The optimization of plasmonic systems—i.e., those based on ‘plasmons,’ the quanta of plasma oscillations—could enable their use in a range of applications (e.g., biomedical and chemical sensing, imaging, and the development of polarization-based plasmonic metadevices). Information about the polarization properties of scattered light from plasmonic metal nanoparticles and nanostructures is key to achieving this.

The acquisition and subsequent analysis of complete polarization information from plasmonic nanostructures represents a crucial step toward the fundamental understanding of a number of recently observed intricate spin-optical effects,^{1} and the optimization of experimental parameters for many practical applications.^{2 }Such studies are, however, confounded by the experimental difficulties inherent in recording the full polarization response from plasmonic nanostructures, and by the complexities of the polarization signals that are obtained from such systems. The scattering signal from plasmonic structures (which is relatively weak) is generally swamped by a large amount of unscattered background light. As a result of this, polarimetric measurements on plasmonic nanostructures must be made in a high numerical-aperture (NA) setting.

To address some of the outstanding challenges in plasmon polarimetry, we have developed a novel spectroscopic system based on a Mueller matrix (i.e., a 4×4 matrix representing the transfer function of any optical system in its interaction with polarized light).^{2} Our system integrates an efficient Mueller-matrix measurement scheme with a dark-field microscopic-spectroscopy arrangement. This arrangement facilitates the detection of weak scattering signals from plasmonic nanostructures. The inherent complexities of measuring polarization in high NA settings (i.e., where one must account for the complex polarization transformation that occurs because of both the focusing and collection geometry of the microscope arrangement) are dealt with by using a robust eigenvalue calibration method, which works over a broad wavelength range.^{2, 3} The experimental system is also enhanced by our inverse analysis models, which extract and quantify the individual polarization properties of plasmonic samples. Using this approach, the experimental Mueller matrix (*M*) is decomposed into a product of a depolarizing and a non-depolarizing matrix.^{3, 4} The latter is then further analyzed to extract the useful polarimetry parameters (i.e., the diattenuation and retardance).

We demonstrated the feasibility and potential utility of our quantitative Mueller-matrix polarimetry on a simple plasmonic system made up of gold nanorods. A schematic of the experimental system (which employs broadband white-light excitation) and the associated calibration results are shown in Figure 1.^{2} We constructed the wavelength-dependent scattering Mueller matrices for this sample—*M*(λ)—by combining 16 spectrally resolved intensity measurements (i.e., the scattering spectra) for four different combinations of the optimized-elliptical polarization-state-generator (PSG) and polarization-state-analyzer (PSA) states. We used the eigenvalue calibration method to determine the exact experimental forms of the system's PSG and PSA matrices (and their associated wavelength dependence), and subsequently used these values to determine the Mueller matrix of the callibrating achromatic quarter waveplate.^{2, 3} The illustrative Mueller matrices exhibit characteristic features of a pure linear retarder over the entire wavelength (λ) range: see Figure 1(c). We also found the values for linear retardance (δ) and linear diattenuation (*d*) of the quarter waveplate and linear polarizer (respectively) to be in close agreement (δ∼1.60rad and *d*∼0.98) with the ideal values (δ=1.57rad and *d*=1) for the entire range: see Figure 1(d). These (and other) calibration results validate the accuracy of our Mueller-matrix measurements.^{2}

**Figure 1. **(a) A schematic of the novel dark-field Mueller-matrix spectroscopic-microscopy system. (b) Dark-field (DF) image of a single isolated gold nanorod. NP: Nanoparticle. (c) The spectral Mueller matrices (over λ=*500- 700**nm*) of a calibrating achromatic quarter waveplate (QWP). The matrices exhibit a characteristic feature of a linear retarder (associated with symmetries in the off-diagonal elements of the lower *3*×*3*block, highlighted in red). (d) The Mueller-matrix-derived linear retardance, δ(λ)(red dotted line), and linear diattenuation, *d*(λ)(blue solid line), of the achromatic QWP and a linear polarizer, respectively. NA: Numerical aperture. PSG: Polarization-state generator. PSA: Polarization-state analyzer.

We subsequently employed the system to record a spectroscopic-scattering Mueller matrix from a single isolated gold nanorod (with a diameter of 14±3nm and a length of 40±3nm): see Figure 2. The scattering spectra from the gold nanorod exhibits two distinct peaks, which correspond to the two electric dipolar plasmon resonances. One of the peaks occurs at λ∼525nm due to transverse resonance along the short axis, and the second occurs at λ∼650nm due to longitudinal resonance along the long axis.

**Figure 2. **(a) The wavelength dependence of the decomposition-derived, non-depolarizing (diattenuation retarder) Mueller matrix, obtained from a single gold nanorod. (b) The decomposition-derived wavelength variation of the linear retardance—δ(λ), red dotted line—and linear diattenuation—*d*(λ), blue solid line—parameters of the gold nanorod.

The corresponding scattering Mueller matrices encode several interesting and potentially useful pieces of information. Despite considerable depolarization effects, which occur due to the high-NA imaging geometry, the decomposed non-depolarizing component (i.e., the diattenuating retarder matrix) shown in Figure 2(a) can be interpreted to extract the spectral diattenuation and retardance—*d*(λ) and δ(λ), respectively—of the plasmonic sample. We interpret these effects via the relative amplitudes and phases of the two orthogonal dipolar plasmon polarizations of the gold nanorod. The *d*(λ) parameter is a measure of the relative amplitudes of the transverse and longitudinal dipolar polarizabilities, and therefore peaks at wavelengths that correspond to their resonances (525 and 650nm for transverse and longitudinal, respectively). The δ(λ) parameter, on the other hand, quantifies the phase difference between the two orthogonal polarizabilities, and thus reaches its maximum value in the spectral overlap region of the two resonances (∼575nm): see Figure 2(b). The *d*(λ) and δ(λ) parameters therefore capture and quantify unique information on the relative strengths and phases (respectively) of the two dipolar plasmon polarizabilities of the gold nanorod. We can use this polarization information to probe, manipulate, and tune the interference of neighboring resonant modes (i.e., orthogonal dipolar modes in plasmonic nanorods), and the resulting spectral line shape of the plasmonic system, via polarization control.^{2}

In summary, we have developed a novel experimental system for recording full 4×4 spectroscopic-scattering Mueller matrices from single isolated plasmonic nanoparticles or nanostructures. Inverse Mueller-matrix analysis on a single plasmonic gold nanorod yields intriguing spectral diattenuation and retardance effects that encode potentially valuable information about the relative strengths and phases of the resonant modes in plasmonic structures. These polarization parameters therefore hold considerable promise as novel experimental metrics for the analysis of a number of interesting plasmonic effects. The parameters could, for example, be used to probe, manipulate, and tune the interference of neighboring modes in complex coupled plasmonic structures, or to study spin-orbit interaction and the spin Hall effect of light.^{1, 5} Results from such studies could be used to develop and optimize novel polarization-controlled plasmonic sensing schemes. We are currently expanding our investigations in these directions. Our next steps include the analysis, interpretation, and control of the plasmonic Fano resonance using Mueller-matrix-based polarization analysis. We are also employing polarization-based weak-measurement schemes to observe and amplify tiny spin-optical effects in plasmonic structures. It is our hope that these studies will aid in the development of novel polarization-controlled optical metadevices for diverse applications.

Shubham Chandel, Subir Kumar Ray, Jalpa Soni, Satyabrata Raj, Nirmalya Ghosh

Indian Institute of Science Education and Research, Kolkata

Mohanpur, India

Shubham Chandel is currently pursuing his doctoral research in the Bio-Optics and Nano-Photonics Group, where his research focuses on the use of polarimetry. He obtained his MSc from the International School of Photonics, India.

Nirmalya Ghosh is a physicist with a specialization in optics and photonics. He is currently an associate professor in the Department of Physical Sciences, and heads the Bio-Optics and Nano-Photonics Group.

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