1. Introduction

Collective oscillation of the conduction electrons in AuNPs by incident electromagnetic field supports a tunable localized surface plasmon resonance (LSPR). This phenomenon that leads to focus and enhance the incident field in AuNPs [1,2], opens a route to numerous practical absorption- and scattering-based applications such as SERS, chemical sensing, drug delivery, photothermal cancer therapy, and new photonic devices. If the field enhancement (FE) is strong enough, it leads to an optical breakdown and generation of electronic plasma around the nanostructure directly in the surrounding materials. When two AuNPs are coupled as a dimer, new plasmonic modes are formed giving rise to a very large FE in its gap. This significant FE is due to a plasmon coupling of the dipolar modes of the individual particle that causes strong induced charge densities of opposite sign at both sides of the gap. The near-field coupling is being tunable by the gap distance and the size of the AuNPs [3–7]. For an ultrashort-pulsed laser interaction, the strong electric FE in the gap distance and around the AuNP dimers attains the required intensity threshold of the incident pulse to achieve optical breakdown. This rather weak pulse leads to a nonlinear absorption of the laser energy and plasma nucleation in the vicinity of the AuNPs. The idea of irradiating the AuNPs by an off-LSPR laser field allows reducing the thermal energy absorption by the AuNPs and inhibits their fragmentation [8]. In this work for the first time, to the best of our knowledge, we study the interaction between off-LSPR ultrashort (less than 300 fs) pulsed laser and AuNP dimers in aqueous medium. We developed a theory to calculate the deposited energy inside the AuNPs and the surrounding induced plasma during the pulse time. We also show the impact of geometrical parameters such as the gap distance and AuNP diameter. Two linear and circular polarizations are considered for the incident laser field. Although we have provided the appendix to show the main equations on modeling involving electro, thermo and plasma dynamics, however, an extensive discussion on the modeling, including equations and details on the parameters used, is presented in another related publication [8–11]. The mathematical model was solved using a finite-element method, using the COMSOL software (www.comsol.com). To model the infinite simulation region with a 3D finite-geometry model, we use anisotropic perfectly matched layers (PML) to avoid the reflective fields from the boundary of the computational domain.

2. Results and discussions

2.1 Field enhancement, absorption, scattering and extinction

In this work, we focus on an AuNP dimer consisting of two identical nanoparticles with different diameters of 60 nm to 120 nm and gap distances between 4 to 16 nm. These AuNPs are characterized by an experimental local dielectric function taken from Johnson and Christy [12]. The environment medium is water which makes the obtained results more practical for biological applications [13]. This nanostructure is irradiated by a linearly and circularly polarized plane wave with a center wavelength λ = 800 nm propagating along the + z axis. The laser has a Gaussian form with ultrashort pulse width of 45-300 fs. This part of the paper is to show the ability of AuNP dimer to create an extremely high field confinement which is a key point to reach the optical breakdown through non-linear absorption of the laser energy occurring in the vicinity of the particle. This process leads to the generation of electronic plasma mostly in the gap distance and around the nanostructure, directly in the water. Electromagnetic interaction between the laser irradiation and the nanostructure-water system is calculated using the Helmohltz equation (See Eqs. (1) - (4) in appendix) [14]. Figure 1(a) and 1(b) show the electric FE (E/E₀) distribution cross sections along the 100 nm AuNP dimer with gap distances of d = 10 nm and a single 100 nm AuNP. The laser light polarization is aligned with the dimer’s longitudinal axis (parallel to particle-pair long axis). Here, E is the magnitude of the electric near field and E₀ is the magnitude of the incident electric field. In this figures the inset shows the cross-section of 2D FE distribution where the color legend shows the magnitude of the FE. A same scale is used for FE to have clear comparison between an AuNP dimer and a monomer. As shown in Fig. 1 for the linearly polarized plane wave, the dimer has a maximum FE (~E/E₀ = 24) occurring in the gap distance. For a 100 nm AuNP monomer the maximum FE (~E/E₀ = 4.5) occurs near the particle surface in the direction of field oscillation. Note that the FE at the external surfaces of the dimer is ~E/E₀ = 6.2 which is more than the maximum FE of AuNP. We also performed the same calculation for a circularly polarized field that lies in the dimer plane (x-y in our case). The reason is that cell transfection or laser ablation, since the AuNP dimer is an asymmetric nanostructure applying a circularly polarized laser field assures equal contribution of all dimers in the same way. In this case the maximum FE has the same nature but with a smaller value (~E/E₀ = 17) for the same configuration. Note that the a small dip exactly in the middle of the gap can be misleading since one expects that the maximum FE should lie precisely at this point. This is true for a nearly-touching regime where the gap distance between the particles is much smaller in comparison to the AuNP radius (few nanometers). For such conditions we face a fully bonding dimer plasmon mode arising from the hybridization of the dipolar modes of the individual nanoparticles. Here, we avoid nearly-touching cases where quantum mechanical effects for dimer plasmons should be considered because very strong nonlinearity [15].

 

Fig. 1 (a) Electric FE distribution cross-sections of 100 nm AuNP dimer with gap distances of d = 10 nm, (b) AuNP monomer for longitudinal polarization at off-resonance wavelength λ = 800 nm. The right inset shows their correspondence FE cross-section on top surface. The color legend on the right shows the magnitude of the FE. (c) and (d) show FEM calculation of scattering, absorption and extinction cross sections as a function of the incident laser wavelength for AuNP dimer and monomer, respectively.

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Figure 1(c) and 1(d) show the calculated absorption, scattering and extinction spectrum for an AuNP and a dimer. Please note that for the off-resonance irradiation (λ = 800nm) the absorption cross-section is quite small. This means the thermal energy absorption by AuNPs is not that much to cause any fragmentation, deformation or melting, therefore the intactness of particles is guaranteed. For λ = 800 nm, the absorption cross-section of AuNP and dimer are σ_abs = 0.05 × 10⁴ nm² and σ_abs = 0.35 × 10⁴ nm², respectively. However, the scattering cross-section (σ_scat = 5.45 × 10⁴ nm²) of the dimer is greater than the cross-section of its AuNP monomer counterpart (σ_scat = 0.35 × 10⁴ nm²). Please note that by changing the dimer gap distance in the range of what we studied in this work for both linear and circular polarization there is only a very small variation in absorption cross section spectrum (See Fig. 8 appendix).

Beside the incident light polarization, the geometric parameters such as gap distance and the AuNP diameters have significant impact on electric FE in the gap distance and surrounding region. It has been shown that the strength of plasmon coupling between a pair of AuNPs falls as a function of the interparticle gap scaled by the particle size with a near-exponential decay trend [16]. To show this in Fig. 2(a) we plot the maximum FE versus the gap distance ranging from d = 4 nm to d = 16 nm. This maximum FE occurs in the gap distance for an 100 nm AuNP dimer immersed in water irradiated by laser field at the wavelength of λ = 800 nm. Here, FE also follows an exponential decay function (FE_max = A₀ + A₁ exp (-A₂d)). For a linear polarization A₀, A₁ and A₂ are 15.3 ± 1.2, 223.1 ± 22.4 and 0.33 ± 0.02 (nm⁻¹). For a circular polarization A₀, A₁ and A₂ are 10.75 ± 0.86, 156.3 ± 16 and 0.33 ± 0.02 (nm⁻¹), respectively. As seen in these equations increasing the gap distance leads to a dramatic decrease of the maximum FE.

 

Fig. 2 Electric FE of (a) 100 nm AuNP dimer versus gap distances, (b) AuNP dimer versus diameter of a gap distance of d = 10 nm at off-resonance wavelength λ = 800 nm.

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In Fig. 2(b) we show the maximum FE versus AuNP diameter for a constant value of gap distance d = 10 nm. As seen in this figure the maximum FE has a maximum around a diameter of 120 nm. It means that, as the AuNP size is increased, the FE increases until about a size of 120 nm, beyond which the enhancement decreases. These results in Fig. 2 are used to justify the energy deposition on surrounding induced plasma in the following sections. The AuNPs up to 120 nm sizes the particles interact coherently with incident light and electric dipole-dipole coupling resonance contributes significantly to field enhancement. However, for larger nanoparticles higher order interaction modes like quadrupoles become important. This higher order couplings lead to incoherent interaction of incident light and the light dipole-dipole interactions are weakened resulting in a reduction in the maximum field enhancement.

2.2 Nanostructures temperature

To ensure that the nanoparticles stay intact, in this part we study the thermodynamics of the AuNP dimer irradiated by femtosecond pulsed laser. Here we show that for the most intense laser pulse in this paper (maximum fluence 50 mJ/cm² and minimum pulse width 45 fs) the AuNPs’ lattice temperature does not reach to the melting and fragmentation temperature and the particles stay intact during irradiation. This is because of the small value on absorption cross-section at λ = 800 nm. In our study the range of the laser pulse width (τ_p = 45-300 fs) is much shorter than the electron-phonon (lattice) coupling time (~1-3 ps), therefore, the AuNP electrons and phonons are heated out of relative equilibrium, each one being described by a distinct temperature T_e and T_l (See Eqs. (5) - (7) in appendix) [17,18]. The conduction electrons in the AuNP first absorb the laser energy through the excitation of a plasmon that decays rapidly, leading to a highly energetic electron population that thermalizes in ~500 fs. After several ps, the electrons and the lattice reach thermal equilibrium. Finally, through phonon-phonon interactions, the thermal energy is transferred to the environment as the Au nanostructures and medium reach their equilibrium temperature. The time required for the last process directly depends on the thermal conductivity and on the heat capacity of the surrounding medium. The two-temperature model (TTM) is used to describe the thermal evolution of the system, which neglects the initial nonthermal electron distribution [19].

Figure 3 illustrates the spatial and temporal temperature increase (T-T₀) of the 100 nm AuNP dimer with gap distance d = 10 nm and its surrounding water. The dimer is irradiated by a 45 fs laser pulse with a fluence of 50 mJ/cm² at the wavelength of λ = 800 nm with both linear and circular polarization. T₀ is considered to be 300 K. Figure 3(a) shows temperature profile at 600 ps after the pulse peak of a linearly polarized field, at which the water temperature reaches its maximum value. The TTM model shows that the absorbed energy is strongly localized in the close proximity of the AuNP at this time. Figure 3(b) presents the temperature profiles as a function of radial distance at 600 ps after the pulse peak in the z = 0 plane. At this time, the water temperature increase (ΔT_m) reaches about 177 K and 110 K at the AuNP surface for linear and circular polarization, respectively. At 600 ps after the pulse peak, for a linear and circular polarization, the lattice temperature increase (ΔT_l) in AuNP is 277 K and 172 K. Our calculations show that the maximum ΔT_l is 553 K and 348 K achieved at 30 ps and 25 ps after the laser pulse peak, for a linear and circular polarization, respectively. Note that the water temperatures at the interface decrease rapidly as we move away from any direction where the thermal penetration depth (defined by the length where temperature drops to around 300 K) is approximately 30 nm. Figure 3(c) shows the electron (ΔT_e) and lattice (ΔT_l) temperature increases of the particle and the water temperature near the particle surface in the gap distance as a function of time after irradiation with a single laser pulse. Due to the smaller electron heat capacity, ΔT_e rises very rapidly during the pulse duration (45 fs) and reaches its maximum value at the end of the pulse. This quick electron temperature rise can be seen as an instantaneous process compared with the time scales of the subsequent electron–phonon and phonon–phonon coupling processes. Thermal equilibrium between the electrons and the lattice occurs approximately 20 and 30 ps after the pulse for circular and linear polarization, respectively. The melting temperature of AuNPs is generally size dependent and has been shown to be lower than the bulk melting temperature (1337 °K) for small particle diameters [19]. This means that for all fluences the ΔT_l is less than the melting point of the AuNP that keep the particle intact during and after the pulse irradiation. The AuNP dimer transfers its energy to its surrounding environment within few nanoseconds (≈2 ns).

 

Fig. 3 (a) Temperature profile in water surrounding of AuNP dimer absorbing a linear 45 fs laser pulse with a fluence of 50 mJ/cm² at wavelength λ = 800 nm 600 ps after pulse peak. (b) Temperature profiles at 600 ps after pulse peak as a function of radial distance in the length of the AuNP dimer at the z = 0 plane. (c) Time dependent temperature evolution of electrons (ΔT_e), lattice (ΔT_l) and water temperature (ΔT_m) at the AuNP-water interface, for a linear (dashed curve) and circular (dotted curve) field polarization. The inset shows the laser intensity profile.

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2.3 Energy deposition in AuNP dimer and surrounding induced plasma

The strong FE in the gap distance and around AuNP dimer leads to an intense near-field scattering effect in the NIR wavelength regime. If the near-field intensity is sufficient, the interaction of a short-pulsed laser with AuNP dimer leads to a nonlinear photoionization of the water molecules. This ionization triggers the generation of electronic nanoscale plasma in the gap distance and around AuNP dimer through various physical processes, including multiphoton ionization, quantum tunneling, and impact ionization [8]. Once created, the plasma continues to absorb laser energy from laser pulse until the free electron density reaches to the critical value of ~10²¹cm⁻³ usually attributed to the onset of optical breakdown [9]. In general, this optical breakdown is associated with shock-wave emission and vapor bubble formation that extends in few picoseconds over a few tens of nanometers from the particle beyond the focal region. Our calculations show that the maximum free-electron density is ~6.5 × 10²² cm⁻³ for a 100 nm AuNP dimer irradiated by a 50 mJ/cm², 45 fs, 800 nm with gap distances 10 nm (See Eqs. (8) and (9) in appendix). To do this we force our computer code to calculate the maximum electron density in the dimer gap distance at the AuNP surface where we have the maximum FE. Compared to the simulation results obtained for a 100 nm AuNP monomer irradiated by a 200 mJ/cm² in water with the same pulse width and wavelength, the maximum values for the free-electron density is ~1 × 10²² cm⁻³ at the surface of the AuNP. As one can see, irradiating the AuNP dimer by three fourths less fluence leads to more plasma density owing to the contribution of the high off-resonant near-field enhancement in the dimer gap distance and the outer surface of AuNPs, despite the extremely small interaction volumes involved.

To compare the energy deposition inside the AuNP dimer and surrounded plasma with its monomer counterpart, we show in Fig. 4 the energy deposition for two different linear and circular polarizations. The energy deposition on plasma is calculated by the numerical integration of the total power absorbed by plasma over the time-interval for pulsed laser. We calculate the total power by numerical volume integration of the ionization and ohmic power density on surrounding medium (See Eqs. (10) and (13) in appendix). The ionization power density is the normalized water excitation energy (see Eq. (1) in [9]) by the ionization rate density. In Fig. 4(a), the nonlinear energy deposition on plasma is potted versus fluence. As an example, for fluence 200 mJ/cm², energy deposition on plasma is ~21 pJ and ~14 pJ for linear and circular polarization, respectively. Figure 4(b) shows the linear energy deposition on AuNP versus fluence that is much smaller than the energy deposition on plasma for both polarizations. Also our calculation show that for the mentioned parameters and configuration the cross point of two energy deposition curves in the AuNP and surrounding plasma occurs at the fluence of 70 mJ/cm² and 0.4 pJ for linear polarization and 104 mJ/cm² and 0.6 pJ for circular polarization.

 

Fig. 4 Energy deposition in pJ on (a) induced plasma around and (b) in 100 nm AuNP monomer and dimer versus laser fluence with 45 fs width at 800 nm. Dashed and circular points show the linear and circular polarization, respectively.

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We also calculated the deposition energy in AuNP dimer for a linear and circular field polarization, respectively. The cross point of two curves (energy deposition in the AuNP dimer and surrounding plasma) occurs at the fluence of 4.5 mJ/cm² and 0.24 pJ for linear polarization and 11.5 mJ/cm² and 0.36 pJ for circular polarization.

We know experimentally that linearly irradiating AuNPs of 100 nm in the range of 100 to 200 mJ/cm² peak fluence, nanobubbles of ~0.5 to 1.5 μm diameters are observed. From Fig. 4, this corresponds to an energy range of 0.5-1.2 pJ absorbed in the AuNP and 2-22 pJ in the plasma. Considering that, the maximum bubble radius is proportional to the total energy deposition in both AuNPs and surrounding plasma (see Eq. (22) in [10]), one should expect that using equivalent total energy range should produce similar nanobubbles diameter. From Fig. 4, dimers irradiatted at fluences of 10-30 mJ/cm2 should produce micron size bubble. This is a significantly smaller fluence range as compared to the use of AuNPs. This is quite remarkable feature especially for nanomedicine applications such as cell optoporation and transfection where a micron size bubble cavitation is required. Please note that for off resonant wavelength (λ = 800 nm) the dimer absorption cross section is around three times less than the on resonant wavelength case (λ = 680 nm). Therefore to avoid reaching the melting point, AuNP dimer with mentioned properties could be irradiated up to 25mJ/cm² by a 45fs pulsed laser (See Fig. 9 in appendix).

In Fig. 5 we investigate the effect of the gap distance between the AuNPs on the energy deposition in the surrounding plasma for different fluences. The gap distance between AuNPs has significant impact on dipole-dipole interaction between the AuNPs and the FE. In order to use the FE to deposit an optimized energy on plasma the gap distances less than 10 nm is strongly recommended. Our calculations show that the ratio between energy deposition on plasma for linear and circular polarization has its minimum value (~3) for 8 nm gap distance.

 

Fig. 5 (a) and (b) energy deposition in pJ in surrounding AuNP dimer plasma versus dimer gap distance with variable fluence 10 – 50 mJ/cm² at 800 nm for the linear and circular polarization, respectively.

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In Fig. 6, we show the impact of AuNP diameter on deposition energy in AuNP dimer surrounding induced plasma where the diameter changes from 60 nm to 120 nm with step 10 nm. The main reason to choose this diameter range is that the maximum FE increasing up to 120 nm diameter and then starts decreasing for larger AuNPs (see Fig. 2).

 

Fig. 6 (a) and (b) Energy deposition in pJ in surrounding AuNP dimer plasma versus laser fluence variable AuNP diameter 60 nm to 120 nm for the linear and circular polarization, respectively.

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Since the energy deposition on plasma is a nonlinear process and photoionization increasing in the form of a power function by increasing the electric field, therefore in dimers with larger AuNPs the deposition energy on plasma per unit volume decreases. In Fig. 6(a) and 6(b) we show the deposition energy in AuNP surrounding plasma by changing the diameter of AuNP for both linear and circular polarisation. The insets show the same quantity in logarithmic scale. Our calculations show that for linear and circular polarization, the energy deposition on plasma has a positive nonlinear increasing rate up to 110 nm and then positive decreasing rate up to 120 nm. To show this variation clearly in Table 1 we include the data regarding the energy deposition in pJ in surrounding AuNP dimer plasma for a linearly polarized 45 fs pulsed laser with fluence of 50 mJ/cm² and AuNP diameter 60 nm to 120 nm. The last column shows the difference energy for successive diameters. As seen in this column numbers have an increasing trend up to 110 nm AuNP and then decreses.

Table 1. Energy deposition in pJ in surrounding AuNP dimer plasma for a linearly polarized 45 fs pulsed laser with fluence of 50 mJ/cm² and AuNP diameter 60 nm to 120 nm.

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For larger diameters the energy deposition on plasma has a negative rate and start decreasing. This trend follows the same trend as FE has in gap distance and around the AuNP dimer. Note that although the calculations are done for a certain value of the gap distance (10 nm); however, the same trend is valid for other gap distances in the range of dimer resonance.

Another key parameter to control the energy deposition in the system is the pulse width. Larger pulse width in the range of few hundred fs along with low fluence leads to use lasers that have reasonable price and occupy less space. In Fig. 7 we show the impact of the pulse width on energy deposition in surrounding plasma. Here, we vary the pulse width from 50 fs up to 300 fs at λ = 800 nm for a 100 nm AuNP dimer with gap distance of d = 10 nm for a linear and circular polarized field.

 

Fig. 7 (a) and (b) Energy deposition in pJ in surrounding AuNP dimer plasma versus laser fluence variable width 50 – 300 fs at 800 nm for the linear and circular polarization, respectively.

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As seen in these figures by increasing the pulse width the energy deposition in surrounding plasma nonlinearly decreases. The energy deposition in AuNP dimer for fluence of 10 mJ/cm² is 0.23 pJ and 0.35 pJ for linear and circular field polarization that increases linearly with the rate of 2.0 × 10⁻² pJ/mJ/cm² and 1.8 × 10⁻² pJ/mJ/cm², respectively for all pulse widths. Our calculation show that for the pulse width greater than 200 fs the cross point of two energy deposition in the AuNP dimer and surrounding plasma curves occurs around 15 mJ/cm² and 30 mJ/cm² for linear and circular polarization, respectively.

Conclusion

In conclusion our calculations show that a significant field enhancement in gap distance and surrounding media significantly reduces the pulsed laser fluence threshold to achieve an optical breakdown in ultrafast regime. This field enhancement leads to nonlinear photon absorption in surrounding water that leads to plasma nucleation. We investigate the impact of different parameters such as gap distance, AuNP diameters and laser pulse width on the energy absorption. At the end our developed tool is capable to carry the calculations for any desired nanostructure configurations. Upon request interested research groups are most welcome to contact the correspondent author for any possible collaboration.

Appendix

In the appendix information material, we provide further explanation on the self-consistent modeling that has been used in the paper including electrodynamical, thermal and plasma dynamics that will provide a deeper insight on the concepts mentioned in the main text.

1. Electromagnetic interaction

Assuming time-harmonic electric field, electromagnetic interaction between the laser and the nanostructure-water system is calculated using Maxwell's equations. The electric field distribution _{$\overrightarrow{E}(r,t)$} is computed using Maxwell's equations as a Helmohltz equation [14]:

(1)$$\nabla \times \left({\mu }_r^{-1}\nabla \times \overrightarrow{E}\right)-k_0^2\left({\epsilon }_r-j\frac{\sigma }{\omega }\right)\overrightarrow{E}=0$$

where the frequency dependant relative complex permittivity, ε_r is calculated from the complex refractive index interpolated from Johnson and Christy [12] in the case of AuNP dimer. Plasma generation inside the water medium will however affect the optical properties, its impact being modeled as a Drude permittivity:

(2)$$\text{ε}={\text{ε}}_{\infty }-\frac{{\text{n}}_e{\text{e}}^2}{{ϵ}_0\text{m}\left({\text{ω}}^2+{\text{jωγ}}_{\text{c}}\right)}$$

where ${\text{ε}}_{\infty }$ is the water permittivity, m is the electron mass and ${\text{γ}}_{\text{c}}$ is the electron scattering time. The relative permeability μr is taken as unity and k0 is the wave number. Equation (1) is solved in a spherical domain 10 times larger than the AuNP with an outer perfectly matched layers shell and a scattering type boundary condition that emulate an infinite domain.

Assuming a non-dissipative host medium, the absorbed and scattered energies by the AuNP are obtained as [20]:

(3)$$Q_{scat}=\frac{1}{2}Re\left[{{\displaystyle \iint }}^{\text{}}{\overrightarrow{E}}_{scat}\times {\overrightarrow{H}}_{scat}·\overrightarrow{n}ds\right]$$

(4)$$Q_{abs}=\frac{1}{2}Re\left[{{\displaystyle \iint }}^{\text{}}{\overrightarrow{E}}_{tot}\times {\overrightarrow{H}}_{tot}·\overrightarrow{n}ds\right]$$

where $\overrightarrow{E}$ and $\overrightarrow{H}$represent the electric and magnetic field vectors, respectively. _{$\overrightarrow{n}$} is an outward-pointing unit vector normal to the surface of the AuNP. The absorption, scattering and extinction cross-section are defined as ${\sigma }_{abs}=Q_{abs}/I_0$, _{${\sigma }_{scatt}=Q_{scatt}/I_0$}and${\sigma }_{ext}={\sigma }_{abs}+{\sigma }_{scatt}$, respectively. Here, $I_0=(1/2){\epsilon }_0{n}_w{E}_0^2$ represents the intensity of the incident laser beam of amplitude E₀ in the surrounding medium. Note that by changing the dimer gap distance there is only a very small variation in absorption cross section spectrum. In Fig. 8 we show scattering, absorption and extinction cross sections as a function of the incident laser wavelength for AuNP 100 nm dimer dimer with gap distances of d = 10 nm (solid curve), d = 20 nm (dotted curve) and d = 30 nm (dashed curve). As seen in this figure decreasing the gap distance leads to a blue shift in extinction peak. The smaller peak on extinction spectrum is due to the role of quadrupole plasmon resonance [21].

 

Fig. 8 Calculation of scattering, absorption and extinction cross sections as a function of the incident laser wavelength for AuNP 100 nm dimer with gap distances of d = 10 nm (solid curve), d = 20 nm (dotted curve) and d = 30 nm (dashed curve). The incident laser field has longitudinal polarization at off-resonance wavelength λ = 800 nm.

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2. Thermal evolution

The intensity of the femtosecond pulses is modeled by a Gaussian curve, as following:

(5)$$I\left(t\right)=\frac{F_L}{\sqrt{2\pi t_{\sigma }}} \text{exp}(\frac{-{\left[t-t_0\right]}^2}{2t_{\sigma }^2})$$

in the above equation, $t_{\sigma }=t_l/2\sqrt{2\ln 2}$ is the pulse width, where t_l is the laser pulse duration defined as the full with at half maximum of the Gaussian temporal profile, t₀ is the position of the center of the peak and F_L is the incident laser energy density (fluence).

The equation governing the thermal evolution of the AuNP and water strongly depends on the irradiation time-regime, depending on how the laser pulse time width compare to the electron-phonon thermalization time (~1-3 ps). Since the pulse width using in this paper is much less than the mentioned values, therefore, thermal evolution of the AuNP and water are governed by the two temperature model (TTM) as following [18]:

$$C_e\left(T_e\right)\frac{\partial T_e\left(r,t\right)}{\partial t}=\nabla ·\left(k_e·\text{ Δ}{T}_e\left(r,t\right)\right)-G·\left[T_e(r,t)-T_l(r,t)\right]+S\left(t\right)$$ (6-a)

$$C_l\left(T_l\right)\frac{\partial T_l\left(r,t\right)}{\partial t}=\nabla ·\left(k_l·\text{ Δ}{T}_l\left(r,t\right)\right)-G·\left[T_e\left(r,t\right)-T_{el}\left(r,t\right)\right]-F$$ (6-b)

$${\rho }_m\left(r\right)C_m\left(r\right)\frac{\partial T_m\left(r,t\right)}{\partial t}=\nabla ·\left(k_m·\text{ Δ}{T}_m\left(r,t\right)\right)+F$$ (6-c)

In above equations, T_e, T_l and T_m are the time dependent temperatures of the electrons, lattice and surrounding medium, respectively. c stands for the heat capacity (c_e, electronic heat capacity; c_l, lattice heat capacity; c_m, medium heat capacity) and k is the thermal conductivity. The second term on the right hand side in (6-a) and (6-b) describes the energy exchange from the electrons to the lattice via electron-phonon coupling with a coefficient G. The thermal conductance G relates the temperature drop at an interface to the heat flux crossing the interface and constitutes the coupling parameter between a particle and surrounding medium energy equations. The term S(t) in equation (6-a) is the absorbed laser energy. Also, the F term on equations (6-b) and (6-c) describes the heat transfer across the interface between AuNP and the surrounding defined as:

(7)$$F=\frac{3h}{R}\left[T_l\left(r,t\right)-T_m\left(r,t\right)\right]$$

here, h is called interfacial thermal conductance and is the fitting parameter for the cooling process of AuNP in aqueous solutions. R is the AuNP radius.

3. Plasma dynamics

Plasma dynamic in water is described by coupling a diffusion equation for both the plasma density (n_e) and the plasma kinetic energy (u) [22,23].

(8)$$\frac{\partial {\text{n}}_{\text{e}}}{\partial \text{t}}+\nabla ·{\text{ j}}^{\text{n}}={\text{S}}_{\text{photo}}+{\text{S}}_{\text{coll}}-{\text{S}}_{\text{rec}}$$

(9)$$\frac{\partial \text{u}}{\partial \text{t}}+\nabla ·j^{\text{q}}=\frac{1}{2}\text{Re}\left(J^{\text{*}}·E\right)-{\text{Q}}_{\text{ei}}-{\text{Q}}_{\text{rad}}-·\tilde{\Delta }{\text{S}}_{\text{coll}}$$

where j is the electronic current, S_photo is the Keldysh photoionization, S_coll is the collision ionization when highly energetic electrons collide with molecules and generate new free electrons, S_rec is the recombination of carriers, Q_ei is the electron-ion energetic coupling, Q_rad is the energetic loss by radiation and $\tilde{\Delta }$ is the effective extraction potential.

4. Energy deposition in AuNP dimer and surrounding plasma

After solving the mentioned coupled differential equations, the energy density deposition rate is evaluated using Eqs. (10) to (13).

(10)$${\text{Q}}_{\text{total plasma}}={\text{Q}}_{\text{plasma}}+{\text{Q}}_{\text{dphoto}}$$

(11)$${\text{Q}}_{\text{plasma}}=\frac{1}{2}{\text{σ}}_{\text{plasma}}\left({\text{T}}_{\text{plasma}}\right){\left|E\right|}^2$$

(12)$${\text{Q}}_{\text{dphoto}}={\text{d}}_{\text{photo}}\frac{{\text{ρ}}_0-\text{ρ}}{{\text{ρ}}_0}\Delta$$

(13)$${\text{Q}}_{\text{AuNP}}=\frac{1}{2}{\text{σ}}_{\text{Au}}{\left|E\right|}^2$$

Q_Au is the optical power density absorbed in the AuNP, σ_Au is the gold conductivity, E is the electric field, Q_plasma is the optical power density absorbed by the plasma through conductive losses (bremsstrahlung process), σ_plasma(T_plasma) is the plasma conductivity that depends on the electronic temperature T_e as given by the Drude theory, Q_dphoto is the power density required to photoionize the water, d_photo is the photoionization rate, ρ₀ is the valence electron density (6.68x10²²cm⁻³) and Δ is the water energy gap between the ground and the quasi-free excited state [9].

The plasma and photoionization power density is space and time integrated from 0 fs to 10 times the pulse duration to yield the total energy deposition. Q_Au and Q_dphoto are integrated over a spherical volume with the radius of 15 × R that includes completely the plasma produced during the irradiation. Q_Au is similarly integrated within the AuNP.

In this paper we are interested in off resonance AuNP dimer excitation to achieve significant energy deposition on nucleated plasma in surrounding material in compare to AuNP monomer off resonance excitation (λ = 800 nm) where we assure that the particles stays intact on pulse duration. On the other hand, on resonance AuNP dimer excitation leads to achieve the same amounts of energy deposition by a pulsed laser with much less fluence compare to off resonance AuNP monomer excitation. To show this, in Fig. 9, we compare energy deposition in the nucleated surrounding plasma for on (λ = 680 nm) and off resonance excitation of 100 nm AuNP dimer with d = 10 nm gap distance. Note that in this case the heat absorption cross section in on resonance (σ_abs= 0.9×10⁴ nm²) irradiation is about three times more than the off resonance (σ_abs= 0.35×10⁴ nm²) one, therefore our calculation show that to avoid reaching the melting point the pulsed laser fluence has to be kept less than 25 mJ/cm².

 

Fig. 9 Energy deposition on induced plasma around 100 nm AuNP dimer versus pulsed laser fluence with 45 fs width for on (λ = 800 nm) and off (λ = 680 nm) resonance. Dashed and circular points show the linear and circular polarization, respectively.

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References and links

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