Cooperative Effects in Plasmonics

Abstract

In this chapter, we present our results on cooperative effects in hybrid plasmonic system involving a large number of fluorophores, e.g., dye molecules or semiconductor quantum dots, situated near a plasmonic nanostructure, e.g., metal nanoparticle. The optical properties of such complex systems are governed by the plasmon-mediated coupling between the fluorophores that leads to drastic changes in the system optical properties. Specifically, we consider in some detail two manifestations of plasmon-assisted cooperative behavior near a spherical nanoparticle: (a) superradiance by an ensemble of emitters and (b) cooperative energy transfer from an ensemble of donors to an acceptor.

Appendix

Here we collect relevant some formulas for the electric field Green dyadic in the presence of metal NP. The Green dyadic satisfies Maxwell equation

$$\begin{aligned} {\varvec{\nabla }} \times {\varvec{\nabla }} \times \hat{\mathbf{G}} - k^2 \varepsilon (r) \hat{\mathbf{G}} = \hat{\mathbf{I}}, \end{aligned}$$

(15.86)

where $\varepsilon (r) =\varepsilon (\omega ) \theta (R-r)+ \varepsilon _{0}\theta (r-R)$ is local dielectric function ($\theta (x)$ is the step-function). The Green dyadic can be split into free space and Mie-scattered parts, $G_{\mu \nu }(\mathbf{r},\mathbf{r}')=G^{0}_{\mu \nu }(\mathbf{r},\mathbf{r}')+G^{s}_{\mu \nu }(\mathbf{r},\mathbf{r'})$, where the free-space Green dyadic is

$$\begin{aligned} G_{\mu \nu }^{0}(\mathbf{r}-\mathbf{r'}) =\biggl ( \delta _{\mu \nu } - \frac{ \nabla _{\mu } \nabla '_{\nu }}{k^2} \biggr ) g(\mathbf{r}-\mathbf{r}'), \end{aligned}$$

(15.87)

with

$$\begin{aligned} g(\mathbf{r})=\frac{e^{ikr}}{4\pi r} \end{aligned}$$

(15.88)

satisfying a scalar equation

$$\begin{aligned} (\triangle + k^2)g(\mathbf{r})= -\delta (\mathbf{r}). \end{aligned}$$

(15.89)

Consider first the free space part. Its near field expression can be obtained in the long wave approximation, i.e. by expanding in $kr\ll 1$. In the first order,

$$\begin{aligned} G^{0}_{\mu \nu }(\mathbf{r})=\frac{1}{4 \pi k^2 r^3} \Bigl [\frac{3 \mathbf{r}_{\mu } \mathbf{r}_{\nu }}{r ^2} - \delta _{\mu \nu } \Bigr ] + \dfrac{ik}{6\pi }\delta _{\mu \nu }. \end{aligned}$$

(15.90)

In the far field limit, i.e., $kr\gg 1$ and $kr'\ll 1$, the free-space part can be expanded via Bessel functions,

$$\begin{aligned} \frac{e^{ik|r-r'|}}{4\pi |r-r'|} = ik \sum _{lm} j_l (kr') h_l(kr) Y_{lm} (r) Y^{*}_{lm}(r'), \end{aligned}$$

(15.91)

which are approximated as

$$\begin{aligned} j_l(kr')=\frac{(kr')^l}{(2l+1)!!}, \quad h_l(kr)=(-i)^{l+1} \frac{e^{ikr}}{kr}, \end{aligned}$$

(15.92)

yielding

$$\begin{aligned} G_{\mu \nu }^{0}(\mathbf{r},\mathbf{r}')=\Bigl (\delta _{\mu \nu }&-\frac{1}{k^2} {\nabla _{\mu } \nabla _{\nu }'}\Bigr ) \frac{e^{ikr}}{r} \biggl [\frac{1}{4\pi } \nonumber \\&- \frac{ikr'}{3} \sum _m Y_{1m} (\hat{\mathbf{r}}) Y^{*}_{1m} (\hat{\mathbf{r}}')\biggr ]. \end{aligned}$$

(15.93)

After differentiation, the far field asymptotics takes the form

$$\begin{aligned} G^{0}_{\mu \nu }(\mathbf{r},\mathbf{r}')= \frac{e^{ik r}}{4\pi r} \biggl [ \delta _{\mu \nu } -\frac{4 \pi }{3} \sum _{m} \hat{\mathbf{r}}_{\mu } Y_{1m}(\hat{\mathbf{r}})\chi _{1m}^{\nu *}(\mathbf{r}') \biggr ], \end{aligned}$$

(15.94)

where we introduced $\chi _{lm}^{\mu }(\mathbf{r})=\nabla _{\mu } [r^{l} Y_{lm}(\hat{\mathbf{r}})$ and $\psi _{lm}^{\mu }(\mathbf{r})=\nabla _{\mu } [r^{-l-1}Y_{lm}(\hat{\mathbf{r}})]$.

Now turn to the scattered part of the Green dyadic derived from solution of Mie problem for electromagnetic wave scattered on single sphere,

$$\begin{aligned} G^{s}_{\mu \nu }(\mathbf{r},\mathbf{r'},\mathbf{k})&=i k \sum _{lm} \bigl [a_{l} N^{\mu }_{lm}(\mathbf{r})N^{\nu }_{lm}(\mathbf{r}') \nonumber \\&\qquad \qquad \quad + b_{l} M^{\mu }_{lm}(\mathbf{r})M^{\nu }_{lm}(\mathbf{r}') \bigr ], \end{aligned}$$

(15.95)

where the first and second terms are electric and magnetic contributions and $a_{l}$ and $b_{l}$ are the Mie coefficients. In the long wave approximation, $kR\ll 1$, the magnetic contribution in Eq.(15.95) can be neglected as $b_{l}\ll 1$ [24–27]. The Mie coefficient $a_{l}$ has a form

$$\begin{aligned} a_{l}=\frac{\varepsilon _{0} j_{l}(\rho _{0}) [\rho j_{l}(\rho )]'- \varepsilon j_{l}(\rho ) [\rho _{0} j_{l}(\rho _{0})]'}{\varepsilon _{0} h_{l}(\rho _{0}) [\rho j_{l}(\rho )]'- \varepsilon j_{l}(\rho ) [\rho _{0} h_{l}(\rho _{0})]'} \end{aligned}$$

(15.96)

where $\rho _{i}=k_{i}R, k_i=\frac{\omega }{c} \sqrt{\varepsilon _i}$, and $i=(\varepsilon ,\varepsilon _0)$. For $kR\ll 1$, it becomes

$$\begin{aligned} a_{l}=-i s_{l} \tilde{\alpha }_{l} k^{2l+1}, ~~ s_{l}=\frac{l+1}{l(2l+1) [(2l-1)!!]^2}, \end{aligned}$$

(15.97)

where

$$\begin{aligned} \tilde{\alpha }_{l}=\frac{\alpha _{l}}{1-i s_{l} k^{2l+1} \alpha _{l}}, \end{aligned}$$

(15.98)

is NP multipolar polarizability that accounts for plasmon radiative decay, and

$$\begin{aligned} \alpha _{l}=R^{2l+1} \frac{l(\varepsilon -\varepsilon _0)}{l\varepsilon +(l+1)\varepsilon _0}, \end{aligned}$$

(15.99)

is the standard NP polarizability. The function $\mathbf{N}_{lm}(\mathbf{r})$ is given by

$$\begin{aligned} \mathbf{N}_{lm}(\mathbf{r}) =\frac{1}{k\sqrt{l(l+1)}} {\varvec{\nabla }} \times \bigl [h^{(1)}_{l}(kr) \mathbf{L} Y_{lm}(\hat{\mathbf{r}})\bigl ], \end{aligned}$$

(15.100)

where $\mathbf{L} = -i (\mathbf{r} \times {\varvec{\nabla }})$ is angular momentum operator. Using the following identity,

$$\begin{aligned} {\varvec{\nabla }} \times \bigl [h^{(1)}_{l}&(kr) \mathbf{L} Y_{lm}(\hat{\mathbf{r}})\bigl ] = i \mathbf{r} k^2 h^{(1)}_l(kr) Y_{lm}(\hat{\mathbf{r}}) \nonumber \\&+ i {\varvec{\nabla }} \bigl [[kr h^{(1)\prime }_{l}(kr) + h^{(1)}_{l}(kr)] Y_{lm}(\hat{\mathbf{r}})\bigl ], \end{aligned}$$

(15.101)

prime standing for derivative, and expanding $h^{(1)}_{l}(kr)=j_{l}(kr)+i n_{l}(kr)$ in $kr$ as

$$\begin{aligned} h^{(1)}_{l}(kr)= \frac{(kr)^{l}}{(2l+1)!!} - i \frac{(2l-1)!!}{(kr)^{l+1}}, \end{aligned}$$

(15.102)

we obtain

$$\begin{aligned} \mathbf{N}_{lm}(kr)= - \frac{1}{k \sqrt{s_{l} (2l+1)}} {\varvec{\nabla }} [\varphi _{l}(kr) Y_{lm}(\hat{\mathbf{r}})], \end{aligned}$$

(15.103)

where

$$\begin{aligned} \varphi _{l}(kr) = \frac{1}{(kr)^{l+1}} - i s_{l} (kr)^{l}. \end{aligned}$$

(15.104)

Thus, for $kr\ll 1$ and $kr'\ll 1$, the scattered part of the Green dyadic has the form

$$\begin{aligned} G^{s}_{\mu \nu }(\mathbf{r},\mathbf{r'},\mathbf{k})&\approx i k \sum _{lm} \bigl [a_{l} N^{\mu }_{lm}(kr)N^{\nu }_{lm}(kr')\bigr ] \\&\approx \sum _{lm} \frac{k^{2l} \tilde{\alpha }_{l}}{2l+1} \nabla _{\mu } \bigl [\varphi _{l}(kr) Y_{lm}(\hat{\mathbf{r}})\bigr ] \nabla '_{\nu } \bigl [\varphi _{l}(kr')Y^{*}_{lm}(\hat{\mathbf{r}}')\bigr ]. \nonumber \end{aligned}$$

(15.105)

This expression can be further simplified by substituting $\tilde{\alpha }_{l}=\bar{\alpha }_{l}+ i s_{l} k^{2l+1} |\tilde{\alpha }_{l}|^2$, where $\bar{\alpha }_{l}= \alpha _{l}|1-is_{l} k^{2l+1}\alpha _{l}|^{-2}$, and keeping the first two powers of $k$

$$\begin{aligned} G^{s}_{\mu \nu }(\mathbf{r},\mathbf{r'}) =&\dfrac{1}{k^2}\sum _{lm} \frac{\bar{\alpha }_{l}}{(2l+1)} \psi _{lm}^{\mu }(\mathbf{r})\psi _{lm}^{\nu *}(\mathbf{r}') \nonumber \\&- i\frac{ks_{1}}{3} \sum _{m=-1}^{1} \Bigl [\tilde{\alpha }_{1} \bigl [\psi _{1m}^{\mu }(\mathbf{r})\chi _{1m}^{\nu *}(\mathbf{r}') + \chi _{1m}^{\mu } (\mathbf{r})\psi _{1m}^{\nu *}(\mathbf{r}')\bigr ] \nonumber \\&-|\tilde{\alpha }_{1}|^2 \psi _{1m}^{\mu }(\mathbf{r})\psi _{1m}^{\nu *}(\mathbf{r}')\Bigr ], \end{aligned}$$

(15.106)

which, after adding the free-space part of the Green dyadic and neglecting plasmon radiative decay, leads to Eq. (15.12).

For $kr\gg 1$, with help of Eqs. (15.92), (15.100), and (15.101), we easily obtain

$$\begin{aligned} \mathbf{N}_{lm}(\mathbf{r})=-\frac{(-i)^{l+1} e^{ikr}}{k \sqrt{l(l+1)}} {\varvec{\nabla }} Y_{lm}(\hat{\mathbf{r}}), \end{aligned}$$

(15.107)

and combining this expression with Eq. (15.103), we obtain the far field Green dyadic (i.e., $kr\gg 1$ and $kr'\ll 1$)

$$\begin{aligned} G^{s}_{\mu \nu }(\mathbf{r},\mathbf{r'})=-\frac{\tilde{\alpha }_1}{3} \sum _{m=-1}^{1} e^{ikr} \bigl [ \nabla _{\mu } Y_{1m}(\hat{\mathbf{r}})\bigr ] \psi _{1m}^{\nu *}(\mathbf{r'}), \end{aligned}$$

(15.108)

where we set $l=1$.