Although photothermal imaging was originally designed to detect individual molecules that do not emit or small nanoparticles that do not scatter, the technique is now being applied to image and spectroscopically characterize larger and more sophisticated nanoparticle structures that scatter light strongly. Extending photothermal measurements into this regime, however, requires revisiting fundamental assumptions made in the interpretation of the signal. Herein, we present a theoretical analysis of the wavelength-resolved photothermal image and its extension to the large particle scattering regime, where we find the photothermal signal to inherit a nonlinear dependence upon pump intensity, together with a contraction of the full-width-at-half-maximum of its point spread function. We further analyze theoretically the extent to which photothermal spectra can be interpreted as an absorption spectrum measure, with deviations between the two becoming more prominent with increasing pump intensities. Companion experiments on individual 10, 20, and 100 nm radius gold nanoparticles evidence the predicted nonlinear pump power dependence and image contraction, verifying the theory and demonstrating new aspects of photothermal imaging relevant to a broader class of targets.

## INTRODUCTION

The challenge of optically detecting individual nano-objects that do not scatter or emit light has inspired the development of a variety of detection techniques capable of distinguishing absorption independently from scattering and emission.^{1,2} One such approach that has been applied to measure small, non-scattering, dielectric, metallic nanoparticles is photothermal imaging.^{3–5} This technique relies on the nanoparticle’s resonant absorption of light and associated heat power dissipated into a locally modified temperature and refractive index gradient in the surrounding medium—a so-called “thermal lens”—through which the nanoparticle can be detected. The thermal lens is large, so that its scattering of a second, ideally non-resonant, (probe) beam of light produces the photothermal signal when interfered against itself. Lock-in detection to the modulated amplitude of the pump provides a further means to isolate this interference component from the total signal that reaches the detector.

More recently, however, photothermal techniques have been used to detect individual plasmonic metal nanoparticles and nanoparticle assemblies that are large enough to both absorb and scatter light.^{6–10} By varying the pump wavelength, photothermal absorption spectra have also been measured from the same nanostructures, independent from scattering.^{11–14} In this size regime, fundamental assumptions made in the original photothermal models are not always valid, and interpreting the photothermal signal has been found to require careful reconsideration.^{6,15} In parallel, other interferometric techniques similar to photothermal imaging have been pioneered to study larger, scattering nanoparticle systems. Two such approaches—coherent brightfield microscopy (COBRI) and interferometric scattering microscopy (iSCAT)—are closely related to photothermal imaging, but do not involve optically heating the target specimen.^{16,17} In both cases, the signal originates from the interference between the field scattered by the (room temperature) target and a reference field without the need for lock-in detection. Often, the reference field is either the transmitted (COBRI) or reflected (iSCAT) probe beam.

While the interferometric nature of COBRI and iSCAT allow for small, weakly scattering particles to be detected, they are fundamentally measures of extinction and, therefore, do not separate absorption from scattering in particles large enough to appreciably scatter. In contrast, photothermal measurements isolate the pure absorption response. However, interpretation of the photothermal signal produced by nano-objects large enough to scatter requires careful consideration of the additional effects of target scattering beyond those induced by the thermal lens.^{5,18–20} Here, we incorporate these effects within a dipole model that includes effects of the heat diffusion dynamics, co-focusing of pump and probe beams, pump modulation, and lock-in detection. Importantly, for large nanoparticles, we find that a new scattering-induced component influences the absorption character of the photothermal signal, and this scattering contribution grows nonlinearly with pump power at fixed pump and probe wavelengths.

As a companion to the presented photothermal scattering model, we acquire experimental data from individual spherical gold nanoparticles, ranging in size from 10 to 100 nm in radius. Despite the small thermo-optic responses of the target and medium, we observe a nonlinearity in the photothermal signal with increasing pump power for the larger nanoparticles. Based on the model presented, we trace the origins of this nonlinearity to a scattering contribution to the photothermal signal that has been disregarded in the small-particle limit, but that becomes important for larger scatterers at higher pump power or smaller pump beam waist.

Below, we present the generalized photothermal model and analyze the signal in the limit of both small and large nanoparticle targets. Furthermore, analysis and exploration are made through a numerical investigation of the pump-wavelength-resolved photothermal spectra and pump-power-dependent photothermal images of plasmonic nanoparticle absorbers/scatterers of varying size. Finally, we present experimental measurements of the photothermal signal acquired from individual gold nanoparticles as a function of pump power and interpret these data from the perspective of the model presented.

## GENERALIZED PHOTOTHERMAL SIGNAL

In photothermal imaging, the signal that reaches the detector is produced by the superposition of a transmitted/reflected probe beam (or reference field) **E**_{pr} and a scattered probe beam **E**_{sca}. The pump field that heats the system **E**_{pu} is removed by spectral filtering before reaching the detector, resulting in the signal

at the probe wavelength *λ*, where *c* is the speed of light in vacuum, and *n*_{b} is the room temperature refractive index of the background medium. Due to the scattering inefficiency of small nano-objects, the |**E**_{sca}|^{2} term can be safely neglected relative to the term linear in **E**_{sca} for the vast majority of targets that have been investigated in the literature using photothermal techniques.^{1} However, for larger targets, such as for plasmonic nanoparticle antennas, the |**E**_{sca}|^{2} term can become important and should be retained. In this section we expose the effects of **E**_{sca} upon the photothermal signal.

Lock-in detection is implemented in photothermal measurements to retrieve the interference signal at the modulation frequency. Here, we will consider the amplitude modulation of the pump beam at frequency Ω, a rate that is typically of the order of kHz to MHz. The outcome of the measurement depends on the order of magnitude of Ω,^{21} and this work specializes in the low modulation regime, where Ω = 100 kHz. The detector locks in only to those signals which vary in-phase and in quadrature with this oscillation frequency, thus projecting out the contribution from the transmitted (and unmodulated) probe, producing the following magnitude |Φ| and phase Ψ,^{19}

of the complex signal Φ = |Φ|*e*^{iΨ}, which are determined from

Lock-in detection, therefore, isolates a part of Φ that stems from the interference between the transmitted/reflected and scattered probes $(\Phi int\u221dRe[Epr\u22c5Esca*])$, but also selects a contribution originating from the photothermally-induced probe scattering (Φ_{sca} ∝ |**E**_{sca}|^{2}). $\Phi intsincos$ is well understood to result in a photothermal signal that is a measure of absorption.^{11} However, the additional effects that $\Phi scasincos$ imparts upon the signal are less well understood, as they would only arise for targets large enough to scatter the probe.

The heating and probe lasers used in our analysis are modeled as focused Gaussian beams. The electric field of a Gaussian beam, which propagates in the +*z* direction, is polarized in the *x* direction, and is focused at the position *z*_{f} and is well approximated by

where *w*_{0} is the beam waist at the focus, $w(z)=w01+(z\u2212zf)2/zR2$ is the beam radius, $R(z)=(z\u2212zf)[1+zR2/(z\u2212zf)2]$ is the radius of curvature of the beam, *ψ*(*z*) = tan^{−1}[(*z* − *z*_{f})/*z*_{R}] is the Gouy phase, and $zR=\pi w02nb/\lambda $ is the Rayleigh length at frequency *ω* = 2*πn*_{b}*c*/*λ*.

The reference field is determined by the transmitted field of the incident probe beam evaluated at the detector position **x**_{d}, which is assumed to be located on the optical axis in the far field. The detected transmitted/reflected probe field is related to the incident probe field via scaling by the Fresnel transmission/reflection coefficients specific to the system and collection geometry. In the following, we set the transmission coefficient to unity. **E**_{sca}(**x**, *t*) = **G**(**x**, **x**_{np}) · *α*_{pt}(*t*)**E**_{G}(**x**_{np}, *t*) represents the scattered field of a dipole, where **G**(**x**, **x**_{np}) = [*k*^{2}**I** + ∇∇]exp(*ik*|**x** − **x**_{np}|)/|**x** − **x**_{np}| is the dipole relay tensor, **x**_{np} is the position of the nanoparticle target, assumed to be at the origin (**x**_{np} = **0**), and *α*_{pt}(*t*) is the time-dependent photothermal polarizability of the nanoparticle’s induced dipole. When evaluated at the detector and nanoparticle, the Gaussian beam is well approximated by

where *w*(*z*) ≈ *w*_{0}(*z*/*z*_{R}) and *e*^{−iψ(z)} ≈ −*i* at the detector (*z* → +*∞*). Using these limiting forms, the probe and scattered fields become

at the detector position $xd=rn\u0302$, where **G**(**x**, **x**′) takes on its far-field form and where *k* = 2*πn*_{b}/*λ* and *z*_{pr} denotes the wavenumber at the focal point of the probe.

As a result, the interference and scattering contributions to the lock-in integral in Eq. (3) are

evaluated on the optical axis (*θ* = *ϕ* = 0°) with *z* ∼ *r* at the detector. Already, a superficial analogy between these expressions and the extinction (*σ*_{ext}(*ω*) = 4*π*(*ω*/*c*)Im[*α*(*ω*)] → *σ*_{abs}(*ω*) in the small particle limit) and scattering [*σ*_{sca}(*ω*) = (8*π*/3)(*ω*/*c*)^{4}|*α*(*ω*)|^{2}] cross sections of a dipole of polarizability *α* are evident. However, there are important differences that will be discussed below, most notably the fact that Eq. (7) involves the photothermal polarizability *α*_{pt} explicitly and not the dipole Mie polarizability *α*. *α*_{pt} is a function of the absorption cross section *σ*_{abs} and pump intensity *I*_{pu}, both of which are functions of the pump wavelength *λ*_{pu} and not the probe wavelength *λ*. Up to this point, we have made no assumptions about the type of material giving rise to *α*_{pt}. Therefore, Eq. (7) is the generalized expression within the dipole limit for the interference and scattering contributions to the lock-in integral in Eq. (3).

### Polarizability model

From these primitive functions, lock-in detection extracts the measured signal, but first a model of the photothermal polarizability *α*_{pt} must be adopted. We choose *α*_{pt} to describe the time- and temperature-dependent response of a spherical target embedded in a background medium of constant refractive index. Note that while a substrate is not included in this model, it may be accounted for by using a more sophisticated polarizability model or through numerical simulation. Specifically, we focus on two extreme cases: (1) the photothermal signal in the thermal lens limit, where the polarizability accounts only for the scattering from the heated medium surrounding a point absorber, and (2) a generalized core–shell model, where the core now includes the radiation-damped response of a nanoparticle scatterer, in addition to the scattering induced by its thermal lens shell.

In either case, since the thermo-optic coefficients of the target and medium are small (*dn*/*dT* ∼ 10^{−4} K^{−1}), the temperature dependence of the target may be approximated at first order by

where *α*_{pt}(*T*_{0}) ≡ *α*(*T*_{0}) is the room temperature Mie polarizability. See the supplementary material for a discussion on the appropriateness of this approximation. We calculate the temperature *T*(**x**, *t*) from the time-dependent heat diffusion equation, assuming the modulated heat power *P*_{abs}(*t*) = *σ*_{abs}(*λ*_{pu})*I*_{pu}(1 + cos Ω*t*)/2 absorbed by a point absorber in the small particle limit or a spherical absorber in the large particle limit. Given that *T*(**x**, *t*) is a function of *I*_{pu}, the photothermal polarizability *α*_{pt} ≡ *α*_{pt}(*I*_{pu}) is, therefore, a nonlinear response function, as it depends upon the pump intensity *I*_{pu} (or pump power $Ppu=Ipu\pi wpu2/2$, where *w*_{pu} is the pump waist evaluated at the nanoparticle) through its temperature dependence. It also encodes the geometry-specific resonant responses of the target through the absorption cross section *σ*_{abs}(*λ*_{pu}), which itself is a function of the linear Mie polarizability *α*(*λ*_{pu}) of the target’s induced dipole moment at room temperature *T*_{0}.

#### Small particle limit

Small (*ka* ≪ 1) metal nanoparticles do not scatter. Instead, they absorb light and dissipate optical heat power into a temperature rise of the surrounding medium. In this limit, it is appropriate to model the optical response as a thermal lens or a large sphere of heated background, with volume *V*_{th,} with the Clausius–Mossotti polarizability

where *ɛ*(*T*(*t*)) is the temperature-dependent dielectric function of the background medium shell, and *ɛ*_{b} is the room-temperature dielectric function of the remaining bulk. Since *ɛ* = *n*^{2} and $\epsilon (T=T0)=\epsilon b=nb2$, $d\alpha /dn|T=T0=(Vth/2\pi )nb$, and *α*_{pt}(*T*_{0}) = 0. Thus,

to lowest order, where the average temperature $T\u0304(t)$ is defined by integrating the temperature rise *T*(**x**, *t*) − *T*_{0} over the volume $Vth=(4/3)\pi rth3$, where $rth=2\kappa /cp\Omega $ is the thermal radius with background medium thermal conductivity *κ* and specific heat capacity *c*_{p}. Specifically,

Note that since the $T\u0304$ depends upon the pump intensity *I*_{pu} and absorption cross section *σ*_{abs}, the photothermal polarizability *α*_{pt} ≡ *α*_{pt}(*σ*_{abs}, *I*_{pu}) is a nonlinear response function that also, in principle, encodes the resonant excitations of the absorbing target through *σ*_{abs}.

when evaluated along the optical axis, where $sin(tan\u22121(x))=x/1+x2$, $I1=[e\u22122cos(1)\u2212sin(1)]/4e$, $I2=[2sin(1)\u2212cos(1)]/4e$, and *λ* = 2*πn*_{b}/*k* is the probe wavelength. This result, which is limited to the thermal lens approximation, is consistent with other photothermal models from the literature, such as Ref. 19, and is identical to what was derived in Ref. 22 after inserting the explicit Gaussian beam form in Eq. (4) into the equations presented therein.

#### Large particle limit

Outside of the small particle limit, larger nanoparticle targets scatter electromagnetic radiation when *ka* ≳ 1. Thus, it is to be expected that their photothermally induced scattered field may contribute appreciably to the total signal, as described by Eqs. (3) and (7). To investigate this size-dependent effect, we adopt an approximate core–shell polarizability model with a large nanoparticle core and a thermal lens of radius *r*_{th} representing the shell.^{23}

Specifically, the core–shell photothermal polarizability^{24} is

where $\epsilon 1\u2261\epsilon 1(T\u03041(t))$ is the dielectric function of the core at $T\u03041(t)$, $\epsilon 2\u2261\epsilon 2(T\u03042(t))$ is the dielectric function of the background at $T\u03042(t)$, and $qi=1/3\u2212(1/3)xi2\u2212i(2/9)xi3$ (*x*_{i} = 2*πr*_{i}/*λ*_{pu}, *r*_{1} = *a*, *r*_{2} = *r*_{th}) is the depolarization factor introduced to account for retardation effects.^{24,25} In the small core radius limit (*x*_{1} = *ka* ≪ 1), Eq. (13) reduces to the Clausius–Mossotti polarizability in Eq. (9). Figure 1 displays a schematic of the core–shell polarizability model and compares it to the true physical system in space (upper) and time (lower). A more complete description of the latter would entail dividing the core and shell into multiple layers to better characterize the full gradient profile of the thermal lens. Such an approach, however, while more accurate, would necessarily involve complicated numerical simulations and would obscure the qualitative physical interpretation of the photothermal signal provided by the core–shell model adopted.

Using the chain rule to calculate the derivatives of *α*_{pt} with respect to temperature, Eq. (8) now becomes

where the average core and shell temperatures $T\u03041(t)=T(|x|=a,t)$ and $T\u03042(t)=(4\pi /Vth)\u222barthT(x,t)r2\u2061dr$ are calculated according to a spherical absorber of radius *a*, where

for *r* ≥ *a*. By substituting these average core–shell temperatures into Eq. (14), the lock-in integration of Eq. (3) results in the generalized photothermal signal components

evaluated along the optical axis with total signal $\Phi sincos=\Phi intsincos+\Phi scasincos$ and

where the $S,C$ terms are defined in the supplementary material.

Thus, we find that introduction of a large nanoparticle core to the polarizability model enhances the photothermally induced scattering contribution $\Phi scasincos$ to the signal that carries a nonlinear dependence upon the pump power *P*_{pu}. Certainly, this term is suppressed by a factor proportional to the square of the thermo-optic coefficients of the core and shell, but it is also enhanced by the resonant responses of the nanoparticle core encoded in the absorption cross section *σ*_{abs}, as well as through the resonances of *∂α*_{pt}/*∂n*_{1} and *∂α*_{pt}/*∂n*_{2} at the pump wavelength in Eq. (18). These resonances of *σ*_{abs} and *dα*_{pt}/*dT* provide additional enhancements of this scattering-like photothermal signal contribution beyond its quadratic pump power dependence.In other words, the importance of $\Phi scasincos$ depends critically upon the properties of the core polarizability, which can be influenced by nanoparticle size and composition, as well as through resonance effects that are all well understood for plasmonic systems. In the small particle limit (*ka* ≪ 1), Φ_{int} in Eq. (16) reduces to that of Eq. (12), and Φ_{sca} in Eq. (17) becomes negligible at low pump powers. Note that the small particle limit can be achieved equivalently by either decreasing the nanoparticle size at a fixed probe wavelength or by increasing the probe wavelength at fixed nanoparticle size. Also, note that resonant scattering of the probe would be filtered out by lock-in detection and, thus, would not contribute to the photothermal signal Φ.

### Photothermal spectra

As reported in the literature^{1} and shown explicitly in Eq. (12) above, the photothermal signal is a measure of absorption in the thermal lensing limit appropriate for small absorbing nano-objects. However, for larger particles, such as for plasmonic nanoparticle antennas, non-negligible scattering contributions arise and call into question some of the approximations made in the thermal lensing limit. Those approximations can change the resulting interpretation.

Through numerical solution of Eqs. (16) and (17), we now investigate the evolution of the photothermal signal when these consequences begin to take effect for gold nanospheres of varying radii (*a* = 10, 20, 100 nm) embedded in a glycerol medium. Figure 2 shows pump wavelength-dependent photothermal spectra, calculated with a fixed probe wavelength of 785 nm, as a function of pump power. In each panel, the black trace is the photothermal signal |Φ| post lock-in detection, together with the sine and cosine parts of its interference (blue trace) and scattering (red trace) components, while the underlying gray and purple shaded spectra correspond to Mie absorption and scattering cross sections (*σ*_{abs,sca}), respectively, including terms up to *ℓ* = 10. The scattering cross section for *a* = 100 nm is reduced by a factor of two to display all traces within the same viewing window. In addition, the photothermal signal and its components are scaled by the area $\pi wpu2/2$ of the pump beam waist, as was done in our prior experiment.^{11}

The upper, middle, and lower rows of Fig. 2(a) correspond to pump powers of 100, 200, and 300 *μ*W, respectively, using a pump beam waist ranging from 216 to 456 nm, depending on the pump wavelength. For the *a* = 10 and 20 nm nanoparticles, the interference contribution (blue) dominates the signal, and Φ closely tracks the absorption cross section (black line shape compared to shaded gray line shape), as can be seen most clearly in Fig. 2(b). These numerical results indicate that photothermal imaging may be used as a proxy for an absorption cross section measurement for small particles at these pump powers, an understanding that has been well established in the literature for lower pump powers.^{6} For the *a* = 100 nm nanoparticle, both interference (blue) and scattering (red) contributions contribute nearly equally to the signal, and the photothermal spectrum only approximately tracks the absorption cross section line shape, with deviations occurring at longer wavelengths. However, as the pump power decreases, the photothermal spectrum of the 100 nm radius particles approaches the line shape of the absorption cross section.

To further investigate the pump power dependence demonstrated in Fig. 2, Fig. 3 displays the evolution of the photothermal signal with varying pump power for the *a* = 10, 20, and 100 nm gold nanoparticles described previously. The pump laser wavelength is fixed at 532 nm, and the probe laser wavelength is 785 nm. For 10 nm (green trace) and 20 nm (red trace) particles, the photothermal signal depends approximately linearly upon pump power. However, for the 100 nm particles (blue trace), a pronounced nonlinearity in pump power dependence is clearly evident. This nonlinear behavior exhibited in the second term of the scattering-like contribution $\Phi scasincos$ in Eq. (17) and calculated in Fig. 3 for realistic experimental parameters (see below) is surprising, given the quadratic dependence upon the small ($\u223c10\u22124$ K^{−1}) thermo-optic coefficients of the nanoparticle and surrounding medium. However, this quadratic dependence clearly becomes relevant in the photothermal response of large particles at higher pump powers.

## EXPERIMENT

To test the predictions of our theoretical model of photothermal imaging and spectroscopy, we imaged samples of gold nanoparticles with nominal radii of 10, 20, and 100 nm with our confocal, photothermal microscope. Figure 4 summarizes their optical extinction and size distribution. The nanoparticle samples were spincast onto separate glass cover slips. Adhesive rubber spacers were used to make wells, which were filled with glycerol and sealed by placing an additional glass cover slip on top, forming a glassgold nanoparticle–glycerol-glass sandwich. The samples were imaged on our photothermal microscope described previously.^{10,11} Briefly, the intensity modulated 532 nm pump and unmodulated 785 nm probe lasers (Coherent, Obis) were collinearly focused onto the sample using a 63×/1.4 numerical aperture (NA) objective (Zeiss, Plan-Apochromat). The pump and probe beam diameters are 260 and 650 nm at full-width-at-half-maximum (FWHM), respectively, measured via a knife edge method. The transmitted probe beam was collected with a 40×/0.6 NA objective (Zeiss, Plan-Neofluoar) and focused onto a Si photodiode (FEMTO, HCA-S 200M-Si), and the photothermally modulated signal was detected with a lock-in amplifier (SRS, SR844) at the pump beam modulation frequency of 100 kHz.

Photothermal images of the 10, 20, and 100 nm radii gold nanoparticles were obtained using a nanopositioning piezo stage (Physik Instrumente, PI-517.2CL) to scan the sample through the focus of the pump and probe laser beams, generating a photothermal image of the gold nanoparticles, as demonstrated in Figs. 5(a)–5(c). The photothermal intensity is measured as the ratio of the maximum lock-in amplitude of each particle to the probe power measured before the microscope, with units of mV/mW. We use this definition of photothermal intensity for the experiments, normalizing to the probe power, because the probe power had to be changed between the different particle sizes to keep the modulated signal within the operating limits of the lock-in amplifier. To ensure that these different probe powers did not introduce any unexpected behavior, we confirmed that the probe power dependence is linear for all probe powers tested (Fig. S1), as is expected from previous literature^{5} and theory. Figure 5(d) shows that the photothermal intensity is clearly size dependent, with the 20 nm radius gold nanoparticles having a maximum signal that is ∼8× that of the 10 nm radius gold nanoparticles at the same pump power. This change in photothermal intensity is consistent with previous experimental results,^{3,4} as well as the theory of the generalized photothermal signal presented above, where the photothermal signal scales with the cube of the nanoparticle radius in the small particle limit, Eq. (12). However, the photothermal intensity increases by only 3.6× [Fig. 5(d)] when increasing the size from 20 to 100 nm radius for a similar pump power, due to additional resonance effects of the 100 nm radius gold nanoparticle at the probe wavelength of 785 nm [Fig. 4(c)], similar to the effect seen in Fig. 2, when comparing the photothermal signal of different gold nanoparticle sizes at a constant pump power and wavelength.

We further investigated the size and pump power dependence of the photothermal signal by recording photothermal images of the 10, 20, and 100 nm radius gold nanoparticles under various pump powers between 20 and 310 *μ*W measured at the sample plane (50–750 kW cm^{−2}). To ensure that the observed trends were robust and that there was no photothermal damage to our nanoparticles under the highest pump powers, we varied the pump powers in random order and repeated each measurement three times. We further verified that no damage occurred by comparing the correlated SEM images of 100 nm radius gold nanoparticles that were either exposed or unexposed to our highest pump power of 310 *μ*W. The size distributions are shown in Fig. S2, and a one-way analysis of variance revealed that there was no statistically significant difference between the exposed and unexposed particle sizes [F(1, 266) = 0.25, p = 0.62].

We first compare the effect of the pump power on the size of the image point spread function (PSF), Fig. 6. We fit the photothermal PSFs with 2D Gaussians to determine the FWHM of the gold nanoparticles as a function of pump power. Example linesections and single particle images are presented in Figs. 6(a)–6(f) for low power (60 *μ*W, 150 kW cm^{−2}) and high power (210 *μ*W, 525 kW cm^{−2}) excitations. The entire power range is summarized in Fig. 6(g). At low power, as the particle increases in size, the FWHM increases due to a convolution of the gold nanoparticle size and the pump beam width (260 nm FWHM). This effect accounts for a broadening of ∼3 and 70 nm for the 20 and 100 nm radius nanoparticles, respectively. Additional broadening of the PSF could be due to changes in the probe beam focus position, which was changed commensurately with the pump beam to maximize the intensity of the photothermal signal for each particle size. For the 10 and 20 nm radius gold nanoparticles, we observe a power independent photothermal PSF FWHM [Figs. 6(a)–6(d) and 6(g)], consistent with theoretical analysis in the small particle limit, Eqs. (12) and (S8). However, in the large particle limit, the photothermal PSF has contributions from both the Φ_{int} and Φ_{sca} terms, leading to a narrowing of the photothermal PSF at higher pump powers [Figs. 6(f) and 6(g)], as described by Eqs. (16) and (S2). Note that the intensity spike in the PSF in Fig. 6(f) is random; some particles exhibit a spike, while some do not.

We next compare the effect of the pump power on the photothermal intensities for the gold nanoparticles extracted from the photothermal images, Fig. 7. The photothermal intensity is defined the same way as in Fig. 5. For the small 10 and 20 nm radius gold nanoparticles, we observe the expected linear pump power relationship as described in the small particle limit, where Φ_{int} is linearly dependent on *P*_{pu} and Φ_{sca} ≪ Φ_{int}.^{3–5} However, for the large 100 nm radius gold nanoparticles, we observe a linear plus quadratic pump power dependence due to the $Ppu2$ term in Φ_{sca}. While the magnitudes of the pump power dependencies are different between theory and experiment, possibly due to the influence of the substrate in the experiment, which is not modeled in the analytical theory, the qualitative trends remain. We note that this nonlinear trend could also be observed by locking into 2 Ω modulation to directly measure the quadratic term in Φ_{sca}, though that is not done in this work. Thus, the major trends expected from the generalized photothermal theory presented above are experimentally supported.

## CONCLUSION

In this article, we assess the scattering effects of large nanoparticles upon the confocal photothermal experiment, using an effective dipole model and companion photothermal spectroscopy and imaging experiments of individual plasmonic gold nanoparticles. For small nanoparticles (*ka* ≪ 1), the pump wavelength-resolved photothermal signal is directly proportional to the absorption cross section, as is well understood in the literature. However, for larger nanoparticles (*ka* ∼ 1), we find the photothermal spectrum to deviate from the absorption spectrum through the addition of a new scattering term Φ_{sca} that depends nonlinearly upon the pump intensity. While Φ_{sca} resembles the scattering cross section *σ*_{sca} of a dipole, i.e., both carry the same *k*^{4} and |*α*|^{2} dependence, the two signals are distinctly different in their underlying polarizabilities: Φ_{sca} is proportional to the photothermal polarizability $\alpha pt(t)=\alpha (T0)+(d\alpha pt/dn)(dn/dT)T0T\u0304(t)$, while *σ*_{sca} depends upon the Mie polarizability *α*(*T*_{0}). As a result of this photothermal scattering term, the photothermal spectrum of larger nanoparticles at higher pump intensities is no longer directly proportional to the absorption cross section. These predictions, which distinguish small and large nanoparticle limits, are evaluated by companion experiments of pump laser power-dependent photothermal images of spherical gold nanoparticles from $\u223c10\u2212100$ nm radius and $\u223c20\u2212300\mu $W pump power (50–750 kW cm^{−2}), where the nonlinearity begins to dominate at pump intensities in excess of 500 kW cm^{−2} for the $\u223c100$ nm radius particles. Excellent agreement between experiment and prediction is shown, highlighting the importance of photothermally induced scattering upon the interpretation of the photothermal signal from larger nanoparticles at higher pump intensities.

## SUPPLEMENTARY MATERIAL

See the supplementary material for definition of the integration terms $S$, and $C$, the second order expansion of the photothermal polarizability, and further analysis of the photothermal image.

## ACKNOWLEDGMENTS

This work was supported by the U.S. National Science Foundation under Grant Nos. CHE-1727092 and CHE-2118333 (D.J.M.), CHE-1727122 and CHE-2118420 (S.L.), and CHE-1728340 and CHE-2118389 (K.A.W.). S.L. also acknowledges support from the Charles W. Duncan, Jr.-Welch Chair in Chemistry (C-0002). The experimental work was conducted in part using resources of the Shared Equipment Authority at Rice University.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

C.A.W. and S.A.L. contributed equally to this work.

**Claire A. West**: Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). **Stephen A. Lee**: Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). **Jesse Shooter**: Investigation (supporting). **Emily K. Searles**: Investigation (supporting). **Harrison J. Goldwyn**: Investigation (supporting). **Katherine A. Willets**: Conceptualization (supporting). **Stephan Link**: Conceptualization (supporting). **David J. Masiello**: Conceptualization (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.