The Poynting vector S represents the amount of power per unit area that a wave carries in the direction of its propagation. The time-averaged Poynting vector S_av (units of W/m²) is given by

An extensive comparison has been provided previously among techniques used to simulate optical features of metallic nanostructures, especially gold nanoparticles. These techniques include Mie theory, the finite element method (FEM), discrete-dipole approximation (DDA), the transition matrix ‘null-field’ method (T-matrix), and the finite-difference time-domain (FDTD) method.^[84]

Mie solution is widely used as a major theory of many prior studies done for studying the treatment of SPR. This theory is an analytical solution simulating the scattering of electromagnetic radiation by spherical particles in Maxwell’s equations.^[85–89]

By using Mie theory for homogeneous spheres Q_sca and Q_ext can be calculated for gold nanospheres. These efficiencies are infinite series as follows:

Mie theory was presented more than 100 years ago. Since then it has been improved widely, but its limitations are significant. The main failure of solution for such an analytical method appears in non-spherical symmetric particles which makes it very limited. Besides, ignoring substrate interaction makes Mie theory more difficult to simulate many experiments.^[90–92]

To obtain the optical properties of different shapes of nanostructures, a more comprehensive mathematical method, which does not have the complexity nor limitation of an analytical solution (Mie theory), is needed. The finite element method (FEM), which is an approximate solution of differential equations, is a powerful numerical solution. FEM is one of the most famous solutions in computational physics which makes it feasible to achieve the optical features of nanostructures with different shapes.^[93–95]

Simulations in this paper have been carried out by using the FEM. Water with a refractive index of 1.33, is considered as a context medium in which nanoparticles have been placed. Also, Johnson–Christy data for the dielectric function of a gold nanoparticle are used at different wavelengths.^[96]

2.3.1. Boundary conditions in FEM

For a full mathematical description of a metal–dielectric interface with the Helmholtz equation (10), a set of boundary conditions (Eqs. (11)–(14)) should be taken into account for the discontinuity of the electrical and magnetic fields at the boundary between medium 1 and medium 2

where , and are the unit normal vector on the boundary, the wave number in a vaccum, and the electric permittivity of the medium respectively.

To truncate the computational domain a perfect electric conductor (PEC) equation (15), a perfect magnetic conductor (PMC) equation (16), and an absorbing boundary conductor (ABC) equation (17), are used as conditions:

and

where H_inc and E_inc are the incident electric and magnetic fields, respectively.

2.3.2. An overview of the FEM

The finite element is a numerical technique that finds an approximate solution to a partial differential equation by decomposing it into a system of simpler equations. This method is widely used today to describe the behavior of an electromagnetic wave incident on an object with dielectric properties different from a surrounding medium. In FEM it is realized by solving the Maxwell equations (18) as a system with complex boundary conditions.

To demonstrate the method, consider a homogeneous spatial domain Ω, where the time harmonic electric field is defined by the Helmhotz wave equation. This is a typical boundary-value problem that has a form

where is a differential operator, ϕ = E is an unknown function to be computed in the region Ω, and f is a forcing term (or excitation). Boundary conditions that enclose domain Ω can be as simple as Neumann (n∇ϕ = S where n is a normal vector and S is a scalar function) or Dirichlet conditions (ϕ = S) or more impedance and radiation conditions. The computational domain Ω is divided into smaller elements, Ω_e, size of which depends on an expected variation of the unknown function. The solution for the unknown function ϕ within each element, is approximated by an expansion with a finite number of basic functions , where c_i are the unknown coefficients of the basis function v_i. The solution in this system of N equations, produces ϕ′ that is a best fit to the unknown function. Variational formulation and weighted residual formulation are two well known methods to solve this problem and find coefficients.

The variational method,^[97–99] also known as the Ritz method is based on formulating the boundary value problem (19) in terms of a functional. It can be shown for the case of self-joint and positive-definite operator L, the functional can be written in the form of

The minimization of the function F_e(ϕ′), produces cofficients c_i so ϕ′ is the best approximation to the solution.

The weighted residual method, is another way to find an approximation solution to Eq. (19). If a trial function ϕ′ is an approximate solution to Eq. (19), then substitution of it into ϕ yields a nonzero residual = ℒϕ′ − f ≠ 0. In order to obtain the best approximation of ϕ′, residual r must be as small as possible at all locations in Ω_e. To force this condition test or weighting functions w_i (where i is the number of unknown degrees of freedom in the approximation), needs to be chosen so weighted residual integrals R_i

The guess coefficient will determine the accuracy of the found approximate solution to Eq. (19) within Ω_e. To summarize the FEM, in order to solve the problems defined by partial differential equations (PDEs), the following steps should be taken:^[100] (i)

discretize the computational domain Ω into smaller elements (triangle or tetrahedral for two-dimensional (2D) or three-dimensional (3D) space);

(ii)

define an interpolation function;

(iii)

for the element, assemble the system of equations by the use of the Ritz or weighted residual method;

(iv)

solve the equations.

Nanostructures in the shapes of sphere and spherical shells are widely used in medical applications. Such shapes in 2 and 3 dimensions are shown in Figs. 1 and 2.

Fig. 1.

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	Fig. 1. Schematics of nanospheres with radius r in three (a) and two dimensions (b).

Figure 2 shows the spherical nanoshell which has a core with a radius of r₁ and a shell with a radius of r₂. Thus, the thickness of the gold shell is t = r₂ − r₁. In the nanostructure, the core contains a material with a constant refractive index like silica (n = 1.44) and gold with a thickness of t covers the core.

Fig. 2.

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	Fig. 2. Schematics of spherical nanoshell with core radius r1 and shell radius r2 in three (a) and two dimensions (b).

As mentioned before scattering the absorption efficiencies of gold spherical nanoparticles can be simulated by Mie theory. So in order to show a very high compatibility between Mie theory and FEM for spherical nanostructure some numerical values and simulated diagrams are presented and compared herein.

Table 1 shows the numerical values of absorption and scattering efficiencies of a sphere with radius 40 nm at some different wavelengths, computed by solving Mie theory and the FEM method.

Table 1.

Table 1.

Table 1. Comparison between the computed numerical values of efficiencies for spherical nanoparticles with radius 40 nm at different wavelengths, obtained by Mie theory and FEM. . λ/nm Qabs Mie Qabs FEM Qsca Mie Qsca FEM 400 2.153206514 2.16×100 0.8997864799 0.90457 450 2.235753233 2.23×100 0.7005297957 0.69783 500 2.575689343 2.58×100 0.7188831486 0.73017 550 3.645651195 3.61×100 2.817775445 2.80392 600 1.044624564 1.04×100 1.714225378 1.73094 650 0.2527850675 2.48×10−1 0.7574652125 0.76647 700 0.1077948317 1.07×10−1 0.4107168010 0.41858 750 0.0678623913 6.83×10−2 0.2530165931 0.26086

Table 1. Comparison between the computed numerical values of efficiencies for spherical nanoparticles with radius 40 nm at different wavelengths, obtained by Mie theory and FEM. .

Figure 3 shows the absorption and scattering efficiencies of such a sphere obtained by solving Mie theory and figure 4 shows the same things achieved by FEM. Figure 5 compares the previous two diagrams with each other. It can be seen from this figure that the FEM is highly compatible with Mie theory. In fact, we can just achieve the absorption efficiencies of a few limited types of nanostructures by using Mie theory while FEM gives us the absorption efficiencies of any different types of nanostructures. Besides, the results of such a limited range of nanostructures obtained by Mie theory are the same as the results obtained by FEM (Fig. 5). So, it would be more logical to use FEM, which is more comprehensive and practical, for all types of nanostructures.

Fig. 3.

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	Fig. 3. Absorption and scattering efficiency spectrum of gold nanosphere with radius 40 nm computed by Mie method.

Fig. 4.

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	Fig. 4. Absorption and scattering efficiency spectrum of gold nanosphere with radius 40 nm computed by FEM method.

Fig. 5.

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	Fig. 5. Comparison between efficiencies of a sphere with radius 40 nm obtained by Mie theory and FEM method.

As shown in Fig. 6, nanoshells are more appropriate options rather than nanospheres in cancer therapy because the wavelength of maximum absorption is placed in the first near-infrared window and they also have much more fractions of absorption efficiency than nanospheres. It can be seen from the diagrams of absorption efficiency of nanoshells that by increasing the refractive index of the core from 1 to 1.5, the absorption peak height gradually decreases and the peak shifts towards longer wavelengths. Scientists usually use silica–Au nanoshells (with n = 1.44 for silica) in order to access more fractions of absorption efficiency and to put the efficiency peak into the near-infrared region. By comparing the absorption efficiencies of nanoshells with different cores in Fig. 6, it is observed that due to the highest absorption efficiency and the absence of inconsistent core with the tissues, the hollow nanoshells with such sizes are the best options in treating the cancerous tissues which are not deep. Therefore, there is no need to use silica nanoshells with toxic cores in medical applications.

Figures 6 and 7 show that by increasing the radius from 20-nm to 40-nm absorption peaks of gold nanospheres are still outside the near-infrared region but absorption peaks of nanoshells are in the second near-infrared window. Thus if we want to eradicate cancerous tissues in the depths of the body, the nanoshells with a shell radius of 40 nm are acceptable options. In nanoshell diagrams, increasing core refractive index leads to the continuous decrease in absorption efficiency and wavelength of maximum absorption shifts to the longer wavelength. Therefore, to eradicate cancerous tissues which are located deep in the body, hollow spherical nanoshells, which have the highest absorption efficiency, are the best options among all nanoshells, which is peculiarly true with silica–Au nanoshells used traditionally in medical applications.

Fig. 6.

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	Fig. 6. Absorption efficiency spectra of gold nanospheres with radius 20 nm and nanoshells with different core materials and radii r1 = 18 nm, r2 = 20 nm.

Fig. 7.

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	Fig. 7. Plots of absorption efficiency versus wavelength for gold nanospheres with radius 40 nm and nanoshells with different core materials and radii r1=38 nm, r2 = 40 nm.

With respect to other spherical nanoshells which have different core materials, we have hitherto proven the superiority of hollow spherical nanoshells. In the following figures, efficiency diagrams of the hollow nanoshells are drawn in two different cases to examine their optical tunability.

To present a more precise discussion, we have constant thickness and variable radius for the first case, while in the second one, we have variable thickness and constant radius.

In the first case, figure 8 shows that by increasing the shell radius (r₂) from 20 nm to 60 nm, the height of the absorption peak decreases gradually and the peak shifts to a longer wavelength, i.e., from the first window to the second window of near-infrared. If we want to use the hollow nanoshells to eradicate deeper cancerous tissues, we should use nanoshells with a larger shell radius, and if we want to kill the cancerous tissues which are not in the deeper part of body, using nanoshells with a smaller radius is more appropriate.

Fig. 8.

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	Fig. 8. Plots of absorption efficiency versus wavelength for hollow nanoshells with different shell radii (r2) and constant thickness of 2 nm.

In the second case, figure 9 shows that by reducing the thickness from 4 nm to 1 nm, the absorption peak first increases and then decreases. The wavelength of maximum absorption, however, shifts to a higher wavelength, i.e., from the first window into the second window of near-infrared. If need be we can shift the absorption peak into the first or second window by adjusting the thickness of the nanoshells.

Fig. 9.

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	Fig. 9. Plots of absorption efficiency versus wavelength for hollow nanoshells with shell radius (r2) of 20 nm and different thickness values.

Like Fig. 10, the geometry of nanorods is considered as a cylinder capped with two hemispheres on each side.

Fig. 10.

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	Fig. 10. Schematis of gold nanorod with length L and width D in three (a) and two dimensions (b).

Fig. 11.

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	Fig. 11. Schematics of nanotube with length L and width D in three (a) and two dimensions (b).

Nanotubes are rod-shaped nanoshells with a core of constant refractive index which is covered in a gold shell with thickness of t. Figure 11 provides the schematics of the nanotube. If an electromagnetic wave is irradiated onto the nanoparticles in the direction of L, the polarization is called longitudinal polarization and if it is irradiated in the direction of D, the polarization is named transverse polarization.^[101]

3.2.1. Longitudinal polarization

Figure 12 shows the spectra of absorption efficiencies for gold nanorods with a diameter of 20 nm and different lengths in the longitudinal polarization. From this figure, we can observe that by increasing the length of nanorod, the peaks of absorption efficiencies shift to longer wavelengths and the efficiencies steadily increase. Thus, if we want to adjust the absorption efficiency of nanorods into the second near-infrared window, the dimensions of nanorods should be larger than 100 nm (as mentioned before this is a disadvantage).

Fig. 12.

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	Fig. 12. Plots of absorption efficiency spectra of nanorods with the same diameter (D = 20 nm) and different lengths.

Figure 13 shows that with the same dimensions we can use nanotubes instead of nanorods, which means that without using dimensions greater than 100 nm, eradicating the deep cancerous tissues is feasible. Although in current medical applications, nanorods are widely used, our findings show that it is useful to use nanotubes in necessary circumstances.

Fig. 13.

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	Fig. 13. Plots of absorption efficiency spectra of nanorods and nanotubes of the same diameter and length, and 2-nm thickness for nanotubes.

Figure 14 shows that by increasing the core refractive index from 1 to 1.5, plasmon peaks shift to longer wavelengths, and the absorption efficiencies decrease. Thus, due to their bigger absorption efficiency, gold nanotubes are more useful for removing tumors than the other nanoshells with different core materials.

In the previous two figures, the superiority of nanotubes with respect to nanorods and other nanoshells with different core materials is observed. Now the optical tunability of gold nanotubes is analyzed in two cases.

To present the discussion more precisely, we have constant thickness and variable length for the first case, but in the second one, we have variable thickness and constant length.

Fig. 14.

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	Fig. 14. Plots of absorption efficiency spectra of gold nanoshells with given dimensions and different core materials.

In the first case, figure 15 shows that by increasing the length from 40 nm to 80 nm, the peaks of absorption efficiencies shift from the first window to the second window of near-infrared and their values increase steadily. Therefore, nanotubes with bigger length can be used in removing tumors in deep tissue.

Fig. 15.

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	Fig. 15. Plots of the absorption efficiency spectra of gold nanotubes with the same diameter and thickness and different lengths.

Fig. 16.

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	Fig. 16. Plots of absorption efficiency spectra of gold nanorod and nanotubes with the same dimensions and different thickness values.

In the second case, figure 16 shows that by reducing the thickness from 4 nm to 1 nm, the peaks of absorption efficiencies shift from the first window into the second window of near-infrared and their values decrease steadily. Therefore, by adjusting the nanotube thickness, the plasmon peak can be situated in the first or second region.

3.2.2. Transverse polarization

Figure 17 shows that increasing the length of nanotube does not induce the absorption peaks to shift, but the value of efficiency first decreases and then increases again.

Fig. 17.

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	Fig. 17. Plots of absorption efficiency spectra of nanotubes with the same diameter and thickness and different lengths in the transverse polarization.

Figure 18 shows that by reducing the thickness of nanotube, the absorption efficiencies increase and plasmon peaks enter into the first infrared window.

Fig. 18.

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	Fig. 18. Plots of the absorption efficiency spectra of gold nanorod and nanotubes with the same dimension and different thickness values for nanotubes.

By comparing the transverse polarization diagram with the longitudinal polarization diagram, it is seen that the longitudinal polarization is more efficient than the transverse polarization because the plasmon peak will have a bigger absorption efficiency and enters into the second near infrared region. Nevertheless, if we want to use the transverse polarization, using the lesser length and thickness is more acceptable for achieving better efficiency.