Quantitative Measurement of the Optical Cross Sections of Single Nano-objects by Correlative Transmission and Scattering Microspectroscopy

The scattering and absorption of light by nano-objects is a key physical property exploited in many applications, including biosensing and photovoltaics. Yet, its quantification at the single object level is challenging and often requires expensive and complicated techniques. We report a method based on a commercial transmission microscope to measure the optical scattering and absorption cross sections of individual nano-objects. The method applies to microspectroscopy and wide-field image analysis, offering fine spectral information and high throughput sample characterization. Accurate cross-section determination requires detailed modeling of the measurement, which we develop, accounting for the geometry of the illumination and detection as well as for the presence of a sample substrate. We demonstrate the method on three model systems (gold spheres, gold rods, and polystyrene spheres), which include metallic and dielectric particles, spherical and elongated, placed in a homogeneous medium or on a dielectric substrate. Furthermore, by comparing the measured cross sections with numerical simulations, we are able to determine structural parameters of the studied system, such as the particle diameter and aspect ratio. Our method therefore holds the potential to complement electron microscopy as a simpler and cost-effective tool for structural characterization of single nano-objects.


S.I. EXPERIMENTAL SET-UP AND PROTOCOL
Our experimental set-up is based on a commercial inverted microscope (Nikon, Eclipse Ti-U), with some customization. Incoherent broadband illumination for micro-spectroscopy is provided by a tungsten-halogen lamp (Nikon, 17005M28). Incoherent illumination with a narrow spectral range for widefield imaging is provided by a multiple light-emitting diode (LED) source (Thorlabs, LED4D067, 405 nm, 455 nm, 530 nm, 660 nm LEDs) coupled to the microscope using a liquid light guide (Thorlabs LLG0538-6). The average emission wavelength λ of the LEDs reported in Fig. 5 of the paper depends slightly on the intensity and has been characterized in house. The illumination is focused on the sample by a 1.34 numerical aperture (NA) oil-immersion condenser lens (Nikon MEL41410, T-C-HNAO). An engineered diffuser (Thorlabs, ED1-C20) inserted in the illumination path in front of the field iris yields a flat angular scattering pattern up to a ±10 • divergence, and provides thereby a quasi-homogeneous illumination intensity in the back focal plane (BFP) of the condenser as assumed by our analytical model of the illumination, see the characterization in Sec. S.VI. A wire-grid polarizing film (MeCan, WGF TM ) can be inserted in front of the condenser and rotated to produce a linear polarization of adjustable orientation in the BFP of the condenser. The sample holder is mounted on an xyz piezoelectric stage (Mad City Labs Nano-LP200), which is used for fine positioning and focusing of the nano-object (NanO). Light is collected by a 40x, 0.95 NA dry objective (Nikon MRD00405, CFI plan apochromat λ series). The angular range of illumination is defined by suitable light stops placed in the BFP of the condenser as depicted in Fig. 1 of the paper. In order to achieve darkfield (DF) illumination of NA df i = 1.1 we cut an anodized aluminum disc of diameter 2f c NA df i = 23.1 mm, calculated based on the condenser focal length f c = 10.5 mm. The disc is mounted on a slider, to permit rapid switching between brightfield (BF) and DF modalities with no need to move the condenser. The aperture diaphragm of the condenser is closed down to NA bf i = 0.95 in BF, matching the objective NA; and to NA df i between 1.2 and 1.3 in DF, excluding the highest NA range of the condenser to reduce deviations from the aplanatic behavior.
Micro-spectroscopy is performed by optically coupling the microscope output to an imaging spectrometer (Horiba Jobin-Yvon, iHR550) equipped with a ruled plane diffraction grating (Horiba, 510 48) of 78 mm side length and 100 lines/mm blazed at 450 nm. The spectrometer output is recorded by a Peltier-cooled back-illuminated charge-coupled device (CCD) sensor (Andor, Newton DU-971N). A magnification of 79 ± 2 from the sample to the sensor was measured shifting the sample by a known amount via the nanometric stage. Since the cross-sections are proportional to the inverse square of the magnification via A in Eq. (7) of the paper, determining its precise value is important for quantitative analysis. The spectrometer is operated first in imaging mode (i. e. with the grating oriented at the zero order of diffraction acting as a mirror) to individuate the NanO and center it with respect to the input slit of the spectrometer. The four detected signals appearing in Eq. (7) of the paper are then acquired in spectroscopic mode (i. e. with the grating oriented at the first order of diffraction). A square detection region A no = 1.0 µm 2 is defined on the scaled sample image, delimited by the input slit and the detected size along the slit on the CCD: This is chosen large enough to accommodate the diffraction-limited image of the NanO, leaving additionally some room for spatial drift over the measurement time (∼ 10 s). A bg corresponds to another similar-sized region of the sensor, displaced along the input slit, so that S df no and S df bg are acquired simultaneously. This is not problematic because DF can offer a large contrast with respect to the background. In BF, conversely, the pixel-to-pixel variations of quantum efficiency (∼ 1 %) are comparable to the difference between S bf no and S bf bg for the NanOs we studied. Therefore the signals must be acquired sequentially using the same region of the sensor and shifting the sample instead, so that an empty area nearby the NanO is imaged. In addition, adjacent regions of the sensor are used in BF to monitor the illumination intensity -which fluctuates ∼ 1 %/min -and scale the S bf bg accordingly. Widefield imaging is performed using a monochrome low noise scientific complementary metal-oxide-semiconductor (sCMOS) camera (PCO Edge 5.5) attached to the Ti-U eyepiece side port. For automated image analysis we rely on the ImageJ Plug-in Extinction Suite [S1] developed within our group. We briefly discuss here the scattering measurements presented in the paper; we refer the reader to previous works [S2, S3] for a detailed description of the technique, also presenting its application to absorbing NanOs. We acquire one BF and one DF image. 2560 frames of exposure time 1 ms were averaged in BF. 4 frames of exposure time 1 s were averaged in DF. Between the acquisition of the two images the sample was laterally displaced by 2.2 µm -just above the characteristic point spread function (PSF) size 3λ/NA for the longest measured wavelength -with the nanometric stage. The BF image serves as reference to create a quantitative scattering image. Extinction Suite individuates in this image all the intensity maxima, which correspond to the PSFs of individual NanOs or unresolved aggregates. The user sets by hand a tolerance level to avoid including small debris or background fluctuations in the analysis. Afterward the image is integrated over a disc (A no ) of radius r psf and an annulus (A bg ) of inner radius r psf and outer radius approximately √ 2r psf , both centered at each intensity maximum. The integrated values over A no and A bg are the scattering signal and the local background, corresponding respectively to S df no /S bf bg and S df bg /S bf bg appearing in Eq. (7s) of the paper. For each exciting wavelength λ we used r psf = 3λ/2NA obj . In the ideal case of an Airy PSF, r psf corresponds approximately to the radius of the second dark ring, and A no contains 91 % of the collected power. However, for real microscopes objectives the PSF is modified by the vectorial nature of light and aberrations. We therefore measured σ sca as a function of r psf on a few well-isolated scatterers and found that, with respect to large r psf values, (79,90,83)% of power is collected at λ = (414, 530, 652) nm. These factors were used to correct the measured σ sca of Fig. 5 of the paper.

S.II. NUMERICAL MODELING
We simulate optical scattering and absorption by an individual NanO using the commercial software comsol Multiphysics R , which implements the finite element method within a convenient user interface. The simulation volume is a sphere centered at the origin of radius λ/[2 min(n 1 , n 2 )], enough to put the boundaries outside the reactive near-field region of the NanO. The sphere is halved into two media: Illumination comes from medium 1 (substrate) and the NanO is placed in medium 2 in contact with the interface. This arrangement reproduces the experimental set-up depicted in Fig. 1 of the paper. The sphere is surrounded by a perfectly matched layer (PML) shell of thickness λ/(n 1 +n 2 ), which absorbs the scattered radiation and thereby mimics an infinitely extended simulation volume. After refining the mesh so that the simulated cross-sections have converged within a few percent tolerance, our model has about 10 5 degrees of freedom and solves in approximately 15 s on a modern workstation (Intel Core i7-8700K CPU, 64 GB RAM).
We solve the electromagnetic problem in its frequency-domain formulation We adopt the so-called scattered field formalism, and decompose the total electric field into the scattered and the exciting field: E tot = E sca +E exc . E exc is given by a plane wave incident on the n 1 to n 2 interface and the resulting transmitted and reflected waves, and is input via its analytical expression derived in Sec. S.III. Such analytical approach improves the model performance by a factor 5 to 10 with respect to the widespread practice of computing E exc numerically with a preliminary solving step. Linearly polarized illumination in the BFP of the condenser is obtained as a superposition of the p and s polarized components with appropriate coefficients depending the azimuth ϕ i . Unpolarized illumination is reproduced instead by averaging results obtained with p and s polarization, since the two components are mutually incoherent. E sca is solved for by the software, and can be used to compute the power scattered P sca and absorbed P abs by the NanO. P sca is the flux of the time-averaged Poynting vector of the scattered field across the surface Σ no of the NanO where n is the outside normal of Σ no . P abs is computed by integrating the power dissipated via resistive (Joule) heating over the volume V no of the NanO where J c = ω Im(ε)E is the conduction current density in the metal. Eventually, σ sca and σ abs can be computed using their definition σ ≡ P/I i and Eq. (S.II). Now, note that σ sca and σ abs depend on the incidence direction (θ i , ϕ i ) of the plane wave E exc , as well as on its polarization state. We reproduce the illumination provided by a high-NA condenser in experiments by averaging σ(θ i , ϕ i ) over the angular range of illumination with the formulae derived in Sec. S.IV B. The weights used in the formulae originate from the analytical description of the microscope illumination reported in Sec. S.IV A, and account as well for the experimental referencing of σ to the measured intensity, that is, the incident intensity projected onto the sample plane. This approach -whereby E exc with different (θ i , ϕ i ) do not interfere -is consistent with the description of the microscope illumination as an incoherent superposition of plane waves adopted by our analytical model of scattering described in Sec. S.V below.

S.III. ELECTRIC FIELD ABOVE AND BELOW A DIELECTRIC INTERFACE
An analytical expression of the exciting electric field E exc is required both for simulating numerically the optical cross-sections of a NanO with the scattered field formalism discussed in Sec. S.II, and for the analytical calculation of the angular distribution of the scattered power, which will be presented in Sec. S.V B.

A. Illumination geometry and plane wave illumination
In this section E exc is derived for the general case discussed in this work, that is, two homogeneous dielectric media separated by an infinite planar interface. The experimental geometry and the notation adopted are presented in Fig. S1. Since in this work we treat the microscope illumination as an incoherent superposition of plane waves, we need to compute E exc above and below the interface for a plane wave impinging from the condenser side, i. e. medium 1. At the interface, the incident (subscript i) wavefront is split into a reflected (r) and a transmitted (t) one. The plane represented in Fig. S1a contains the wavevectors k of the three waves and is called plane of incidence. The directions parallel (p) and perpendicular (s) to the plane of incidence constitute the natural basis to decompose E, because the p and s components do not mix upon transmission and reflection. It is convenient to introduce the notation p ≡ E p and s ≡ E s .
Consider now the spherical unit vectors shown in Fig. S1b, whose expressions in Cartesian coordinates read Comparing these to the wavevectors and fields of the three waves in Fig. S1a, one finds that the two vector triads have always the same directions: ( k, p, s) = (− r, θ, ϕ). Therefore, the wavevector of the incident wave is k i = −k i r(θ i , ϕ i ), and equivalent expressions hold for the reflected (i → r) and transmitted (i → t) waves; the respective wavenumbers are k i = k r = n 1 k 0

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and k t = n 2 k 0 . As for the field orientations, one has p i = θ(θ i , ϕ i ) and s i = ϕ(ϕ i ) for the incident wave, and equivalent expressions hold for the reflected and transmitted ones. The propagation direction k of these waves is thus identified by a polar angle θ and an azimuth ϕ. The polar angles are related through the law of specular reflection θ r = π − θ i and Snell's law n 1 sin θ i = n 2 sin θ t . The planar character of the problem implies ϕ i = ϕ r = ϕ t for the azimuths. Note that the alternative formulation θ r = θ i and ϕ r = ϕ i + π of the law of specular reflection -albeit more common in literature -would not allow the identification ( k, p, s) = (− r, θ, ϕ) for the reflected wave. The transmission and reflection amplitudes are  Figure S1. (a) Two homogeneous dielectric media with refractive indices n 1 and n 2 are separated by the plane z = 0. A plane wave E i incident on the interface splits into a reflected wave E r and a transmitted wave E t . The plane of incidence ϕ = ϕ i is represented, so that the p and s component of the fields are, respectively, parallel and perpendicular to the drawing plane. The field orientation depicted corresponds to n 1 > n 2 and θ b < θ i < θ c , where θ b and θ c indicate Brewster's and critical angle, respectively. The exciting field induces an electric dipole moment p in a NanO placed near the interface, which forms an angle Θ with respect to the normal to the interface. expressed by the Fresnel coefficients for p and s polarization. Under the assumptions of non-magnetic (µ 1 = µ 2 = 1), non-absorbing (ε 1,2 = n 2 1,2 ) media they are We have now established a formalism which permits to write a compact expression of the field components where the time dependence, given by a common factor e −iωt , with ω = k 0 c and the vacuum light velocity c, has been omitted. Note that E i ∈ R when the phase of E i is taken as reference. We then use Eq. (S6) to obtain the expression of E exc (x, y, z) we were after. E exc in medium 1 is the coherent superposition of the incident and reflected fields whereas E exc in medium 2 coincides with the transmitted field We have highlighted that the fields (S7) and (S8) depend on the direction of incidence solely, as all other variables can be expressed as a function of (θ i , ϕ i ) via the relations between angles reported above. Eq. (S8) holds as well for total internal reflection (TIR), which occurs for n 1 > n 2 and θ i > θ c = arcsin (n 2 /n 1 ). In this case θ t = n 2 /n 1 is a complex number and k t,z = −k t cos θ t becomes purely imaginary: The resulting real exponential in Eq. (S8) describes an evanescent decay of the field moving away from the interface.

B. Unpolarized and linearly polarized illumination
For unpolarized light, the p and s components of E exc , E p and E s , have the same amplitude and no phase relation. Accordingly, the expression of E exc given by Eq. (S7) and Eq. (S8) was used for simulating the optical cross-sections under unpolarized illumination of the gold and polystyrene spheres investigated in the paper. However, using polarized light is often useful, e. g. to study the optical properties of anisotropic NanOs such as the gold rods investigated in the paper. Under polarized illumination, E exc is a coherent superposition of E p and E s , with coefficients depending on the direction of incidence. Let us derive the analytical form of the coefficients for linearly polarized illumination.
In our experiments, a linear polarization state can be created in the back focal plane (BFP) of the condenser lens; the azimuth ψ identifies the illumination direction as in Fig. S2. In terms of the Cartesian unit vectors, the electric field in the BFP is The illumination intensity is assumed to be homogeneous over the BFP, so that E bfp is independent of the position (ρ, ϕ) in the plane. As shown in Fig. S2, the basis unit vectors of Cartesian and cylindrical coordinates are related by a rotation around z by the angle ϕ represented by the rotation matrix R z (ϕ) E bfp can be thereby expressed in terms of the cylindrical versors Let us now calculate E i corresponding to the direction of incidence (θ i , ϕ i ) in the front space of the condenser, where the spherical coordinates (r, θ, ϕ) are used. As illustrated by Fig. S1b, the condenser focuses the collimated illumination onto the sample plane by converting the propagation direction − z into − r, so that the resulting wavevector is k i = −k i r(θ i , ϕ i ). Concurrently, the radial component ρ(ϕ i ) of the field is rotated into a polar one θ(θ i , ϕ i ) = p i , while the tangential component ϕ(ϕ i ) = s i is left unchanged. Thus namely, a coherent superposition of the p and s fields defined by Eq. (S6). Moreover, since the p and s components are not mixed upon reflection or transmission, an analogous expression holds for the reflected (i→r) and transmitted (i→t) waves. We can eventually write via Eq. (S12) E exc above and below the interface, namely with E 1,p , E 1,s given by Eq. (S7) and and E 2,p , E 2,s given by Eq. (S8). This expression for E exc was used for simulating the optical cross-sections under linearly polarized illumination of the gold rods investigated in the paper.

S.IV. ANALYTICAL DESCRIPTION OF THE MICROSCOPE ILLUMINATION
In many micro-spectroscopy experiments, such as those presented in this work, the sample is illuminated by a high NA condenser lens. The illumination NA range in BF modality is typically chosen to match the NA range of the microscope objective used, to provide the best diffraction-limited spatial resolution, while in DF modality the NA range of condenser and objective are chosen to have no overlap. Describing such excitation as a plane wave impinging perpendicularly onto the sample plane -a common approach to numerical modeling in the nanoplasmonics field -can provide insight as to the position and linewidth of the resonant modes of the investigated system, but is not suited to accurately calculate the measured cross-section magnitude. For example, the axial polarization component introduced by illumination at non-zero angles of incidence is not accounted for.
We will therefore develop in the following section S.IV A a mathematical description of the high NA, incoherent microscope illumination employed in our experiments. This description is an essential part of the method described in Sec. S.IV B to compute the cross-sections under microscope illumination, which are directly modeling the measurements. It is also used in the analytical model of scattering presented in Sec. S.V B, which we can use to calculate the scattering parameters of our quantitative method.

A. Angular dependence of the exciting power and intensity
Generally speaking, lens systems transform positions (ρ i , ϕ i in cylindrical coordinates) in the back focal plane (BFP) to directions (θ i , ϕ i in spherical coordinates) impinging onto the front focal plane (FFP) as depicted in Fig. S1b -and vice versa. Consistently with the rest of this work, the subscript i stands for "incidence" and denotes coordinates and quantities related to the illumination. We consider axially-symmetric optical elements only, so that ϕ i is conserved in the transformation. Aplanatic lens systems (aplanats) are of paramount importance for microscopy, because they are free of both off-axis coma and spherical aberration. An aplanat is defined by a specific relation between position in the BFP and angle of incidence in the FFP where f is the effective focal length and n is the refractive index of the medium filling the FFP. In literature Eq. (S14) is referred to as Abbe's sine condition. In this work we assume that the condenser is an aplanat focusing the illumination onto the sample placed at the FFP: This is indeed an adequate description for a scientific-grade condenser lens such as that used in our experiments. From a geometric point of view, an aplanat maps the flat wavefront of a collimated beam in the BFP into a hemispherical surface Σ (often called the front principal "plane") of radius f n converging at the front focal point F as depicted in Fig. S3a. In our case, the illumination Figure S3. Sectional view of a typical Köhler illumination set-up for micro-spectroscopy experiments.
The sketch depicts the main geometrical features associated with an aplanatic condenser lens (double arrow) discussed in the text. The angle of incidence θ i in the magnified insets (b) and (c) is chosen larger that the illumination range in (a) for clarity.

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originates from an incoherent source and therefore does not form a coherent wavefront in the BFP. Its coherence length in the BFP is given by the diffraction limit of the illuminated region in the FFP, which is an image of the field diaphragm in Köhler illumination. We use illuminated regions with sizes of about 300 µm, three orders of magnitude larger than the diffraction limited point spread function (PSF) size for the full condenser NA. Accordingly, the coherence length in the BFP is about three orders of magnitude smaller than the size of the BFP. We can therefore treat the coherence areas in the BFP as small elements of area dΣ = (f n) 2 dΩ i corresponding to a well-defined direction of incidence (θ i , ϕ i ), of a plane wave propagating towards F . Note that the effective condenser focal length f n, in our apparatus 10.5 mm, is much larger than the illuminated region, so that the illumination in the BFP of the condenser is still collimated to a small angular range, in our case about 0.03 radians. The illumination impinging on the sample is then the incoherent superposition of these plane waves. Now, the small element of area dA bfp corresponding to dΣ shrinks towards the edges of the BFP, as illustrated by the geometrical construction in Fig. S3b. Equivalently, this result can be obtained as a consequence of the sine condition (S14) The factor thereby introduced is sometimes referred to as the aplanatic apodization cosine.
We want now to investigate the dependence on θ i of the power dP pw and intensity dI pw carried by the plane waves (PW) in our description. With dΣ and dA bfp related via Eq. (S15), the same power crosses them by geometrical construction in a ray picture dP pw = dP bfp = I bfp dA bfp = I bfp cos θ i dΣ (S16) where I bfp is the illumination intensity over the BFP, which is assumed to be constant.
Eq. (S16) shows that dP pw decreases as θ i increases, ruled by the aplanatic apodization cosine. Let us call A ffp the illuminated region of the FFP. As discussed before, under the Köhler illumination scheme employed in our set-up, A ffp is defined by the field aperture and is the same for all directions of incidence. This implies that the plane wave elements have a wavefront of size A pw , which according to the geometry given in Fig. S3c is given by A pw = cos θ i A ffp , reduced compared to A ffp by the beam squeezing cosine. This expression of A pw can be used to calculate the intensity of the plane wave where the expression (S16) of dP pw has been substituted in the last equality. Eq. (S17) shows that dI pw is independent of the illumination direction: This is a non-trivial consequence of an exact compensation between the aplanatic apodization cosine and the beam squeezing cosine under the Köhler illumination scheme. S11

B. Angular averaging of the simulated cross-sections
In our numerical simulations, we compute the cross-section σ pw under plane wave illumination (in this work we use superscripts consistently to indicate the illumination condition), obtained using the plane wave expressions of E exc derived in Sec. S.III. On the other hand, what is measured in micro-spectroscopy experiments is the cross-section under microscope (m) illumination σ m . In this section, we will use the mathematical description of the microscope illumination developed in Sec. S.IV A to express σ m in terms of σ pw , to relate the simulated to the measured cross-sections.
Let us begin with the definition of the cross-section as the power removed from the exciting electromagnetic mode(s) by a given optical process divided by the incident intensity Importantly, referencing occurs with respect to the intensity I ffp traversing the FFP, to reproduce the way S bf bg is measured in experiments, where the microscope objective images the sample plane onto the sensor. In the second equality of Eq. (S18), power and intensity are expressed as the sum over the contributions of the plane waves within the illumination cone. Note that while the removed power is proportional to the plane wave intensity dI pw , the corresponding reference intensity is proportional to the power dP pw traversing A ffp , so that a cos θ i factor appears at the denominator, see also Fig. S3c. Effectively, the way referencing is performed in experiments results in σ m > σ pw , as if the NanO shadow projected onto the FFP was measured (dashed orange line in Fig. S3c). By virtue of this analogy, we call the cos θ i factor in Eq. (S18) the long shadow cosine. dI pw given by Eq. (S17) can be rewritten using Eq. (S15) as By substituting the first expression into Eq. (S18) we obtain a formula of simple interpretation: σ m is the average of σ pw over the illuminated region A i of the BFP, weighted by the inverse long shadow cosine to account for the experimental referencing to I ffp . Now, since we compute σ pw as a function of the direction of incidence, the integration must be recast over the angular variables using the last form of Eq. (S19) S12 σ pw (θ i , ϕ i ) is a discrete set of data, interpolated with constant steps ∆θ i and ∆ϕ i from the sparse simulated values. The integral (S21) is therefore approximated by the discrete sum where we have highlighted that, under linearly polarized illumination, the cross-sections depend on the polarization direction ψ in the BFP of the condenser defined as in Fig. S2. Under unpolarized illumination (ψ \), two incoherent plane waves having equal intensity and orthogonal polarization -say, p and s -correspond to each direction of incidence, so that σ pw = (σ p + σ s )/2 and we have Often one can use the symmetries of the investigated system (excitation + NanO) to reduce the integration/summation domain in the formulae above. This is useful as it cuts the number of simulations required for averaging with a given accuracy, and hence the total computation time. In particular, symmetry with respect to the optical axis z allows to reduce the domain of ϕ i ; let us illustrate this through a few, experimentally relevant examples. We assume the NanO has at least one mirror plane containing z, and orient the axes so that x lies on such plane. If the NanO admits • one mirror plane (x = 0), then the domain can be reduced to ϕ i ∈ [0, π) for excitation polarized along x or y, or unpolarized. The whole ϕ i ∈ [0, 2π) domain must still be simulated for a generic exciting polarization. This is for instance the case of tetrahedra or sphere heterodimers.
• two orthogonal mirror planes (x = 0 and y = 0), then the domain can be reduced to ϕ i ∈ [0, π) under a generic linearly polarized illumination, and to ϕ i ∈ [0, π/2) for excitation polarized along x or y, or else unpolarized. This is for instance the case of rods, cubes, and sphere homodimers.
• z as a continuous axis of rotation, then σ m (ψ) = σ m (ψ \), and any single value of ϕ i (N = 1) can be used to compute Eq. (S23). This is for instance the case of spheres, disks lying flat on a substrate, and cones or pillars standing on a substrate.
Note that the presence of optical interfaces ⊥ z is irrelevant for these symmetry considerations.
Carrying out the directional averaging just described at every simulated wavelength would be rather burdensome; therefore, in this work, we simulated the angular dependence σ pw (θ i , ϕ i ) for a single wavelength (at resonance, when present) and computed the factor σ m /σ pw | θ i =0 . Then the full spectrum was simulated at normal incidence (θ i = 0) only, and multiplied by this factor. This approach, whereby the directional response of the system is assumed to be independent of λ is applied throughout this work. Specifically, the scattering parameters η l and ζ also are computed for a given form of α. In fact, if multiple modes of different geometry were present in the same spectrum, the quantitative method would have to be applied separately to them (different angular scaling factor, and different scattering parameters).
Somewhat surprisingly, due to the long shadow effect described above, σ m is larger than σ pw even in the most symmetric case of a sphere surrounded by a homogeneous medium. Indeed, this system is isotropic so that σ pw does not depend on (θ i , ϕ i ) and can be taken out of the integral (S21), yielding This formula was used for the gold spheres investigated in the paper. With the illumination ranges used (NA bf i from 0 to 0.95 corresponding to θ bf i from 0 to 38.7 • ; and NA df i from 1.10 to 1.28 corresponding to θ df i from 46.4 to 57.4 • ) one has σ bf abs = 1.12 σ pw abs and σ df sca = 1.63 σ pw sca respectively. In a recent work [S4] we verified experimentally on absorption-dominated gold spheres the dependence of σ m ext on the angular range of illumination predicted by Eq. (S24).

S.V. CALCULATION OF THE SCATTERING PARAMETERS
Eq. (7) of the paper express the cross-section amplitudes as functions of the experimental signals, and contain the scattering parameters where l ∈ [BF, DF] denotes the illumination modality. In Eq. (S25) the scattering parameters are expressed in terms of which is the power scattered within the solid angle of detection Ω d defined in spherical coordinates by the angular ranges θ ∈ [θ d , θ d ] and ϕ ∈ [0, 2π) owing to the axial symmetry of our detection geometry. In Eq. (S26) P l no is the angular distribution of the power scattered by the NanO to the far-field. Note that in this work we reserve the calligraphic font P to angular power distributions (units W/sr), whereas integrated powers (units W) are indicated by the usual italic font P .
In general, P l no depends both on the electromagnetic excitation and on the optical response of the NanO. Writing an explicit form of excitation and response requires some assumptions, which are listed right before Eq. (10) of the paper. Nonetheless, the expressions of P l det , and hence of η l and ζ, we derive in this section, still apply to many experimentally relevant cases. One assumption we make in our treatment is the dipole approximation, which is accurate in the electrostatic regime, where the NanO is much smaller than the excitation wavelength. Within this approximation, higher-order multipolar terms of the charge distribution induced in the NanO are neglected, and only an electric dipole moment is considered. Schematically, our description of the scattering process consists of two steps: (i) a set of dipoles are excited at the NanO; (ii) these dipoles radiate. Now, (i) precedes (ii) and is independent from it, meaning that all interactions of the dipole with the field it radiates are neglected in this description. Firstly, this restricts the validity of these calculations to weak scatterers, consistent with the small-NanO limitation imposed by the dipole approximation. Secondly, in presence of an interface near the NanO, scattering of the reflected radiation is not taken into account. This is expected to be a suitable approximation for the measurements presented in this work, where the NanO is placed either in a homogeneous medium or at a moderately mismatched glass/air interface, creating only weak reflections of the scattered field.
Let us first present in Sec. S.V A some results on the radiation emitted by oscillating dipoles, which we will need in the following. In Sec. S.V B we will then carry out calculations for common types of NanOs.

A. Dipole radiation near a dielectric interface
Lukosz and Kunz have derived in a series of papers [S5-S8] the angular distribution P(θ, ϕ) of the power radiated by an oscillating electric dipole p placed at a distance z no from a planar dielectric interface. We report here for convenience their results recast into the notation used throughout this work. The expressions have been simplified by assuming z no = 0, since in our experiments the NanO always lies on the substrate and is described as point-like in the electrostatic regime.
The overall power distribution P(θ, ϕ) = P p + P s described by Eq. (S27)-(S30) is represented in Fig. S4 for an interface of glass with air, water, and index-matching immersion oil, and for dipoles having different orientations. Discontinuities of the distribution occur in the denser medium at the critical angle (θ 1,c = 41.1 • in air and θ 1,c = 61.0 • in water) as well as along the interface, where propagation is impeded. These plots highlight how significantly the directionality of the emission is affected by the presence of an interface, even at a moderate refractive index mismatch. Simple phenomenological methods used sometimes in literature such as the effective medium approximation are therefore inadequate for the quantitative analysis presented in this work.
Normalization of the radiated power distributions P(θ, ϕ) is normalized to the total power radiated P tot expressed in units of P iso ≡ lim zno→∞ P tot (z no ), i. e. the power radiated into an isotropic medium of refractive index n 2 4π P(θ, ϕ) dΩ = P tot P iso .
P iso is related to the total power P vac radiated in vacuum (n 2 = 1) through The quartic frequency dependence of Eq. (S32) is characteristic of dipole emission and rules the scattering spectrum of small dielectric particles, which display no material (e. g. plasmonic) resonances. Interestingly, Ref. [S6] demonstrates that the two dipoles with orientation ⊥ and to the interface radiate two orthogonal electromagnetic modes in the  Figure S4. Normalized angular distribution P(θ, 0) × (P iso /P vac ) of the power radiated by an oscillating electric dipole placed at a planar θ = 90 • dielectric interface. Medium 1 (above) and medium 2 (below) have refractive indices n 1 = 1.52 and n 2 as indicated in the legend of (b). The dipole is placed in medium 2 at a distance z no = 0 from the interface. Note that the radial scale changes between panels. The orange arrows/dot indicate the dipole orientation: far-field, so that the power emitted by an arbitrarily-oriented dipole can be decomposed into their independent contributions P tot P iso (n, Θ) = p e,⊥ (n) cos 2 Θ + p e, (n) sin 2 Θ .
The coefficients in Eq. (S33) can be expanded into a Taylor series in powers of 4πz no /λ 2 . In the limit z no = 0 adopted in this work, we retain only the zero-order terms of the expansion, whose coefficients are reported in Ref. [S5] p m,⊥ = 2 5 p m, = 1 5 where the m and e subscripts refer to a magnetic and an electric dipole respectively.

B. Parameters of some experimentally relevant nano-objects
In order to compute the integral (S26) we need first to break down P no into dipolar contributions. To begin with a simple description, one could represent the NanO as a single dipole p induced by the exciting field E exc where α is the polarizability tensor of the NanO and ε m the permittivity of the embedding medium. Then, in terms of the dipolar power distribution introduced in the previous section where P is the normalized angular distribution of the power scattered by p. However, the microscope illumination cannot be accurately represented by a single analytical expression of E exc due to its incoherent character -that is why the sign ∼ was used in Eq. (S36). As discussed in Sec. S.IV A, the microscope illumination is an incoherent superposition of plane waves, whose analytical form we derived in Sec. S.III. Consequently, the NanO is not just a single dipole, but rather a collection of dipoles oscillating with unrelated phases, and P no is the sum of the powers radiated by all the dipoles. The individual dipolar contributions have the form (S36) and their amplitude is proportional to P iso ∝ p 2 ∝ |E exc | 2 , that is, to the intensity of the exciting plane wave. Now, we found in S11 that the illumination intensity does not depend on θ i , and therefore all the dipolar contributions have the same weight in the sum. In the dipole model we outlined, Eq. (S26) reads where a constant factor n 2 ω 4 /(12πε 0 c 3 0 ) in front of the integral has been omitted. In fact, the scattering parameters Eq. (S25) are defined as power ratios, which also renders them independent of NanO size (within the dipole approximation) and composition (assuming an isotropic material response) -these properties affect only the absolute value of P det . In Eq. (S37) we highlighted that both the amplitude p and the orientation (Θ, Φ) of p depend on the direction of incidence of E exc given by θ i ∈ [θ l i , θ l i ] and ϕ i ∈ [0, 2π) owing to the axial symmetry of our illumination geometry.
Let us now consider the specific relevant scenario where all dipoles have a fixed orientation (Θ, Φ) regardless of the exciting polarization. This is relevant for instance at the longitudinal plasmon mode of an elongated NanO, where the dipoles are mostly excited along the longitudinal direction. Then P(θ, ϕ) does not depend on (θ i , ϕ i ), and Eq. (S37) can be factorized into an excitation integral and a detection integral, P l det = L l exc L det , with L l exc is proportional to the total radiated power, and L det is the power normalized to P iso , emitted within the detection range Ω d . The excitation and detection parameters, given by Eq. (S25), can be written as with L det (0, π) = P tot /P iso given by Eq. (S33). In the rest of the section, the integrals (S37) or (S39)-(S39) are computed for some diagonal forms of α corresponding to relevant symmetries of the resonant modes of NanOs.

Uniaxial polarizability parallel to the interface
Let us begin by considering a uniaxial α parallel to the interface -say along x without loss of generality: (α xx , α yy , α zz ) = (α, 0, 0). This describes well, for instance, the longitudinal resonant mode of elongated particles like rods lying on the substrate, as well as the bonding mode of dimers, which is linearly polarized along the particle separation. As a result of the anisotropy of α, only the x component of E exc excites dipoles of amplitude The orientation of these dipoles is (Θ = π/2, Φ = 0) regardless of (θ i , ϕ i ), and therefore we can use the separated form (S38)-(S39) of P l det . Substituting the expression of E 2,x given by Eq. (S8) and Eq. (S13-2), the excitation integral (S38) reads with an explicit dependence on the linear polarization ψ of the excitation in the BFP of the condenser. Note that t p , t s , and θ t in the integrand of Eq. (S42) are all functions of θ i , given by Eq. (S5) and Snell's law respectively. We have omitted the constant factor (ε 0 ε m αE i ) 2 in front of the integral since we will take a ratio of this expression: In fact, ζ(ψ) is obtained by substituting Eq. (S42) into Eq. (S40). The detection integral (S39) is computed via Eq. (S29) and (S30), as collection occurs only in medium 2 The fourfold symmetry (with respect to the x = 0 and y = 0 planes) of P 2 has been exploited to reduce the azimuth integration range to ϕ ∈ [0, π/2]; these symmetry considerations are useful in order to reduce the time required for numerical integration. Note that the integration range of L det has changed from [θ obj , π] to [0, α obj ] because the integration now runs over θ 2 instead of θ -see Fig. S1b for the definition of these angles. η bf = η df is computed by substituting Eq. (S43) to the numerator of Eq. (S40), while the denominator is is P tot /P iso Θ=π/2 = p e,⊥ , which is given by Eq. (S34).
The expressions of ζ and η l we just derived were used for the quantitative analysis of the gold rods investigated in the paper.
Unpolarized illumination While η l is independent of the excitation for this choice of α, ζ(ψ) depends on the polarization state in the BFP of the condenser. Let us then compute ζ(ψ \) under unpolarized illumination. The incoherent p and s components of E exc excite two orthogonal sets of dipoles which yield an independent contribution to the total radiated power, so that the detection integral (S38) reads The two dipole moments in Eq. (S44) have amplitude p p = ε 0 ε m α|E 2,p,x | and p s = ε 0 ε m α|E 2,s,x |. Using the expressions (S8) for the fields (with amplitude E i / √ 2 instead of E i since each component carries half of the total incident intensity) one finds namely Eq. (S42) without the ψ-dependent term, which corresponds to interference between the p-and s-excited radiation. Since this term disappears for ψ = π/4 one has Although one has L l exc (π/4) = 1 2 L l exc (ψ) + L l exc (ψ + π/2) , we emphasize that in general and in particular for unpolarized illumination ζ(ψ \) = ζ(0) + ζ(π/2) /2. Homogeneous environment In the special case n = 1 representing a homogeneous optical environment, a closed-form expression of the scattering parameters can be derived. This scenario is of practical interest, as it corresponds to a NanO either covered by a fluid matching the refractive index of the substrate, or embedded in a solid matrix, or suspended within an optical trap. For n = 1 one has t p = t s = 1 and θ t = θ i , so that Eq. (S42) simplifies to note that the extrema in the final expression are inverted with respect to the integral. As for η l , the denominator is P tot /P iso n=1 = 1 since there is no interface now. The numerator is the integral of P 2 in a homogeneous environment, given by Eq. (S29) for n = 1 and θ 1 = θ 2 η l =L det (0, α obj ) = α obj 0 dθ 2 sin θ 2 2π 0 dϕ 3 8π cos 2 θ 2 cos 2 ϕ + sin 2 ϕ = 1 8 4 − 3 cos α obj − cos 3 α obj . (S49)

Uniaxial polarizability perpendicular to the interface
(α xx , α yy , α zz ) = (0, 0, α) describes well, for instance, the longitudinal mode of pillars or cones standing on a substrate. Reasoning along the same lines as in Sec. S.V B 1, one has p = ε 0 ε m α |E 2,z | and the excitation integral reads where we have exploited the axial symmetry with respect to z for the integration over ϕ i . Similarly, the detection integral is as P 2,s ∝ sin 2 Θ vanishes for Θ = 0 along with the ϕ dependence of P 2,p , see Eq. (S29) and Eq. (S30). ζ and η l are computed by substituting the integrals (S50) and (S51) into Eq. (S40), while the denominator of η l is P tot /P iso Θ=0 = p e, given by Eq. (S34). For axially-symmetric modes like the ones considered in this and the next two sections, the powers in the definition (S25) of the scattering parameters do not depend on ψ. Therefore, ζ and η l are the same for every exciting polarization as well as for unpolarized illumination. Indeed, the latter case is computed by averaging any two orthogonal polarization directions as done in Eq. (S46).
Homogeneous environment With considerations analogous to Sec. S.V B 1, the integrals (S50) and (S51) simplify to

Isotropic planar polarizability parallel to the interface
The plasmonic resonance of a thin disc lying on the substrate, or the transverse mode of a rod standing on it, is described well by (α xx , α yy , α zz ) = (α, α, 0). One then has p = ε 0 ε m α |E 2,x | 2 + |E 2,y | 2 and ζ is computed using Not surprisingly, this result differs only by a constant factor from L exc (ψ \) calculated for (α xx , α yy , α zz ) = (α, 0, 0), which is given by Eq. (S42) without the polarization-dependent term. η l on the other hand is the same found for (α xx , α yy , α zz ) = (α, 0, 0): In fact, for Θ = π/2 the azimuthal orientation of the dipole is irrelevant for η l due to the axial symmetry of the collection. Homogeneous environment In accordance with the considerations just made for the general case, ζ is calculated using Eq. (S48) without the polarization-dependent term and η l is given by Eq. (S49).

Isotropic polarizability
The isotropic optical response of a sphere is described by a scalar polarizability α. This case is inherently more complicated than the previous ones, inasmuch as the direction of the dipoles is not fixed: p = ε 0 ε m αE 2 (θ i , ϕ i ). Therefore P(θ, ϕ) depends on (θ i , ϕ i ) via (Θ, Φ), and the integral (S37) cannot be separated into excitation and detection parts as in the previous cases. As discussed above, when z is an axis of symmetry of the object, ζ and η l do not depend on the exciting polarization: Calculations in this case are simplified by assuming unpolarized illumination. Indeed, once the symmetry breaking introduced by a linear polarization is removed, the whole problem (excitation + NanO) has axial symmetry, and therefore we can limit ourselves to consider a single plane of incidence.
Choosing then ϕ i = 0 without loss of generality, E exc = E 2 given by Eq. (S8) reduces to We emphasize that for unpolarized illumination E 2,p and E 2,s are incoherent, and therefore E 2 is not their sum. They excite two incoherent dipoles of amplitude The integrand in Eq. (S37) is thus the sum of the powers radiated by the p-and s-polarized dipoles where P(θ, ϕ) has only a twofold symmetry with respect to ϕ because p p is not parallel to the interface. The real part appearing in the definition of Θ for the p dipole (hence Θ p ) fixes p p z for θ i ≥ θ 1,c , where only the imaginary part of θ t varies; we will discuss soon the reasons and the effects of this approximation. Eventually, ζ and η l are computed by substituting Eq. (S57) into Eq. (S25). Note that the explicit expressions (S29)-(S30) of P have to be evaluated only for the denominator of η l ; all other powers in Eq. (S25) are integrals over Ω d = 4π and can be simplified using Eq. (S31). These expressions of ζ and η l were used for the quantitative analysis of the gold and polystyrene spheres investigated in the paper.
To summarize, differently from the previous cases, for a sphere P no depends on the illumination. This is illustrated by Fig. S5, where the detection parameter is computed using a 1 • wide illumination cone. A discontinuity of η 1 • (θ i ) at θ i = θ 1,c separates two regimes: • For θ i < θ 1,c one has Θ p = π/2 − θ t ; as the p dipole progressively aligns to − z (where the objective is placed) less scattering is collected, see • For θ i ≥ θ 1,c one has Θ p = 0; having both p and s dipoles a fixed orientation, the trend is ruled by their relative amplitudes. Specifically, |t p | decreases more rapidly than |t s |, and therefore the relative contribution to P obj of the s dipole -which is favorably oriented with respect Ω obj -increases with θ i .
Polarization in the total internal reflection regime A peculiar property of the evanescent wave is the elliptical polarization of its p component. Indeed, in the plane of incidence, choosing ϕ i = 0 for simplicity, E t,p according to Eq. (S6p) is written as Therefore, E t,p describes an ellipse of eccentricity 1 − ( p t ) 2 x /( p t ) 2 z = having its major axis along z. In general 1 ≥ ≥ n 2 /n 1 for θ 1,c ≤ θ i ≤ π/2. At θ i = θ c the ellipse degenerates into a line ( = 1) perpendicular to the interface, while decreases for larger values of θ i up to grazing incidence, where the polarization is almost circular ( 1) for highly mismatched interfaces (n 1 n 2 ). In Eq. (S57) we have set p p z in the TIR regime, thereby neglecting the elliptic polarization of the exciting evanescent wave. This approximation cannot be easily avoided since the analytical expressions of P(θ, ϕ) presented in Sec. S.V A specifically refer to a dipole oscillating along a given direction. On the other hand, its effect on the calculated scattering parameters is expected to be only a few %; let us present here a qualitative argument to estimate the error in η l . The largest error occurs for a quasi-circular polarization state, which is generated at near-grazing incidence for a highly-mismatched interface, and can be decomposed into an x and a z component of almost equal magnitude. If any effect of coherence between the two components is neglected, they excite two incoherent dipoles.
Our approximation then consists in replacing the x dipole with a second z one. The effect of rotating by 90 • a dipole out of two can be assessed in Fig. S5, by comparing θ i = 0 where p p , p s ⊥ z, and a certain θ i value in the TIR range where p p z instead in our approximation. For a glass/air interface this value must be θ i = 51 • , where |t p | = |t s | and thus p p = p s (as also is for θ i = 0). The collected fraction of p-excited scattering is therefore underestimated by η l .022 as can be observed in Fig. S5; that is, 17 % of η l θ i =51 • . Now, η l results from a p contribution underestimated by the amount calculated above, and an s contribution which is not affected by our approximation. The s polarization contributes approximately twice as much because |t s | = 1.5 |t p | at near-grazing incidence and due to the more favorable orientation of the s dipole with respect to the collection. Therefore the overall underestimate of η l is slightly above 5 %. Furthermore, we emphasize that this argument refers to a worse-case scenario, whereas a significantly smaller error is expected in typical experimental conditions where 1. near-grazing incidence angles are hardly accessed ( θ df i = 59 • is the maximum value used in our high-NA measurements); 2. the glass/water and glass/air interfaces commonly used are only moderately mismatched, resulting in rather large values of .

S.VI. CHARACTERIZATION OF THE EXPERIMENTAL ILLUMINATION
The analytical calculations presented in the previous sections assumed that the condenser lens is a perfect aplanat with a homogeneous illumination in the BFP. In this section we discuss how to realize these conditions using a commercial microscope, the extent to which they are verified in our experimental set-up, and how to correct for eventual deviations.
We produced a homogeneous intensity over the BFP by inserting an engineered diffuser (Thorlabs, ED1-C20) in the illumination path before the field diaphragm. According to the manufacturer's specifications, the angular distribution of the transmitted intensity displays a top-hat pattern, namely approximately flat for |θ| < 11 • and steeply dropping off to zero outside this central range.
We measured the illumination power in the BFP, P bfp , by scanning a photodiode (Thorlabs, power meter PM100 + Si sensor S120B) along a diameter of the circular region corresponding to the condenser maximum aperture. To increase the spatial resolution, the sensor was covered by a dark mask with a circular opening of 2 mm diameter, corresponding to 0.07 NA. The results are reported in Fig. S6a for scans along and across the orientation of the image of the bulb filament, which is visible without the diffuser. The power is rather flat over most of the NA range of the condenser, and displays a drop larger than 10 % only for NA i > 1.2. The profile across the filament is less flat, suggesting that the filament image is not completely S26 erased by the diffuser. This brings about a slight dependence on ϕ i of the illumination on the sample plane, which we have neglected in our analytical description. To obtain an even flatter illumination in the BFP of the condenser, one can introduce a 4:3 beam expander (Nikon MEL59500 T-C High NA Common Condenser Lens Unit) in front of the condenser, at the expense of the total illumination power. In comparison, the standard diffuser from Nikon along with the 4:3 expander provides the profile shown in Fig. S6b, where P bfp reduces to half of the peak power at NA i 1.2, and to less than one third at NA i 1.34. The NA dependence of the power P i impinging on the sample is additionally affected by the dependence of the transmission of the condenser on NA i , which is expected to be lower at high NA due to stronger reflections from the internal optical interfaces at higher angles of incidence. We therefore we measured directly P i (NA i ) after the condenser. To do so, we pierced a circular aperture of 1.8 mm diameter in a dark mask placed in the BFP of the condenser, so that the illumination is a light needle of well defined NA i . The mask was mounted on a slider to allow a continuous variation of NA i along the direction of the bulb filament. Absolute NA reading was provided by a ruler (white stripe of paper in the top left corner of Fig. S7, left) calibrated with an interpolation of the known NAs of four different microscope objectives and the condenser itself. A half-ball lens attached to the front lens of the condenser as shown in Fig. S7 (left) transmits the light needle with the same efficiency for all NA i values, mimicking the n = 1.52 immersion medium. The transmitted P i is measured with a photodiode (Thorlabs, power meter PM100 + Si sensor S120B). The data shown in Fig. S7 (right) refer to the spectral band λ = (600 ± 20) nm; no relevant differences were observed across the whole investigated spectral range 400 nm to 750 nm. The overall trend resembles the one in Fig. S6, but the reduction of P i towards large NAs is somewhat stronger. For instance, at NA i = 1.20 ± 0.05, P i is reduced by 25 % with respect to the peak value, whereas in the BFP only about 10 % are seen. Note however the error bars, indicating that each data point incorporates some contributions from larger values of NA i too.
In order to minimize systematics when adopting our quantitative method, one should avoid the largest NA range for illumination -NA i > 1.2 for our set-up according to Fig. S7 (right). Furthermore, the angular dependence of the illumination can be characterized experimentally as just described, and taken into account in the analysis as we will now discuss. The main effect of a reduction of P i at large NAs is to reduce the DF illumination with respect to the intensity that would be produced by the idealized system. As a result ξ ≡ I bf i /I df i is underestimated and so is σ df sca , which is proportional to ξ. Now, I l i appearing in the definition of ξ is the intensity of the illumination traversing the sample plane, which the experimental cross-section are referenced to. In these supporting information we have called the same quantity I l ffp , and derived its expression in Eq. (S18)-(S20). We can introduce the angular dependence of P i in the integral to calculate a generalized ξ factor, namelỹ It is easy to see thatξ reduces to Eq. (3) of the paper for P i (θ i ) = 1. To compute the integrals in Eq. (S62) one can use an analytic form of P i (θ i ). We used the function which provides a good fit to the experimental data as shown in Fig. S7. For the illumination ranges NA bf i ∈ [0, 0.95] and NA df i ∈ [1.1, 1.3] -which we used for the polystyrene bead experiment -Eq. (S62) and (S63) yieldξ = 2.54. With respect to the uncorrected value ξ = 1.88, this brings about a 35 % increase in σ df sca , which is included in the quantitative measurements in Fig. 5 of the paper. This effect is suppressed when the largest illumination NAs are not included: For instance reducing the DF range to NA df i ∈ [1.1, 1.2] -which we used for the gold rod experiment -the correction amounts only to 17 %.
Another consequence of a reduced DF illumination is the underestimate of ζ ≡ P bf sca /P df sca calculated for the idealized illumination. A larger ζ results in a smaller σ bf sca according to Eq. (7a) of the paper; however, this scattering correction is negligible for absorption-dominated NanOs, such as the metal nanoparticles we investigated in this work. We have not included any corrections to ξ and ζ in the analysis of the gold spheres and rods in order to retain the simplicity of the analytical approach, and considering that the discrepancy with simulations is governed by other, larger systematics. We will show in future works how to increase the accuracy of our quantitative method with various improvements both in the experiment and in the analysis procedure, including corrections as discussed above.

S.VII. ADDITION OF DAMPING TO THE PERMITTIVITY
The lifetime of the localized surface plasmon resonances (LSPRs) sustained by metal NanOs is limited by both radiative and non-radiative processes, corresponding respectively to scattering and absorption of light. Non-radiative relaxation mechanisms can be further distinguished: Let us call intrinsic those mechanisms present in a defect-free infinite crystal, i.e. collisions of electrons with other electrons and the ionic lattice; all other channels, such as collisions of electrons with lattice defects and the specimen surface, are lumped together into the denomination of extrinsic. In numerical modeling, radiative damping is automatically accounted for by the electromagnetic solver, whereas non-radiative damping is included via the material properties, specifically via Im ε.
In this work ε(ω) of gold was taken from the experimental data sets by Johnson and Christy [S9], McPeak et al. [S10], and Olmon et al. [S11]. These were measured via spectroscopic ellipsometry on thin films deposited in vacuum, barring the last one, which characterized (among other samples) a single crystalline optically-thick layer. The measured ε(ω) are therefore expected to include some amount of extrinsic damping, due to sample imperfections (especially for Ref. [S9]) and surface contributions (which affect reflection measurements). Nonetheless, the simulated spectra consistently exhibit sharper LSPRs than our experiments, indicating the NanOs we measured have a larger extrinsic damping. We ascribe this to surface contributions (collisions of electrons with the surface become frequent for a particle of size comparable to the electron mean free path, which is about 40 nm in gold), to chemical interface damping (due to molecules chemically adsorbed onto the NanO surface, see Ref. [S12]) and, for the gold spheres only, to crystalline defects observed in transmission electron microscopy (TEM), see Fig. 3a of the paper. Additionally, also the radiative damping resulting from numerical simulations is inaccurate to some extent, since it refers to NanOs of idealized shape and homogeneous material.
In order to reproduce the experimental cross-section spectra, and hence perform a more reliable optical sizing, the material properties used in numerical models must represent as accurately as possible the measured NanOs. We therefore modified the permittivity as in a previous work by our group [S13]. First, the aforementioned experimental data sets were fitted with an analytical expression ε(ω) in the spectral range of interest (350 nm to 900 nm). Second, the expression thereby found was modified to include the additional damping exhibited by each individual NanO. To carry out the first step, we split the material response into the contributions of free and bound electrons (ε = ε f + ε b ) and provide the analytical expression of the two terms.
Free electrons The Drude model describes the free electron response as where γ 0 is the angular rate of electron relaxation, σ 0 is the real DC conductivity, ω p is the plasma frequency, and ε ∞ is the high frequency limit of ε(ω). The low energy part of the permittivity -say for λ > 700 nm -is fitted well by Eq. (S64) for the data sets by Johnson and Christy and McPeak et al., as will be shown below. On the other hand, a satisfactory agreement could not be reached with the data set by Olmon et al.. Now, in the standard Drude model, the frequency dependence of the damping and the charge carrier mass are neglected. In the last equality of Eq. (S64) ε f is decomposed into two poles; to include such dependencies, following Ref. [S14], we can split the pole at ω = −iγ 0 into two (or possibly more) distinct poles of the same form Note that this expression preserves causality as well as the asymptotic behavior of Eq. (S64) (ω −1 at low frequencies and ω −2 at high frequencies), and reduces to the original expression for β = 0 or β = 1. A good fit to the data set by Olmon et al. was obtained by adding a small contribution β ∼ 10 −4 of a pole corresponding to a much higher scattering rate, see the fitting parameters reported in Table S1 (top).
Bound electrons The analytical model developed by Rosei [S15] and Guerrisi et al. [S16] describes the response of bound electrons in gold. Following Eq. (9) of Ref. [S16], the absorption is the sum of the contributions of the main optical transitions t active in the spectral range of interest Specifically, X indicates the interband transition from the d band to the Fermi surface near the X point of the Brillouin zone, while L + 4 and L + 5+6 indicate the transitions near the L point from the 4d and 5d+6d bands, respectively. The transition strengths A t in Eq. (S66) are related to the square of the corresponding dipole transition matrix elements P t via  where i = 4 or 5 + 6. We fixed the values of the effective electron masses to the ones reported in Ref. [S13], and used P X , P L +

5+6
, and P L + 4 as fitting parameters. The joint density J t of each transition appearing in Eq. (S66) can be expressed as where the occupation probabilities of the p and d bands are given by the Fermi-Dirac distributions using the electron temperature T = 300 K and defining the origin of the energy scale at the Fermi level. The functions k t are modeling the inverse density of states close to the respective S31 points in the Brillouin zone, and the transition energies ω X 7 , ω X 6 , ω L + 5+6 , ω L + 4 , and ω L − 4 are left as free fitting parameters. The integration limits in Eq. (S68) near the X point are ω X = −20k b T / and Near the L point the minimum of the integration interval is where the lattice parameter of gold is a = 4.08 Å. Note that some typos have been corrected with respect to the corresponding Eq. (A22) in Ref. [S13]. The maximum of the integration interval is In order to reproduce in the model the broadening of the transition energies, we convolute Im ε b obtained via (S66) with a Lorentzian function where b is an fitting parameter scaling the broadening relative to the Drude broadening. time. When the modified Drude model (S64) including an additional pole is used, γ 0 in Eq. (S74) and Eq. (S75) is substituted with (1 − β)γ 1 + βγ 2 . After determining Im ε b with this model, Re ε b is determined via the Kramers-Krönig transform using the causality of the material response.
The analytical model just outlined with the parameters reported in Table S1 provides a good fit to all the considered data sets. This is shown in Fig. S8 for the data set by Johnson and Christy and McPeak et al., which we have used in this work for the optical sizing of the gold spheres and the gold rods, respectively. The two experimental permittivities feature a few noteworthy differences. Re ε b for McPeak et al. displays a clearer signature of the interband transitions below 400 nm and follows more closely the theoretical spectrum. It exhibits as well a stronger free electron contribution, Re ε f being about 10 % larger in absolute terms than Johnson and Christy, which entails a blue shift of the LSPR by about 5 nm for the gold spheres and 20 nm for the gold rods, see Fig. 3d and Fig. 4e of the paper. Lastly, Im ε by McPeak et al. is lower for wavelengths longer than 500 nm, resulting to sharper LSPRs, as will be shown in the following. All these observations point towards a better crystalline quality of the film measured by McPeak et al., and indeed their work focuses mainly on optimizing deposition procedures.
As discussed at the beginning of this section, metal NanOs often exhibit experimentally a larger damping than numerical simulations using bulk data sets. To better reproduce experiments, one can modify the analytical ε(ω) found by fitting. Specifically, we have increased the damping of the free electrons (to account for their additional collisions with the NanO surface and crystal defects) by replacing γ 0 in Eq. (S64) with where v f = 1.4 × 10 6 m/s is the Fermi velocity and g an adimensional factor which parametrizes the magnitude of the extrinsic damping of the NanO. When the modified Drude model (S65) is used, both γ 1 and γ 2 are increased in this manner. It is customary in literature to factor out a R −1 size dependence in Eq. (S76), because surface scattering is assumed to be dominant extrinsic damping mechanism in small NanOs, and for small spheres it is expected to scale as R −1 , both in a classical billiard-type picture and in the quantum box model proposed first by Kawabata and Kubo [S17]. In the spectral range of interest Eq. (S64) reduces to so that Re ε f is unaffected whereas Im ε f increases linearly with g/R, as observed in Fig. S8.
We are now able to assess the effect of the additional damping on our numerical simulations. In Fig. S9 the full width at half maximum (FWHM) of the main dipolar LSPR is plotted for the gold spheres and rods (longitudinal mode) investigated in this work as a function of g/R. For a fixed particle geometry (average deduced from electron microscopy), the FWHM increases linearly with the damping to a good approximation, except for Olmon et al., where the modified Drude model (S65) was used. The simulated FWHM can be compared to the experimental linewidth represented by the horizontal dashed lines, so that the intersection of these lines with the fit of the simulated data point individuates graphically the estimated value of g/R for each particle.
For spheres (Fig. S9a) we compared the three data sets considered in this work. The linewidth at g = 0 reflects the bulk damping γ 0 of each data set, and is smaller for McPeak et al. because of its smaller Im ε at λ lspr . Consequently, a larger value of g is required to reproduce the experimental linewidth when McPeak et al. is used. Values of g around unity are in line with previous experimental reports in literature [S13, S18, S19].
For rods (Fig. S9b), we considered only the data set by McPeak et al., which provides the best agreement with the experimental spectroscopy, see Fig. 4e of the paper. On the other hand, different environments are compared, with the interesting finding that g/R is systematically larger in air than in oil for the same rod. In fact, a dependence on the S34 particle surroundings cannot be explained appealing to surface damping alone, and suggests that other mechanisms such as chemical interface damping play as well a role in gold rods, as recently reported by other groups, see Ref. [S12]. Notably, the average g/R in oil is significantly smaller for rods than for spheres: 30 µm −1 versus 140 µm −1 (it seems more meaningful to refer to g/R rather than g when comparing different particle shapes). When correlated to the TEM images reported in the paper, which display a high crystalline quality of rods (Fig. 4a) but not of spheres (Fig. 3a), this suggests that -in contrast to what is typically assumed in literature -for spheres of the investigated size a major portion of the extrinsic damping is due to crystal defects rather than surface scattering. In fact, the values of g we have found are larger than the theoretical estimate g s = 0.73 for surface damping alone calculated via the quantum-box model in Ref. [S17]. and λ lspr = 695 nm in oil), placed on a n = 1.52 glass substrate.