded object from incomplete k-space data is a well posed problem, and it was recently shown that this incomplete spectrum approach can be used to reconstruct undersampled MRI images with similar quality to compressed sensing approaches. Here, we apply this incomplete spectrum approach to the field-to-source inverse problem encountered in quantitative magnetic susceptibility mapping (QSM). The field-to-source problem is an ill-posed problem because of conical regions in frequency space where the dipole kernel is zero or very small, which leads to the kernel's inverse being ill-defined. These "ill-posed" regions typically lead to streaking artifacts in QSM reconstructions. In contrast to compressed sensing, our approach relies on knowledge of the image-space support, more commonly referred to as the mask, of our object as well as the region in k-space with ill-defined values. In the QSM case, this mask is usually available, as it is required for most QSM background field removal and reconstruction methods. We tuned the incomplete spectrum method (mask and band-limit) for QSM on a simulated dataset from the most recent QSM challenge and validated the QSM reconstruction results on brain images acquired in five healthy volunteers, comparing incomplete spectrum QSM to current state-of-the art-methods: FANSI, nonlinear dipole inversion, and conventional thresholded k-space division. Without additional regularization, incomplete spectrum QSM performs slightly better than direct QSM reconstruction methods such as thresholded k-space division (PSNR of 39.9 vs. 39.4 of TKD on a simulated dataset) and provides susceptibility values in key iron-rich regions similar or slightly lower than state-of-the-art algorithms, but did not improve the PSNR in comparison to FANSI or nonlinear dipole inversion. With added (ℓ1-wavelet based) regularization the new approach produces results similar to compressed sensing based reconstructions (at sufficiently high levels of regularization). Incomplete spectrum QSM provides a new approach to handle the "ill-posed" regions in the frequency-space data input to QSM.

Nonlinear Monopole and Dipole Acoustic Radiation of a Weakly Charged Droplet Oscillating in a Uniform Electrostatic Field

Nonlinear Monopole and Dipole Acoustic Radiation of a Weakly Charged Droplet Oscillating in a Uniform Electrostatic Field

Insights into solvent effects on molecular properties, physicochemical parameters, and NLO behavior of brinzolamide, a bioactive sulfonamide: A computational study

In this study, quantum chemical calculations performed on brinzolamide (BZ), a sulfonamide derivative widely used as an antibacterial and antifungal drug, are presented. The molecular geometry optimizations of BZ were calculated by using both B3LYP functional and HF method with the 6-31G (d, p) basis set via Gaussian software. Since solvent phase behaviors are extremely important especially for drug molecules, calculations were renewed for BZ in ten different solvent environments. Universal solvation model SMD (Solvent Model based on Density) was utilized for solvent phase simulations. Mulliken and natural atomic charges, frontier molecular orbital (FMO) energies, quantum chemical descriptors were studied in detail according to solvent and method variation. In addition, the partition coefficient (logPow) value, which is an important parameter for the activity of the drug candidates, was theoretically estimated according to the relevant formula through the computed values for the water and 1-octanol phases for each methodology. The slightly lipophilic character of the BZ molecule was supported by the obtained values and the molecular lipophilicity potential map prepared by means of Molinspiration Galaxy 3D Structure Generator v2018.01 beta program. In the nonlinear optical (NLO) property analysis, dipole moment (μtot), mean polarizability (αtot), anisotropic polarizability (Δα), and mean first-order hyperpolarizability (βtot) values of BZ and the solvent effect on these values have been reported. Last, the interaction energies between donor and acceptor orbitals were calculated by the analysis of natural bond orbitals, and the obtained results were summarized.

A broadband intermodulation reference generator based on artificial nonlinear dipole

AbstractAccurate passive intermodulation (IM) test calls for precise IM reference for verifying and evaluating the test error from the tester, which makes the IM calibrator source with a wide working band and dynamic range important. This letter reports on a low‐cost and broadband IM generator with a compact area of 5 × 5 cm2. The broadband IM source is designed and demonstrated on the coplanar waveguide structure, where an artificial nonlinear dipole is used to generate IM reference. In special, the dipole is constructed by a diode (HSMS 2822) and two independent copper islands with optimized size. By controlling the bias voltage across the diode, the IM magnitude level can be changed accordingly. The IM tunable dynamic range is over 47.5 dB that covers GSM 800 MHz, GSM 900 MHz, DCS 1800 MHz, and TD‐LTE band. Meanwhile, the obtained IM has an improved IM‐bias flat response.

Blip up-down acquisition for spin- and gradient-echo imaging (BUDA-SAGE) with self-supervised denoising enables efficient T2 , T2 *, para- and dia-magnetic susceptibility mapping.

To rapidly obtain high resolution T2 , T2 *, and quantitative susceptibility mapping (QSM) source separation maps with whole-brain coverage and high geometric fidelity. We propose Blip Up-Down Acquisition for Spin And Gradient Echo imaging (BUDA-SAGE), an efficient EPI sequence for quantitative mapping. The acquisition includes multiple T2 *-, T2 '-, and T2 -weighted contrasts. We alternate the phase-encoding polarities across the interleaved shots in this multi-shot navigator-free acquisition. A field map estimated from interim reconstructions was incorporated into the joint multi-shot EPI reconstruction with a structured low rank constraint to eliminate distortion. A self-supervised neural network (NN), MR-Self2Self (MR-S2S), was used to perform denoising to boost SNR. Using Slider encoding allowed us to reach 1 mm isotropic resolution by performing super-resolution reconstruction on volumes acquired with 2 mm slice thickness. Quantitative T2 (=1/R2 ) and T2 * (=1/R2 *) maps were obtained using Bloch dictionary matching on the reconstructed echoes. QSM was estimated using nonlinear dipole inversion on the gradient echoes. Starting from the estimated R2 /R2 * maps, R2 ' information was derived and used in source separation QSM reconstruction, which provided additional para- and dia-magnetic susceptibility maps. In vivo results demonstrate the ability of BUDA-SAGE to provide whole-brain, distortion-free, high-resolution, multi-contrast images and quantitative T2 /T2 * maps, as well as yielding para- and dia-magnetic susceptibility maps. Estimated quantitative maps showed comparable values to conventional mapping methods in phantom and in vivo measurements. BUDA-SAGE acquisition with self-supervised denoising and Slider encoding enables rapid, distortion-free, whole-brain T2 /T2 * mapping at 1 mm isotropic resolution under 90 s.

Structure and principles of self-assembly of giant “sea urchin” type sulfonatophenyl porphine aggregates

Principles of molecular self-assembly into giant hierarchical structures of hundreds of micrometers in size are studied in aggregates of meso-tetra(4-sulfonatophenyl)porphine (TPPS4). The aggregates form a central tubular core, which is covered with radially protruding filamentous non-branching aggregates. The filaments cluster and orient at varying angles from the core surface and some filaments form bundles. Due to shape resemblance, the structures are termed giant sea urchin (GSU) aggregates. Spectrally resolved fluorescence microscopy reveals J- and H-bands of TPPS4 aggregates in both the central core and the filaments. The fluorescence of the core is quenched while filaments exhibit strong fluorescence. Upon drying, the filament fluorescence gets quenched while the core is less affected, showing stronger relative fluorescence. Fluorescence-detected linear dichroism (FDLD) microscopy reveals that absorption dipoles corresponding to J-bands are oriented along the filament axis. The comparison of FDLD with scanning electron microscopy (SEM) reveals the structure of central core comprised of multilayer ribbons, which wind around the core axis forming a tube. Polarimetric second-harmonic generation (SHG) and third-harmonic generation microscopy exhibits strong signal from the filaments with nonlinear dipoles oriented close to the filament axis, while central core displays very low SHG due to close to centrosymmetric organization. Large chiral nonlinear susceptibility points to helical arrangement of the filaments. The investigation shows that TPPS4 molecules form distinct aggregate types, including chiral nanotubes and nanogranular aggregates that associate into the hierarchical GSU structure, prototypical to complex biological structures. The chiral TPPS4 aggregates can serve as harmonophores for nonlinear microscopy.

Synthesis, single-crystal X-ray diffraction, NLO and DFT studies of centrosymmetric 4-amino-3,5-dimethyl-1H-pyrazolium citrate monohydrate salt

An organic salt compound 4-amino-3,5-dimethyl-1H-pyrazole (ADMP) has been combined with citric acid and crystallised at room temperature, forming a new molecular salt 4-amino-3,5-dimethyl-1H-pyrazolium citrate (ADMPCA) as a monohydrate. The compound was then structurally characterised by single-crystal X-ray diffraction (XRD), spectral analysis of FTIR, NMR and UV, and thermal studies. The structural analysis has shown that the ADMP molecules participate in the dominant hydrogen-bonding patterns forming a ladder-like structure via charge-assisted N+–H···O, O–H···O and N–H···O hydrogen bonds. The Hirshfeld surface analysis has been performed to understand the importance of intermolecular interactions in the stability of the salt structure. Theoretical studies were performed employing the DFT/B3LYP/6-311++G(d,p) level of theory to investigate Non-linear optical (NLO) behaviour, first-order hyperpolarisability, dipole moment and polarisability. The molecular properties, such as natural charge analysis and MESP analysis, were also explored in the present study.

Toy model of harmonic and sum frequency generation in 2D dielectric nanostructures

Optical nonlinearities of matter are often associated with the response of individual atoms. Here, using a toy oscillator model, we show that in the confined geometry of a two-dimensional dielectric nanoparticle a collective nonlinear response of the atomic array can arise from the Coulomb interactions of the bound optical electrons, even if the individual atoms exhibit no nonlinearity. We determine the multipole contributions to the nonlinear response of nanoparticles and demonstrate that the odd order and even order nonlinear electric dipole moments scale with the area and perimeter of the nanoparticle, respectively.

Electronic dephasing in mixed quantum–classical molecular systems using the spin-boson model

Pure electronic dephasing is investigated using the spin-boson Hamiltonian in mixed quantum–classical environment. The spin-boson model used here is a composite system made up of a quantum subsystem, an electronic 2-level subsystem linearly coupled to harmonic vibrations, interacting with a classical bath. Experimental results for a multitude of molecular systems indicate that the zero-phonon line (ZPL) profile is determined by electronic dephasing, which is not accounted for in the multimode Brownian oscillator (MBO) model due to the unphysical contribution from the MBO bath modes to the ZPL profile. Mixed quantum–classical dynamics formalism of non-equilibrium systems is employed to assess the contribution of the bath modes to pure electronic dephasing by probing the ZPL profile when coupled to a classical bath in the mixed quantum–classical condensed systems. Pure electronic dephasing is discussed in the context of mixed quantum–classical dynamics formalism which starts with mixed quantum–classical Liouville equation in a mixed quantum–classical environment. It is noteworthy, however, that the fundamental difference between the fully quantum MBO model and the mixed quantum–classical Brownian oscillator, is that the zero-phonon line calculated by the former shows unphysical asymmetry on the low-energy side as it has not been observed in real systems, whereas the ZPL reported herein eliminates this asymmetry. A systematic approach using matrix mechanics is developed to treat this phenomenon. To this end, a closed-form expression of linear and nonlinear optical electronic transition dipole moment time correlation functions in a dissipative media are derived. Linear absorption spectra and 4-wave mixing signals at various temperatures showing a sound thermal broadening, temporal decay, and accurate pure dephasing further ratify the applicability and correctness of the mixed quantum–classical dynamics approach to spectroscopy and dynamics are computed.

High-order nonlinear dipole response characterized by extreme ultraviolet ellipsometry

Polarization engineering and characterization of coherent high-frequency radiation are essential to investigate and control the symmetry properties of light–matter interaction phenomena at their most fundamental scales. This work demonstrates that polarization control and characterization of high-harmonic generation provides an excellent ellipsometry tool that can fully retrieve both the amplitude and phase of a strong-field-driven dipole response. The polarization control of high-harmonic generation is realized by a transient nonlinear dipole grating coherently induced by two noncollinear counterrotating laser fields. By adjusting the ellipticity of the two driving pulses simultaneously, the polarization state of every high-harmonic order can be tuned from linear to highly elliptical, and it is fully characterized through an energy-resolved extreme ultraviolet polarimeter. From the analysis of the polarization state, the ellipsometry indicated that both the amplitude and phase of the high-harmonic dipole scale rapidly with the driving laser field for higher-order harmonics, and, especially, for gases with a small ionization potential. Our experimental results were corroborated by theoretical simulations. Our findings revealed a novel high-harmonic ellipsometry technique that can be used for the next generation of high-harmonic spectroscopy and attosecond metrology studies because of its ability to provide single-digit attosecond accuracy. Our work also paves the way to precisely quantify the strong-field dynamics of fundamental processes associated with the transfer of energy and angular momentum between electron/spin systems and the symmetry-dependent properties of molecules and materials.

Second Harmonic Scattering of Molecular Aggregates

A general model is developed to describe the polarization-resolved second harmonic scattering (SHS) response from a liquid solution of molecular aggregates. In particular, the molecular spatial order is introduced to consider the coherent contribution, also known as the retarded contribution, besides the incoherent contribution. The model is based on the description of a liquid suspension of molecular dyes represented by point-like nonlinear dipoles, locally excited by the fundamental field and radiating at the harmonic frequency. It is shown that for a non-centrosymmetrical spatial arrangement of the nonlinear dipoles, the SHS response is very similar to the purely incoherent response, and is of electric dipole origin. However, for centrosymmetrical or close to centrosymmetrical spatial arrangements of the nonlinear dipoles, the near cancellation of the incoherent contribution due to the inversion symmetry rule allows the observation of the coherent contribution of the SHS response, also known as the electric quadrupole contribution. This model is illustrated with experimental data obtained for aqueous solutions of the dye Crystal Violet (CV) in the presence of sodium dodecyl sulfate (SDS) and mixed water-methanol solutions of the dye 4-(4–dihexadecylaminostyryl)-N-methylpyridinium iodide (DiA), a cationic amphiphilic probe molecule with a strong first hyper-polarizability; both CV and DiA form molecular aggregates in these conditions. The quantitative determination of a retardation parameter opens a window into the spatial arrangements of the dyes in the aggregates, despite the small nanoscale dimensions of the latter.