Open Access
Add to BibSonomy
Abstract
Controlling temperature distribution at the micro/nano-scale brings new applications in many fields such as physics, chemistry and biology. This paper proposes a photothermal metasurface that employs polarization and wavelength multiplexing to regulate various temperature distributions at the micro/nano-scale. Such a photothermal metasurface is numerically validated by the finite element method. Firstly, the inversion algorithm is used to calculate the thermal power density distribution, which is decided by a given temperature distribution. Then, based on the bottom-up design method, (a) the library of absorption cross sections of gold nanoparticles is established by resizing nanoparticles; (b) the single pixel is constructed for wavelength and polarization multiplexing; (c) the overall structure of a photothermal metasurface is optimized and established. Finally, four given temperature distributions, combining the multiplexing of two orthogonal polarizations and two wavelengths, are achieved in the same area. The simulation results well confirm the feasibility of photothermal multiplexing. Such photothermal metasurface provides solutions for flexible control of temperature distribution at the micro/nano-scale.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Unlike the propagation of light which is governed by the wave equation, thermal conduction satisfies the diffusion equation. In many cases, the temperature field is more difficult to manipulate than the light field especially at micro/nano-scale. For example, one can shape the distribution of light with a chosen pattern by a Spatial Light Modulator, but it is challenging to shape the distribution of temperature as small heat sources tend to produce a Gaussian-like temperature distribution [1]. As a basic tool in the field of science and technology, nano-size heat sources have many important applications. Nanoheaters can be used in electrothermoplasmonic nanotweezer [2,3], optofluidic control [4] and plasmon-assisted nano-chemistry [5]; Nanoheaters can even be used in molecular surgery [6], photothermal therapy [7–9], photothermic vaporization [10,11] and optical memory [12–14]. In addition, due to the small heating volume, the micro/nano-scale heating also has the characteristics of rapid response [15]. In those applications, controlling the temperature distribution accurately at micro/nano-scale plays a key role.
Optical metasurfaces are artificial structures composed of subwavelength metallic or dielectric building blocks, which offer the possibility of controlling light efficiently at the micro/nano-scale [16]. Optical metasurfaces even have optical properties that natural materials do not possess. As a result, this technology has enabled new applications in many areas, such as optical holography [17–20], two-dimensional optical element [21–23], electromagnetically induced transparency [24], perfect absorber [25] and manipulating light [26–28]. So far, there have been relatively few studies using metasurfaces to control temperature field compared to control light field.
The photothermal effect provides a fast, non-contact, and precise method for controlling temperature distribution. There are generally two solutions to do it by using of photothermal effect on nanoparticle arrays: (1) adjusting the distribution density of nanoparticles under uniform light irradiation [29]; (2) shaping the beam to alter the spatial light intensity distribution on uniformly distributed nanoparticle arrays [1,15]. The first solution is easy to implement, but lacks flexibility, as it generally achieves only one predefined temperature distribution. The second solution can flexibly adjust the spatial and temporal distribution of temperature. However, there are also several disadvantages. Firstly, a more complex optical path is needed. Secondly, due to the limitation of light diffraction, the spatial resolution of thermal power density is relatively poor. Finally, when the numerical aperture is small, too large speckle particles will cause uneven temperature distribution. In order to balance the high resolution and flexibility of temperature distribution, the first solution is adopted and developed in this paper. Gold nanoparticles are employed as heat sources to ensure high spatial resolution of thermal power density, and the limitation of having only one pattern of temperature distribution is broken by the wavelength and polarization multiplexing of light.
In this paper, the conversion method between the temperature distribution and the thermal power density distribution is introduced, which is the theoretical basis for accurately adjusting the temperature field. Then, gold nanoparticles with multi-order absorption cross sections are size-screened at two target wavelengths (650 nm and 750 nm). By optimizing the structure of a single pixel unit and determining structural parameters, a photothermal metasurface capable of achieving wavelength and polarization multiplexing is designed. Finally, a finite element simulation model is established and the proposed scheme is verified numerically.
2. Materials and methods
2.1 Theoretical model
Plasmon resonance of noble metal nanoparticle can enhance the light absorption. The heat generation of noble metal nanoparticle can be flexibly controlled by the light field [30].
For spherical nanoparticles with radius $R$ in a uniform background medium, the temperature rise $\Delta T$ around the particles can be obtained by analogy to the Coulomb potential generated in a uniform medium with an effective dielectric constant of $\kappa$ [31],
Here, $r$ represents the distance from the center of the nanoparticle, while $\varDelta {T}_{N\!P}$ denotes the temperature rise of the nanoparticle due to the total absorbed power $Q$:
where ${\sigma }_{abs}$ is the absorption cross section of nanoparticle and $I$ is the intensity of the incident light. For particles with arbitrary shapes, the dimensionless heat capacity coefficient $\beta$ is introduced, and the temperature rise of the particles can be modified as [32]: where ${R}_{eq}$ is the equivalent radius and the size is equal to the radius of the sphere with the same volume as the particle under study.When considering an interface $\mathcal {D}$, which separates two isotropic media with thermal conductivity of ${\kappa }_{1}$ and ${\kappa }_{2}$ respectively, the $N$ nanoparticles are distributed in the form of a square lattice at this interface. $d$ is set as the lattice constant and the thermal power density (heating power per unit area) in the interface is $q(\mathbf {r})$. The thermal power density distribution determines the temperature rise distribution $\varDelta T(\mathbf {r})$. For the $i-th$ nanoparticle, the temperature rise is $\varDelta {T}_{i}$, and the thermal power ${Q}_{i}={d}^{2}{q}_{i}$. Since the heat diffusion equation is linear, the total temperature distribution can be obtained by superimposing the individual temperatures of the $N$ particles [33]. The temperature increase $\varDelta {T}_{i}$ of the $i-th$ nanoparticle is determined by two terms [34],
The first term of $\varDelta {T}_{i}^{S}$ on the right side of Eq. (4) is the contribution of the $i-th$ nanoparticle itself to temperature rise, and the second term $\varDelta {T}_{i}^{ext}$ is the contribution of the remaining $N-1$ nanoparticles around the $i-th$ nanoparticle. $\bar {\kappa }=({\kappa }_{1}+{\kappa }_{2})/2$, which is the average thermal conductivity of the two environmental media [1,34,35]. ${r}_{ij}=\left | {\mathbf {r}}_{i}-{\mathbf {r}}_{j}\right |$, ${\mathbf {r}}_{i}$ is the coordinate vector of the $i-th$ nanoparticle. After defining the vectors of $\mathbf {T}={({T}_{i})}_{i\in \left [ 1,N\right ]}$ and $\mathbf {Q}={({Q}_{i})}_{i\in \left [ 1,N\right ]}$, the Eq. (4) can be rewritten in matrix form [29],
where $\mathbb {A}$ [29] is an $N\times N$ matrix withIf a target temperature distribution $T$ is given, the corresponding thermal power distribution can be achieved by reversing Eq. (5),
In other words, as long as the inverse matrix $\mathbb {A}$ is obtained, the thermal power (or the thermal power density) distribution corresponding to the temperature distribution can be achieved.
A schematic diagram of the conversion between temperature distribution and thermal power density distribution is shown in Fig. 1. It can be seen that a uniform thermal power density distribution leads to a non-uniform temperature distribution due to thermal diffusion. If a uniform temperature distribution is to be achieved, the thermal power density distribution needs to be weaker in the middle and stronger at the edges, which can well counteract the effects of thermal diffusion. In addition to uniform temperature distribution, more complex temperature distributions such as linear temperature gradients and parabolic temperature distributions can also be achieved [29].
Fig. 1. (a) Uniform thermal power density distribution. (b) The temperature distribution corresponding to (a). (c) The heat power density distribution corresponding to (d). (d) Uniform temperature distribution.
2.2 Construction of photothermal metasurface
To be more practical, we assume that the metasurface is fabricated with electron beam lithography. The gold nanoparticles of the metasurface are arranged in the square lattice and resized according to the calculated thermal power density distribution in a two-dimensional plane. At the current state of the art, the resolution limit of e-beam lithography can reach 1 nm, the size of a single isolated feature can reach 5 nm, and the spacing of periodic array structures can reach 30 nm [36–38]. The shape of nanoparticles is shown in Fig. 2(a), where $l$, $w$, $h$, and $r$ represent the length, width, height, and radius of curvature of corner and edge of the particles, respectively.
Fig. 2. (a) Brick-like gold nanoparticle. (b) Schematic diagram of plasmonic photothermal metasurface composed of an array of gold nanoparticles at the air-glass interface. Absorption cross sections of 18-order particles as functions of wavelength for the target wavelengths of (c) 650 nm and (d) 750 nm (the ordinal number of the 18-order particles is listed along the ordinate). (e) and (f) show the absorption cross sections at wavelengths of 650nm and 750nm of the 18-order particles in (c) and (d), respectively.
For the brick-like gold particles, different length-width ratios correspond to different wavelengths of absorption peak, and the angle between the orientation of the long-side of gold nanobricks and the polarization direction of the irradiated light also affects the absorption. The scheme of the photothermal metasurface is shown in Fig. 2(b). By multiplexing of the wavelengths and polarizations, the gold nanobrick array can generate multiple heat patterns in the same area. Each gold nanoparticle is associated with a specific area, which is equivalent to a pixel. The basic idea of adjusting the temperature distribution is to use a set of gold nanoparticles whose absorption cross sections are proportional to the required thermal power density at all pixels. In other words, the thermal power density at all pixels will be mapped to a set of gold nanoparticles with multi-order absorption cross sections linearly distributed from small to large.
The absorption cross section of nanoparticles are calculated by using the finite element method (COMSOL Multiphysics). The dielectric functions of materials are shown in Table 1. The particles were screened by controlling the size. Where $h=16$nm [39], $r=5$nm [40,41], these two parameters are fixed and based on the experimental results in the literature. $l$ and $w$ take values from 16 nm to 100 nm. Any two adjacent values of $l$ or $w$ are set to be at least 2 nm apart and $w\le l$. Here, 650 nm and 750 nm are selected as the wavelengths of multiplexing. In order to reproduce the set thermal power density distribution more accurately and continuously, the order of varying the absorption cross section of the gold nanoparticles should be as large as possible. But too many orders can lead to oversized nanoparticles. In this paper, 18 orders are chosen, which is a reasonable compromise. The structural parameters of the particles are optimized according to the absorption cross section. The black dotted boxes in Fig. 2(c) and Fig. 2(d) indicate the absorption cross sections near 650 nm and 750 nm respectively, and they are also plotted in Fig. 2(e) and Fig. 2(f). For the two groups of nanoparticles, the absorption cross sections of these 18-order particles increase linearly at the target wavelength, while remain constant and small at non-target wavelength. This configuration ensures that there is no significant temperature crosstalk between the two groups of particles during wavelength multiplexing, and each only heats up at its own target wavelength. (See Supplement 1 for details of structural parameters of gold nanoparticles).
Table 1. Material parameters
In the following, the selected particles with multi-order absorption cross sections are used to achieve the target thermal power density distribution. First, the maximum thermal power in the region is divided by the maximum absorption cross section to calculate the incident light intensity, and then the thermal power of the remaining pixels in the region is divided by the calculated incident light intensity to get the absorption cross section of the particle at these pixels. The original thermal power density distribution is then transformed into the corresponding absorption cross section distribution. For each pixel, by consulting the multi-order absorption cross section list, the required absorption cross section value is replaced with the closest value in the list and the corresponding size information is obtained. Finally the particle size distribution map is obtained. Here, we choose nanobricks with two orientations whose long-sides are perpendicular to each other to reduce the crosstalk of polarization multiplexing. The four distribution maps are combined together to form the photothermal metasurface including wavelength and polarization multiplexing.
3. Results and discussion
In order to realize wavelength and polarization multiplexing, each pixel needs to contain more than one gold nanoparticle. The right side of Fig. 3(a) is a schematic diagram of the overall structure of the photothermal metasurface. The left side of Fig. 3(a) is a partial enlarged view. The area surrounded by the black dotted lines is a complete pixel unit, which specifies the relative positions of four gold nanobricks corresponding to two wavelengths and two polarizations. All gold nanobricks are arranged in a square lattice. The lattice constant $d$ is set to 200 nm. The side length of each pixel is $2d$.
Fig. 3. (a) Schematic diagram of the photothermal metasurface model and the enlarged view of pixels. The photothermal metasurface consists of 2592 pixels and the side length of each pixel is 400 nm. (b) Four patterns of temperature distribution for multiplexing. The temperature rise is set to 20 K in Latin letters area and 0 K in other area. (c) The four thermal power density distributions corresponding to the temperature distributions in (b). Panels of (b) and (c) are labeled with laser wavelength and a double-headed arrow indicating polarization direction.
The photothermal metasurface consists of $48\times 54\times 4=10368$ gold nanoparticles, which corresponds to the area of $19.2{\mathrm{\mu} \mathrm{m}}\times 21.6{\mathrm{\mu} \mathrm{m}}=414.72\mathrm {{\mu m}^{2}}$. As shown in Fig. 3(b), the patterns of the four temperature distributions are designed as the letters "S", "C", "N" and "U". The target temperature rise in the letter area is uniform and set to 20 K. The four temperature distributions correspond to four different illumination conditions (wavelength / polarization): 650 nm / lateral polarization, 650 nm / longitudinal polarization, 750 nm / lateral polarization, and 750 nm / longitudinal polarization.
Based on the four temperature distributions, the matrix $\mathbb {A}$ is easily derived by Eq. (6), and the thermal power density distribution is finally calculated by Eq. (7). As shown in Figs. 3(b) and 3(c): in the area of letters, although the temperature distribution is uniform, the thermal power density is different at different locations. The reasons are as follows. There are a large number of heat sources around the particle in the central region, resulting in significant thermal aggregation effects and a lower thermal power density required. In the edge region, the situation is reversed. The heat sources around the particle have decreased, so a higher thermal power density is required to maintain the same temperature.
For a point source, the intensity of electromagnetic wave decays by $1/{r}^{2}$ and the temperature decays by $1/r$, the latter decaying more slowly. So even if there is no obvious optical coupling between adjacent particles, it can still have thermal coupling. This property allows the calculation of the optical properties of arrays to be reduced to the study of the optical properties of individual particles, thus greatly reducing the required computer memory. This is why the thermal power density can be directly mapped to the absorption cross section of the individual gold nanoparticle, without considering the coupling of light. However, the calculation of the temperature field generated by a large number of nanoheaters cannot be simplified like light field. When designing the temperature field, the contribution of other particles needs to be taken into account, as presented in Eq. (4). The following simulation of multiphysics field coupling also confirms that it is reasonable to treat the light and heat fields in this way when designing a photothermal metasurface.
The temperature distribution of the photothermal metasurface is finally simulated using the finite element method with coupled optical and thermal fields (COMSOL Multiphysics). The ambient temperature $T$ is set to 293K. The material parameters are shown in Table 1. The four temperature distributions under four illumination conditions are shown in Fig. 4. The light intensities from left to right are $49.0\mathrm {\mu W/{\mu m}^{2}}$, $54.5\mathrm {\mu W/{\mu m}^{2}}$, $52.2\mathrm {\mu W/{\mu m}^{2}}$, and $53.6\mathrm {\mu W/{\mu m}^{2}}$, respectively. As can be seen: the edge of Latin letters is clear; the temperature distributions in the area of letters are uniform and continuous, and the designed temperature rise of 20 K is achieved; outside the area of the letters, the temperature drops rapidly. In addition, there are almost no crosstalks between the temperature distributions for multiplexing of two wavelengths and two polarizations.
Fig. 4. Temperature distributions of photothermal metasurface under four different illumination conditions.
Although our work mainly focuses on proposing a photothermal metasurface with polarization and wavelength multiplexing in theory, the impact of manufacturing errors on its functionality must be discussed. Manfrinato et al. have achieved designed patterns at a length scale of 1 nm using aberration-corrected electron-beam lithography in different materials [36,38]. Referring to this excellent manufacturing technology, in our scheme, the size parameters of the particles are designed to have a minimum difference of 2 nm. Due to the size sensitivity of the absorption cross section, some particles with adjacent numbers in the 18-order particles are most affected. The manufacturing errors may affect the uniformity of temperature in a pattern, but by comparing particle sizes, it is unlikely to affect polarization and wavelength multiplexing due to significant size differences (See Supplement 1).
The multiplexing of wavelength and polarization can break the limitation that there is only one temperature distribution in the same area, which enables metasurface a flexible platform for many applications, especially suitable for controlling chemical reactions and fabrications at the micro/nano-scale. In addition, the proposed metasurface has potential application in the fields of encrypted information storage and anti-counterfeiting. When information is "written" on the metasurface, the metasurface must be illuminated by light with corresponding wavelength and polarization, and then the encrypted information can be "read" by measuring the infrared light field (temperature field distribution). This highly complicated signal reading process greatly improves the security of information.
4. Conclusions
In summary, a photothermal metasurface composed of gold nanoparticles is proposed and verified. This photothermal metasurface provides a platform for precise control of temperature distributions at the micro/nano-scale. Since the surface plasmon resonance of gold nanoparticles is sensitive to the wavelength and the polarization, different temperature distributions can be obtained by multiplexing the wavelength and the polarization. Finally, the model is validated numerically using the finite element method. The proposed photothermal metasurface provides a non-contact, fast-response heating management method. It has potential applications in controlling chemical reactions, fabricating nanoparticles, and encrypting information and anti-counterfeiting.
Funding
National Natural Science Foundation of China (12174155); Guangdong Science and Technology Department (2020B1212060067); Natural Science Foundation of Guangdong Province (2019A1515011578).
Acknowledgments
The authors thanks Prof. Sheng Lan of South China Normal University for helpful discussion.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. L. Durdevic, H. M. Robert, B. Wattellier, et al., “Microscale temperature shaping using spatial light modulation on gold nanoparticles,” Sci. Rep. 9(1), 4644 (2019). [CrossRef]
2. J. C. Ndukaife, A. V. Kildishev, A. G. A. Nnanna, et al., “Long-range and rapid transport of individual nano-objects by a hybrid electrothermoplasmonic nanotweezer,” Nat. Nanotechnol. 11(1), 53–59 (2016). [CrossRef]
3. C. Hong, S. Yang, and J. C. Ndukaife, “Stand-off trapping and manipulation of sub-10 nm objects and biomolecules using opto-thermo-electrohydrodynamic tweezers,” Nat. Nanotechnol. 15(11), 908–913 (2020). [CrossRef]
4. G. L. Liu, J. Kim, Y. Lu, et al., “Optofluidic control using photothermal nanoparticles,” Nat. Mater. 5(1), 27–32 (2006). [CrossRef]
5. L. Cao, D. N. Barsic, A. R. Guichard, et al., “Plasmon-assisted local temperature control to pattern individual semiconductor nanowires and carbon nanotubes,” Nano Lett. 7(11), 3523–3527 (2007). [CrossRef]
6. M. J. Kogan, N. G. Bastus, R. Amigo, et al., “Nanoparticle-mediated local and remote manipulation of protein aggregation,” Nano Lett. 6(1), 110–115 (2006). [CrossRef]
7. P. K. Jain, I. H. El-Sayed, and M. A. El-Sayed, “Au nanoparticles target cancer,” Nano Today 2(1), 18–29 (2007). [CrossRef]
8. S. Lal, S. E. Clare, and N. J. Halas, “Nanoshell-enabled photothermal cancer therapy: impending clinical impact,” Acc. Chem. Res. 41(12), 1842–1851 (2008). [CrossRef]
9. X. Huang, P. K. Jain, I. H. El-Sayed, et al., “Plasmonic photothermal therapy (pptt) using gold nanoparticles,” Lasers Med. Sci. 23(3), 217–228 (2008). [CrossRef]
10. N. J. Hogan, A. S. Urban, C. Ayala-Orozco, et al., “Nanoparticles heat through light localization,” Nano Lett. 14(8), 4640–4645 (2014). [CrossRef]
11. M. Gao, C. K. Peh, H. T. Phan, et al., “Solar absorber gel: localized macro-nano heat channeling for efficient plasmonic au nanoflowers photothermic vaporization and triboelectric generation,” Adv. Energy Mater. 8(25), 1800711 (2018). [CrossRef]
12. P. Zijlstra, J. W. Chon, and M. Gu, “Five-dimensional optical recording mediated by surface plasmons in gold nanorods,” Nature 459(7245), 410–413 (2009). [CrossRef]
13. Q. Dai, M. Ouyang, W. Yuan, et al., “Encoding random hot spots of a volume gold nanorod assembly for ultralow energy memory,” Adv. Mater. 29(35), 1701918 (2017). [CrossRef]
14. Y. Chu, H. Xiao, G. Wang, et al., “Randomly distributed plasmonic hot spots for multilevel optical storage,” J. Phys. Chem. C 122(27), 15652–15658 (2018). [CrossRef]
15. C. Liu, G. Tessier, S. I. Flores Esparza, et al., “Reconfigurable temperature control at the microscale by light shaping,” ACS Photonics 6(2), 422–428 (2019). [CrossRef]
16. A. Arbabi, Y. Horie, M. Bagheri, et al., “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015). [CrossRef]
17. X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4(1), 2807 (2013). [CrossRef]
18. L. Huang, X. Chen, H. Mühlenbernd, et al., “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4(1), 2808 (2013). [CrossRef]
19. G. Zheng, H. Mühlenbernd, M. Kenney, et al., “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10(4), 308–312 (2015). [CrossRef]
20. D. Wen, F. Yue, G. Li, et al., “Helicity multiplexed broadband metasurface holograms,” Nat. Commun. 6(1), 8241 (2015). [CrossRef]
21. D. Lin, P. Fan, E. Hasman, et al., “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014). [CrossRef]
22. A. A. High, R. C. Devlin, A. Dibos, et al., “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015). [CrossRef]
23. S. Wang, P. C. Wu, V.-C. Su, et al., “Broadband achromatic optical metasurface devices,” Nat. Commun. 8(1), 187 (2017). [CrossRef]
24. Y. Yang, I. I. Kravchenko, D. P. Briggs, et al., “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5(1), 5753 (2014). [CrossRef]
25. Y. Yao, R. Shankar, M. A. Kats, et al., “Electrically tunable metasurface perfect absorbers for ultrathin mid-infrared optical modulators,” Nano Lett. 14(11), 6526–6532 (2014). [CrossRef]
26. E. Karimi, S. A. Schulz, I. De Leon, et al., “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Sci. Appl. 3(5), e167 (2014). [CrossRef]
27. Y. F. Yu, A. Y. Zhu, R. Paniagua-Domínguez, et al., “High-transmission dielectric metasurface with 2π phase control at visible wavelengths,” Laser Photonics Rev. 9(4), 412–418 (2015). [CrossRef]
28. J. B. Mueller, N. A. Rubin, R. C. Devlin, et al., “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017). [CrossRef]
29. G. Baffou, E. B. Ure na, P. Berto, et al., “Deterministic temperature shaping using plasmonic nanoparticle assemblies,” Nanoscale 6(15), 8984–8989 (2014). [CrossRef]
30. G. Baffou and R. Quidant, “Thermo-plasmonics: using metallic nanostructures as nano-sources of heat,” Laser Photonics Rev. 7(2), 171–187 (2013). [CrossRef]
31. G. Baffou, R. Quidant, and C. Girard, “Heat generation in plasmonic nanostructures: Influence of morphology,” Appl. Phys. Lett. 94(15), 153109 (2009). [CrossRef]
32. G. Baffou, R. Quidant, and F. J. García de Abajo, “Nanoscale control of optical heating in complex plasmonic systems,” ACS Nano 4(2), 709–716 (2010). [CrossRef]
33. P. Zijlstra, J. W. Chon, and M. Gu, “Effect of heat accumulation on the dynamic range of a gold nanorod doped polymer nanocomposite for optical laser writing and patterning,” Opt. Express 15(19), 12151–12160 (2007). [CrossRef]
34. G. Baffou, P. Berto, E. Bermúdez Ure na, et al., “Photoinduced heating of nanoparticle arrays,” ACS Nano 7(8), 6478–6488 (2013). [CrossRef]
35. G. Baffou, R. Quidant, and C. Girard, “Thermoplasmonics modeling: A green’s function approach,” Phys. Rev. B 82(16), 165424 (2010). [CrossRef]
36. V. R. Manfrinato, A. Stein, L. Zhang, et al., “Aberration-corrected electron beam lithography at the one nanometer length scale,” Nano Lett. 17(8), 4562–4567 (2017). [CrossRef]
37. C. Vieu, F. Carcenac, A. Pepin, et al., “Electron beam lithography: resolution limits and applications,” Appl. Surf. Sci. 164(1-4), 111–117 (2000). [CrossRef]
38. V. R. Manfrinato, F. E. Camino, A. Stein, et al., “Patterning si at the 1 nm length scale with aberration-corrected electron-beam lithography: tuning of plasmonic properties by design,” Adv. Funct. Mater. 29(52), 1903429 (2019). [CrossRef]
39. R. Marty, G. Baffou, A. Arbouet, et al., “Charge distribution induced inside complex plasmonic nanoparticles,” Opt. Express 18(3), 3035–3044 (2010). [CrossRef]
40. P. Dombi, A. Hörl, P. Rácz, et al., “Ultrafast strong-field photoemission from plasmonic nanoparticles,” Nano Lett. 13(2), 674–678 (2013). [CrossRef]
41. H. Qian, S.-W. Hsu, K. Gurunatha, et al., “Efficient light generation from enhanced inelastic electron tunnelling,” Nat. Photonics 12(8), 485–488 (2018). [CrossRef]
42. X. Chen, Y. Chen, M. Yan, et al., “Nanosecond photothermal effects in plasmonic nanostructures,” ACS Nano 6(3), 2550–2557 (2012). [CrossRef]
43. P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
44. G. Chen and P. Hui, “Thermal conductivities of evaporated gold films on silicon and glass,” Appl. Phys. Lett. 74(20), 2942–2944 (1999). [CrossRef]
