A study of two-photon florescence in metallic nanoshells

Metallic nanoparticles have the ability to absorb and emit energy in the form of coherent electron oscillations. Photons can couple to the free surface electrons in a metallic nanoparticle causing oscillations in a group of surface electrons. The quantized particles of these oscillations are called surface plasmon polaritons (SPPs). SPP resonance properties of metallic nanoparticles are dependent on the polarizability which in turn depends on the size, shape, and material of the particle itself [1–14]. Therefore, optical properties of metallic nanoparticles can be fine-tuned by altering these dimensions. This allows for the construction of specifically engineered particles with the desired optical properties that are needed for a specific task. For example [1], several types of gold nanorods were designed with dimensionalities that were resonant to wavelengths of 700–1060 nm. Although many different types of metallic shapes have been studied [2–10], metallic nanoshells (MNSs) and metallic nanorods seem to be promising due to manipulations of the SPP electric field. These structures have tunable geometry via the aspect ratio [11–14].

Recently there is a considerable interest to study the two-photon spectroscopy in metallic nanoparticles to minimize damage to bodily tissue due to small scattering within tissue. Metallic nanoparticles can be used in macrophage tracking within the human body [7]. This would involve the direct implantation of the metallic nanoparticles into the cells, or the consumption or absorption of the particle by the cells. In either case the goal is to have a particle or particles within the body of the active cell. The photo-luminescing properties of metallic nanoparticles would allow tracking of the concentration of macrophages throughout the body, and the use of two-photon processes could increase the accuracy of measurements and decrease potential negative effects on the body tissue.

In this paper we have developed a theory of the two-photon florescence (TPF) of a metallic nanoshell (MNS) in the presence of quantum emitters (QEs). The MNS is made of a metallic nanosphere as a core and a dielectric material as a shell. A very underexplored area of research is study of two-photon optical properties for MNS systems. Recently some experiments on two-photon emission have been done. For example, Xiao et al [15] measure two-photon florescence in an MNS system which is made of MNS and QEs. The MNS is made of the Au nanosphere core and the silicon dioxide (SiO₂) shell. The dielectric shell of the MNS is surrounded by various concentrations of Cadmium-Selenium quantum dots as QEs. Their experiments show that two-photon florescence in the MNS changes dramatically in the presence of quantum emitters. They also found that as the amount of quantum emitters in proximity to the gold-MNS increases, so did the relative intensity of the two-photon florescence.

In the first section1, we have surveyed the literature on the TPF on metallic nanoparticles and metallic nanoshells. In the second section 2, a theory of the two-photon florescence for a metallic nanoshell in the presence of an ensemble of quantum emitters has been developed. The dipole-dipole interaction (DDI) between QEs is calculated using many-body theory and mean field approximation. An analytical expression of the two-photon florescence is derived in the presence the dipole-dipole interaction. In the third section 3, we have compared our theory with experiments of a metallic nanoshell which is made for Au nanosphere core and the SiO₂ shell. The metallic nanoshell is surrounded by various concentrations of Cadmium-Selenium quantum dots as quantum emitters. We have also calculated the TPF as a function of the probe detuning for the different values of the DDI coupling. The effect of geometrical parameters of the QE and the MNS has also been investigated. Finally, we have summarized findings our paper in the fourth section 4.

To study the two-photon process, we apply a probe field on the MNS system. With the coupling of the probe photons and surface plasmons present in the metallic nanosphere, the SPP field is created at the interface between the core and shell. The intensity of the SPP field is huge when the probe field energy is in resonance with the SPP energy. The SPP field along with the probe field falls on the ensemble of QEs. Induced electric dipoles are created in each QE and these dipoles interact with each other via the DDI. Recently, Singh and Black [16] have studied the DDI in one-photon photoluminescence of QEs in metallic nanohybrids made of the metallic nanoprism and QEs. Using this theory, we have derived the expression of the DDI field. An analytical expression of the two-photon florescence for the MNS is also derived in the presence the DDI field.

We consider an MNS which is made from a metallic core and an outer dielectric shell. An ensemble of QEs is deposited on the surface of the MNS. This system is called a MNS system. The MNS system is deposited on a background material which can be chemical, biological dielectric materials. A schematic diagram of the MNS system is depicted in figure 1.

Zoom In Zoom Out Reset image size

To study the TPF for the MNS system, we applied a strong probe field with frequency ω_p and amplitude E_p in the MNS system. With the coupling of the probe photons and surface plasmons present in the metallic nanosphere, SPP field is created at the interface between the core and shell. The SPP field along with the probe field falls on the ensemble of QEs. Induced electric dipoles are created in each QE and these dipoles interact with each other via the dipole-dipole interaction. The electric field produced by the induced dipole of the ensemble of QEs is called the DDI field and it is denoted as $E_{ddi}^q$. This field has been evaluated later in the paper.

To calculate an analytical expression for the TPF, we need to find the expression of the second order susceptibility ($\chi _{MNS}^{\left( 2 \right)}$). Let us first calculate the second order polarization, $P_{MNS}^{\left( 2 \right)}$ for the MNS in the presence of the probe field $E_p^{}$ and the DDI field, $\,E_{ddi}^q$. The total field falling on the MNS is $E_T^m = E_p^{} + E_{ddi}^q$. Following the method of Hamura et al [17], the expression of the second order polarization is given as

$$\begin{equation}P_{MNS}^{\left( 2 \right)} = { \in _0}\chi _{MNS}^{\left( 2 \right)}\left( {{\omega _p},{\omega _p}} \right)\left( {E_p^{} + E_{ddi}^q} \right)\left( {E_p^{} + E_{ddi}^q} \right)\end{equation} \tag{ 1 }$$

where $\chi _{MNS}^{\left( 2 \right)}$ is the second order susceptibility for the MNS due to the total field $E_T^m$. The expression of the $\chi _{MNS}^{\left( 2 \right)}$ is calculated in [17, 18] and it is written as

$$\begin{equation}\chi _{MNS}^{\left( 2 \right)}\left( {{\omega _p},{\omega _p}} \right) = \left( {\frac{\alpha }{{24\left( {2{\omega _p}} \right){\gamma _m}}}} \right){\left[ {\chi _{MNS}^{\left( 1 \right)}\left( {2{\omega _p}} \right)} \right]^2}\end{equation} \tag{ 2 }$$

where α is a constant, ${\gamma _m}$ is the decay rate and $\chi _{MNS}^{\left( 1 \right)}$ is the first order susceptibility for the MNS and it is expressed as

$$\begin{equation}\chi _{MNS}^{\left( 1 \right)}\left( {2{\omega _p}} \right) = \frac{{P_{MNS}^{\left( 1 \right)}\left( {2{\omega _p}} \right)}}{{{ \in _0}E_T^m}}\end{equation} \tag{ 3 }$$

where $P_{MNS}^{\left( 1 \right)}\left( {2{\omega _p}} \right)$ is the first order polarization induced in the MNS due to the probe field.

Let us evaluate $P_{MNS}^{\left( 1 \right)}$. We consider that the refractive index of the dielectric shell and the metallic nanosphere core are denoted as ε_s and ε_m, respectively. The dielectric constant for the background material is taken as ε_b. The radius of the shell is denoted as R_s and the radius of the metallic sphere is R_m. Solving the Maxwell's equations in the quasi-static approximation [19–24] one can find the following expression for the polarization of the MNS as

$$\begin{equation}P_{MNS}^{\left( 1 \right)}.{{\,}} = 4\pi { \in _0}{ \in _b}R_s^3{g_l}{\zeta _m}E_T^m\end{equation} \tag{ 4 }$$

where

$$\begin{equation}{\varsigma _m} = \left[ {\frac{{{ \in _{ms}} - { \in _b}}}{{{ \in _{ms}} + 2{ \in _b}}}} \right],\begin{array}{*{20}{c}} &{{\in _{ms}} = {\in _m}\left[ {\frac{{R_s^3 + R_m^3\left( {\frac{{{ \in _s} - { \in _m}}}{{{ \in _s} + 2{ \in _m}}}} \right)}}{{R_s^3 - 2R_m^3\left( {\frac{{{ \in _s} - { \in _m}}}{{{ \in _s} + 2{ \in _m}}}} \right)}}} \right].} \end{array}\end{equation} \tag{ 5 }$$

The constant g_l is called the polarization parameter and it has values g_l = 1 and g_l = −2 for $P_{MNS}^{\left( 1 \right)}\parallel E_T^m$ and $P_{MNS}^{\left( 1 \right)} \bot E_T^m$, respectively. Here $E_T^m = E_p^{} + E_{ddi}^q$ is the total electric field falling on the MNS. The paramter ${\zeta _m}$ is called the MNS polarization factor which takes into account of the shape and size of the MNS. Surface plasmon polaritons are present on the interfaces of the core and shell due to the interaction between the electric fields and the surface plasmons located on the surface of the metallic sphere.

For the dielectric constant ${\in _m}$ appearing in equation (5), we used the following expression of the dielectric constant as

$$\begin{equation}{ \in _m}{{ = }}{ \in _\infty }\left( {1 - \frac{{\omega _{PL}^2}}{{\omega _p^2 + i{\gamma _m}{\omega _p}_{}^{}}}} \right)\end{equation} \tag{ 6 }$$

where ${\in _\infty }$ is the dielectric constant of metal when light frequency ${\omega _p}$ is very large and ${\omega _{PL}}$ (${\lambda _{PL}}$) is the plasmon frequency (wavelength) and ${\gamma _m}$ is decay constant responsible of the heat loss in metals.

Introducing equation (4) into equation (3) we get the expression of the first order susceptibility as

$$\begin{equation}\chi _{MNS}^{\left( 1 \right)}\left( {2{\omega _p}} \right) = 4\pi { \in _0}{ \in _b}R_s^3{g_l}{\zeta _m}\left( {2{\omega _p}} \right)(E_T^m/E_T^m).\end{equation} \tag{ 7 }$$

The expression of the second order susceptibility can be obtained by introducing equation (7) into equation (2) as

$$\begin{align}\chi _{MNS}^{\left( 2 \right)}\left( {{\omega _p},{\omega _p}} \right) = \left( {\frac{{\alpha {{\left( {4\pi { \in _0}{ \in _b}R_s^3{g_l}} \right)}^2}}}{{24\left( {2{\omega _p}} \right){\gamma _m}}}} \right){\left[ {{\zeta _m}\left( {2{\omega _p}} \right)} \right]^2}.\end{align} \tag{ 8 }$$

Note that the second order susceptibility depends on the polarization factor ${\zeta _m}$.

Now, we calculate the two-photon florescence from the MNS in terms of the second order susceptibility $\chi _{MNS}^{\left( 2 \right)}$. The intensity of the TPF can be evaluated by using the method of [18]. Following the method of this reference, the intensity of the TPF is calculated as

$$\begin{equation}{I_{MNS}} = \frac{{{\omega _p}{\omega _p}}}{{2c{ \in _0}\sqrt {{ \in _b}} }}{\left| {\chi _{MNS}^{\left( 2 \right)}\left( {{\omega _p},{\omega _p}} \right)E_T^m\left( {{\omega _p}} \right)E_T^m\left( {{\omega _p}} \right)} \right|^2}.\end{equation} \tag{ 9 }$$

The expression of the TPF can be obtained by introducing equation (8) into equation (9) as follows.

$$\begin{align}{I_{MNS}} &= \left( {\frac{{{\omega _p}{\omega _p}}}{{2c{ \in _0}\sqrt {{ \in _b}} }}} \right)\left( {\frac{{\alpha {{\left( {4\pi { \in _0}{ \in _b}R_s^3{g_l}} \right)}^2}}}{{24\left( {2{\omega _p}} \right){\gamma _m}}}} \right)\nonumber\\ &\quad\times{\left| {{{\left[ {{\zeta _m}\left( {2{\omega _p}} \right)} \right]}^2}\left( {E_p^{} + E_{ddi}^q} \right)\left( {E_p^{} + E_{ddi}^q} \right)} \right|^2}.\end{align} \tag{ 10 }$$

Note that the TPF depends on the DDI field $E_{ddi}^q$ emitted by the QEs and the polarization factor ${\zeta _m}$.

Let us calculate the DDI field ($E_{ddi}^q$) appearing in equation (10) due to the ensemble of QEs. Each dipole in the ensemble interacts with other dipoles via the DDI. A theory of the DDI between metallic nanoparticles has been calculated by Singh and Black [16] using the many-body theory. Following the method of [16], the DDI field is found as

$$\begin{equation}E_{ddi}^q = \frac{{\lambda {P_q}}}{{\left( {3 \times 4\pi { \in _0}{ \in _b}} \right)r_q^3}}\end{equation} \tag{ 11 }$$

where λ is a constant and it is directly related to the density of QEs. Here $P_q^{}$ is the polarization for the QE and $r_q^{}$ is the distance between the centre of the QE and the surface of the MNS.

We calculate $P_q^{}$ appearing in equation (11). We consider that the refractive index of QE is denoted as ε_q. and the radius of the QE is taken as $R_q^{}$. Two electric fields are falling on the QE: the probe field $E_p^{}$ and the SPP field $E_{spp}^{}.$ The polarization $P_q^{}$ in the QE is induced by these two fields. Following the method of [19, 20], we can calculate the expression of the polarization of the QE as

$$\begin{equation}{P_q} = 4\pi { \in _0}{ \in _b}R_q^3{g_l}{\zeta _q}\left( {E_p^{} + E_{spp}^{}} \right)\end{equation} \tag{ 12 }$$

where ${\varsigma _q}$ is called the QE polarization factor and it is found as

$$\begin{equation}{\varsigma _q} = \left[ {\frac{{{ \in _q} - { \in _b}}}{{{ \in _q} + 2{ \in _b}}}} \right].\end{equation} \tag{ 13 }$$

With the help of equations (12) and (11), one can calculate the DDI field from the ensemble of QEs as

$$\begin{equation}E_{ddi}^q = \Phi _{ddi}^p\left( {E_p^{} + E_{spp}^{}} \right),\begin{array}{*{20}{c}} &{} \end{array}\Phi _{ddi}^p = \frac{{\lambda {g_l}{\zeta _q}R_q^3}}{{3r_q^3}}.\end{equation} \tag{ 14 }$$

In equation (14), $E_{spp}^{}$ is the SPP electric field produced by the MNS. It can be calculated from polarization $P_{MNP}^{\left( 1 \right)}\left( {2{\omega _p}} \right)$ which is calculated in equation (4). The SPP field is calculated from [19, 20] and the expression of $E_{spp}^{}$ is written as

$$\begin{equation}{E_{spp}} = \frac{{P_{MNP}^{\left( 1 \right)}}}{{4\pi { \in _0}{ \in _b}r_{}^3}}\end{equation} \tag{ 15 }$$

where r is the distance at which the SPP field is measured. Introducing the expression $P_{MNP}^{\left( 1 \right)}$ from equation (4) into equation (15) we get the expression of the SPP field as follows.

$$\begin{equation}{E_{spp}} = \Pi _{spp}^{}{E_p},\,\Pi _{spp}^{} = \frac{{R_s^3{g_l}{\zeta _m}}}{{r_{}^3}}.\end{equation} \tag{ 16 }$$

Note that the SPP field depends on r⁻³. Here $\Pi _{spp}^{}$ is called the SPP parameter which depends on the polarization parameter ζ_m. Note that the polarization parameter has the largest value when the denomination is very small for a certain probe frequency. This frequency is called the SPP resonance frequency and it is denoted as ω_sp. We can say that the SPP field ${E_{spp}}$ and the SPP parameter $\Pi _{spp}^{}$ have huge values when the probe frequency is in resonance with the SPP frequency ω_sp. This is because both above terms depend on the polarization factor.

Now, we can calculate the DDI field by introducing the SPP field given by equation (16) into equation (14) and we get

$$\begin{equation}E_{ddi}^q = \Phi _{ddi}^{}{E_{p\,}},\,\Phi _{ddi}^{} = \left( {\Phi _{ddi}^p + \Phi _{ddi}^{spp}} \right),\,\Phi _{ddi}^{spp} = \Phi _{ddi}^p\Pi _{spp}^{}.\end{equation} \tag{ 17 }$$

Note that the DDI field made of two terms. The first term ($\Phi _{ddi}^p$) is the DDI field induced by the probe field and the second term ($\Phi _{ddi}^{spp}$) is the DDI field induced by the SPP field. The second term is the new term and it depends on the SPP parameter $\Pi _{spp}^{}$ which has a huge value when the probe frequency is in resonance with the SPP resonance energy. Therefore, the second term is many times larger than the first in this case.

We want to make a comment on the second DDI term ($\Phi _{ddi}^{spp}$). In the condensed matter physics literature, the DDI is calculated as follows. When an external probe light falls on the ensemble of particles, induced dipoles are created in these particles. These induced dipoles interact with each other and create the DDI field. Therefore, this DDI field calculated in the literature is due to the external probe field. Therefore, in our theory the DDI field induced by the external probe field is nothing but the first DDI term $\left( {\Phi _{ddi}^p} \right)$. On the other hand, in the present paper, we have found the second DDI term. The origin of this term is as follows. The external probe field induces the SPP field in the MNS. This SPP field is an internal field produced by the MNS. This internal SPP field falls on the ensemble of QEs and it induces dipoles in the QEs. Therefore, induced dipoles in the ensemble of QEs interact with each others and produce the second DDI term ($\Phi _{ddi}^{spp}$). Therefore, the second DDI field is induced by the internal SPP probe field and that is why it is a new DDI term.

Finally we can calculate the TPF by substitutng the DDI field from equation (17) into equation (10) and we get

$$\begin{equation}{I_{MNS}} = {I_0}{\left| {{{\left[ {{\zeta _m}\left( {2{\omega _p}} \right)} \right]}^2}{{\left( {{{{\Omega }}_p} + {{{\Omega }}_p}\Phi _{ddi}^p + {{{\Omega }}_p}\Phi _{ddi}^{spp}} \right)}^2}} \right|^2}\end{equation} \tag{ 18 }$$

where

$$\begin{equation}{I_0} = {\left( {\frac{{\alpha {{\left( {\pi R_s^3{g_l}} \right)}^2}}}{{48{\gamma _m}{ \in _b}c}}{{\left( {\frac{\hbar }{\mu }} \right)}^2}} \right)^2},\,{{{\Omega }}_p} = \frac{{\mu E_p^{}}}{\hbar }.\end{equation} \tag{ 19 }$$

Equation (18) is expressed in terms of the Rabi frequnecy ${{{\Omega }}_p}$ which is defined in equation (19). It is interesting to note that the expression of the TPF depends on the DDI field induced by the probe field ($\Phi _{ddi}^p$) and it also depends on the DDI field induced by the SPP field ($\Phi _{ddi}^{spp}$). The $\Phi _{ddi}^{spp}$ term is many times larger than other terms when the prove frequnecy is resonant with the SPP resonance frequency.

According to equation (18), our theory predicts that the intensity of the TPF in the MNS is enhanced in the presence of both DDI fields relative to the florescence of the MNS in isolation. Physics behind the enhancement is the presence of the dipole-dipole interaction between the ensemble of QEs. It is also found that as the concentration of the QE increases, the both DDI fields also increase. This in turn, increases the TPF as function of the concentration of QEs. Finally, our theory shows that the TPF can be tailor made via the altering of the metal compound and the size of the metallic core as well as the dielectric shell. Finally, we have compared our theory with experiments of Xiao et al [15] in the next section.

We have compared our theorical findings to experimental results obtained by Xiao et al [15] where they have measured the two-photon luminescence of an MNS systems. The MNS is made of the SiO₂ shell and a gold (Au) core. The MNS shell surrounded by various concentrations of Cadmium-Selenium quantum dots as QEs. A probe laser was directed at the sample, then the luminescence of the sample was measured for differing wavelengths of the probe field. The TPF of the MNS was measured with and without the ensemble of QEs. They also measured the TPF in the MNS by varying the concentration of the quantum dots

The physical parameters for the plasmon frequency for Au was taken as ω_PL = 9 eV [25], and the size of the MNS was taken as 40 nm in radius. The dielectric constant for Au when the frequency is very large is $\in _\infty ^{}$ = 8. The dielectric constant of the QD is 6.25 [26], and the dielectric constant for the spacer layer is $\in _s^{}$ = 1.25. The plasmon decay rate for the metal is γ_m/ω_PL = 0.5.

In figure 2, the experimental data of the TPF of the SiO₂/Au MNS of various concentrations of QD's is plotted. The intensity is measured in arbitrary units (A.U), as a function of the probe field wavelength. The triangles, crosses and open circles correspond to concentrations of QD's of 0.061, 0.037 and 0.026 respectively [15]. The solid circles are the TPF data for SiO₂/Au MNS alone without QEs. We have calculated the TPF for SiO₂/Au MNS using our theory. In the same order, the solid line, the dashed line, the dashed-dotted line, and the dotted line represent theoretical curves for data represented by triangles, crosses, open circles, and solid circles, respectively. Theoretically, it is found that as the concentration of QDs increases, there is a distinguishable increase in the intensity of the TPF. Our theoretical predictions are consistent with the experiments. One can see from figure 2 that there is a good agreement between theory and experiments.

Zoom In Zoom Out Reset image size

Physics of enhancement with QDs concentration can be explained as follows. As the concentration of QD increases the DDI field also increases. The TPF expression in our theory directly depends on the DDI field parameters ($\Phi _{ddi}^p$) and ($\Phi _{ddi}^{spp}$). It is also noted that as the concentration of QDs increases the distance between them also decreases. The DDI field depends inversly on the distnace bewteen QDs and MNS. These are two possibel reasons for the enhancement of the TPF enhancement with QD concnetrations.

In figure 2, the experimental maximums occurred very near to the theoretically predicted maximums and displayed a similar trend in that as the concentration of QD's increased the peak shifted to the longer wavelength of the probe laser. Another interesting relation is the increased sharpness of the peaks as the concentration of QD's increased. There is some discrepancy between the theory and the experimental data near the fringes of the probe field wavelengths, specifically near the longer end of the spectrum, in which our theory predicts a more gradual drop off in intensity, but the experimental data shows a much sharper decline. A possible explanation for this difference may be approximations made in equation (2) to get analytical expressions of the TPF.

Now we study the effect of the DDI coupling on the TPF intensity theoretically for a general MNS doped in an ensemble of QEs. The results are plotted in figure 3 as function of the probe detuning. Here the probe detuning is defined as $\delta = e_p^{} - e_{sp}^{}$ where $e_p^{}$ is the energy of the probe field and $e_{sp}^{}$ is the SPP resonance energy. The solid line, dash line and dash-dotted line correspond to $\Phi _{ddi}^{} = 0.2$, $\Phi _{ddi}^{} = 0.5$ and $\Phi _{ddi}^{} = 1.0$, respectively. Three-dimensional results are also plotted in figure 4 as function of the probe detuning and the DDI coupling parameter $\Phi _{ddi}^{}$. The parameters used are ω_PL = 9 eV [25], $\in _\infty ^{}$ = 9, $\in _s^{}\,=\,\in _b^{} = 1.2$, γ_m/ω_PL = 0.5 and $R_m^{}/R_s^{} = 0.95$. One can see that the TPF intensity increases as the DDI coupling increases. The DDI coupling increases as the density of the QEs increases. This means that the TPF increases as the concentration of the QEs. This is consistent with experimental data presented in figure 2.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Next, we study the effect of QE geometrical parameter such as the size (i.e. radius) on the TPF of the MNS. The effect of the size of the QE is shown in figure 5 where the TPF is plotted as function of the QE size ratio (R_q/r_q) where R_q is the diameter of the QE and r_q is the distance between the centre of the QE and the surface of the MNS. When the QE size ratio is equal to one, this means that the surface of the QE is in contact with the surface of the MNS. The detuning parameter for this curve is taken as zero (i.e. δ = 0). One can see that as the size of the QE increases the TPF also increases. This means that the DDI and SPP fields are very sensitive to the distance between the QE and the MNS. This is because the TPF depends on the DDI term ($\Phi _{ddi}^{}$) which in turn depends on the QE size ratio (R_q/r_q) as shown in equation (14).

Zoom In Zoom Out Reset image size

Further, we investigate the effect of the geometrical parameter of the core–shell of the MNS on the TPF. In figure 6, we have shown the effect of the core–shell size ratio (i.e. R_s/R_m) where R_s is the radius of the dielectric shell and R_m is the radius of the metallic nanosphere. The TPF is plotted as function of the detuning parameter for different values of the core–shell ratio. Here the dash-dotted line, the dash line and the solid line correspond to the R_s/R_m = 0.85, R_s/R_m = 0.90 and R_s/R_m = 0.95 respectively. The DDI coupling parameter is taken as $\Phi _{ddi}^{} = 1$ for this figure.

Zoom In Zoom Out Reset image size

One can see from figure 5 that as the core–shell size ratio decreases the peak of the TPF shifts to right side of the zero detuning. In the same time the height of peaks of the TPF is also decreases. This shift is because the position of the TPF peak depends the SPP frequency. As shown in equation (18), the TPF depends on the polarization paramter ${\zeta _m}$ which in turn depends on the core–shell size ratio (R_s/R_m) (see equation (5)). The SPP resonance frequnecy is calculated from the polarization paermater $\,{\zeta _m}$. Therefore, as the core–shell size ratio changes so does the SPP frequency. That is why the peak of the TPF shifts to the right due to the shift in the SPP frequency.

Finally, we make comment on our theory. We have developed an new method to study the two-photon florescence for the metallic nanoshell doped in the ensemble of QEs. This is the first time the dipole-dipole interaction has been included for the study of the TPF for MNS systems. This type of theory does not exist in plasmonic literature. It is well known fact that to develop a new method (theoretical or experimental) one must use different existing techniques and physics from previous literature. That is exactly what we did here. This is the way science and technology progresses. An analytical expression of the two-photon florescence is derived in the presence the DDI coupling for this complex system. This analytical expression will be useful for the experimentalists working in plasmonics to explain their experiments and also help them to plan new types of experiments.

Our theory also predicts that the intensity of the two-photon florescence is enhanced in the presence of quantum emitters relative to the fluorescence of the metallic nanoshell in isolation. Physics behind the enhancement is the presence of the dipole-dipole interaction between the ensemble of quantum emitters. It is also found that as the concentration of quantum emitters increases, the dipole-dipole field also increases. This in turn, increases the two-photon florescence as function of the concentration. This prediction is consistent with experiments. In our opinion it is a new, original and very interesting work on the TPF for MNSs. This work will help in the enhancement of the field of nonlinear plasmonics and will help to develop new types of nonlinear plasmonic nano-devices for nanotechnology and nanomedicine.

In conclusion, we have developed a theory of the TPF of a metallic nanoshell in the presence of QEs. The MNS is made of a metallic nanosphere as a core and a dielectric material as a shell. The high intensity SPP field along with the probe field falls on the ensemble of QEs. Induced electric dipoles are created in each QE and these dipoles interact with each other via dipole-dipole interaction. We found that DDI field is made of two terms where the first term is induced by the probe field and the second terms is induced by the SPP field. An analytical expression of the two-photon florescence is derived in the presence of two DDI fields. Our theory predicts that the intensity of the TPF in the MNS is enhanced in the presence of DDI fields relative to the florescence of the MNS in isolation. Physics behind the enhancement is the presence of the dipole-dipole interaction between the ensemble of QEs. It is also found that as the concentration of the QE increases, the TPF also increases. Finally, we have compared our theory with experiments of Xiao et al [15] in which the two-photon florescence of MNS which is made for Au nanosphere core and the silicon dioxide shell. The MNS shell is surrounded by various concentrations of Cadmium-Selenium quantum dots as QEs. It is found that there is a good agreement between theory and experiment.

In summary, we have developed a theory of the two-photon florescence for a metallic nanoshell in the presence of quantum emitters. Our theory predicts that the intensity of the two-photon florescence is enhanced in the presence of quantum emitters. It is also found that as the concentration of quantum emitters increases, the two-photon florescence also increases. Our theory is in consistent with experimental data.

One of authors (MRS) is thankful to the Natural Sciences and Engineering Research Council of Canada (NSERC) for the research grant.