A dipole near a sphere: decay rates, quenching and enhancement

A metal nanoparticle next to a fluorescent molecule can make it brighter by an order of magnitude or switch it off completely, and the difference is a few nanometres of distance. Both effects come out of the same calculation: the dipole's own field, reflected by the sphere, acts back on the dipole. The sphere's response is the set of transfer matrices already built for the plane wave; only the source changes, from a wave arriving from infinity to a singular outgoing wave at the dipole's position. Ruppin's 1982 paper did this for a molecule beside a small silver sphere, Chew's 1987 paper gave the closed expressions used ever since, and Kim, Leung and George extended them to a dipole inside a coated sphere.

The dipole's field as outgoing multipoles

An electric dipole $\mathbf{p}$ at $\mathbf{r}_d$ radiates the field of a point source, which, seen from the sphere's centre, is a sum of outgoing multipoles $\mathbf{H}_{pL}(k, \mathbf{r})$ for $r > r_d$ and of regular ones for $r < r_d$ , with known coefficients (the expansion of the free-space Green dyadic). A dipole therefore enters the transfer-matrix recursion as a jump in the amplitudes at the radius $r_d$ : between the inside and the outside of the shell that contains it, $A_{pL}$ and $B_{pL}$ change by the dipole's own multipole coefficients. The two closing conditions are regularity at the centre, $B_{pL}(1) = 0$ , and no incoming wave from infinity, $A_{pL}(N+1) = 0$ . Solving the $2\times 2$ system per channel gives the field everywhere, and in particular the reflected field at the dipole, which is what decides its fate.

Unlike a plane wave, which excites only $m = \pm 1$ , a dipole off the centre excites every $m$ for each $\ell$ , and a dipole close to the surface excites high $\ell$ strongly, because the near field of a point source is rich in short-wavelength harmonics. The sums below therefore converge slowly at small distances, and the engine extends the number of orders until they do.

Radiative and non-radiative rates

The power the dipole radiates to the far field, relative to the same dipole in the unbounded host, is a sum over orders of the squared total exterior field amplitudes at the dipole's radius. For a dipole outside a sphere at distance $r_d$ from the centre, with $z = k_h r_d$ and the Riccati–Bessel functions $\psi_\ell$ , $\xi_\ell$ , Chew's results read, for the radial (perpendicular) and tangential (parallel) orientations,

\frac{\Gamma^\mathrm{rad}_\perp}{\Gamma_0} = \frac{3}{2 z^4}\sum_{\ell=1}^{\infty} \ell(\ell+1)(2\ell+1)\,\bigl|\psi_\ell(z) + a_\ell\,\xi_\ell(z)\bigr|^2, \qquad \frac{\Gamma^\mathrm{rad}_\parallel}{\Gamma_0} = \frac{3}{4 z^2}\sum_{\ell=1}^{\infty} (2\ell+1)\left[\bigl|\psi_\ell'(z) + a_\ell\,\xi_\ell'(z)\bigr|^2 + \bigl|\psi_\ell(z) + b_\ell\,\xi_\ell(z)\bigr|^2\right],

where $a_\ell$ , $b_\ell$ are the sphere's (composite) Mie coefficients in the convention in which the total exterior field of order $\ell$ is $\psi_\ell + a_\ell\xi_\ell$ . A radial dipole drives electric multipoles only; a tangential one drives both. Far from the sphere ( $z \to \infty$ ) the brackets oscillate about $\psi_\ell$ and the rates tend to $\Gamma_0$ with a decaying ripple of period half a wavelength, the spherical version of Drexhage's oscillations.

An absorbing sphere also takes power that never reaches the far field: the dipole's near field drives currents in the metal. The non-radiative rate is this absorbed power, obtained either from Poynting's theorem on a surface just inside the sphere, or as the total rate minus the radiative one, where the total follows from the imaginary part of the reflected field at the dipole, $\Gamma^\mathrm{tot}/\Gamma_0 = 1 + \tfrac{6\pi}{k^3}\,\mathrm{Im}\,[\hat{\mathbf{p}}\cdot \mathbf{E}_\mathrm{refl}(\mathbf{r}_d)]/|\mathbf{p}|$ in the normalisation of Novotny and Hecht. PyStratify computes the three rates separately and reports the total as their sum, which is a check in itself: for a lossless sphere the non-radiative rate must vanish to machine precision.

Quenching, enhancement and the best distance

Two opposite trends meet at the surface of a metal sphere. The radiative rate is enhanced near the particle's plasmon resonance, because the sphere acts as an antenna that radiates the dipole's near field efficiently, and the enhancement reaches further out for a larger sphere. The non-radiative rate grows without bound as the dipole approaches the surface, like $d^{-3}$ in the quasi-static limit for a local metal, because the high- $\ell$ multipoles that a nearby dipole excites are dark: they store the energy in the metal and dissipate it. The quantum yield of the coupled system,

q = \frac{\Gamma^\mathrm{rad}}{\Gamma^\mathrm{rad} + \Gamma^\mathrm{nr} + \Gamma_0(1 - q_0)/q_0},

therefore collapses within the first few nanometres (this is quenching), rises to a maximum at a distance of the order of the particle's radius, and returns to the intrinsic yield $q_0$ far away. The maximum sits further out for a larger sphere, which Anger, Bharadwaj and Novotny measured by scanning a gold particle over a single molecule: an 80 nm sphere enhances the fluorescence by a factor of several at about 5 nm, a 20 nm sphere mostly quenches. Fluorescence is the product of two factors, the excitation rate, proportional to the local plane-wave intensity $|\mathbf{E}|^2$ at the molecule, and the quantum yield; both are sphere calculations, one with a plane wave and one with a dipole, and the calculator runs them with the same inputs.

Classic paper, computed live

Enhancement and quenching of fluorescence

P. Anger, P. Bharadwaj and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006), doi:10.1103/PhysRevLett.96.113002. Fig. 1a: quantum yield of a molecule beside an 80 nm gold sphere, against distance.

Decay rates and quantum yield of a radial dipole at 650 nm, 0.5–100 nm from an 80 nm gold sphere in free space: quenching at contact, radiation enhanced beyond.

Inputs: Gold ε = −12.99 + 1.09i at 650 nm (printed), i.e. n + ik = 0.1511 + 3.6073i; free space; intrinsic quantum yield 1.
Agreement: Times the plane-wave intensity at the molecule (a second run, the near field along the polarization), the fluorescence peaks at 3.5 near z = 12 nm, as in Fig. 1b; 5.3, 2.2 and 1.04 for 100, 60 and 20 nm spheres. 12 of 12 checks within tolerance
Try: Shrink the sphere to 20 nm: the quantum yield collapses over a much longer range, and Fig. 1b's enhancement disappears.

Open in calculator Recomputation script

Classic paper, computed live

A molecule beside a small silver sphere

R. Ruppin, J. Chem. Phys. 76, 1681 (1982), doi:10.1063/1.443196. Figs. 1–2: lifetime of a molecule against its distance from a silver sphere of radius 10 nm.

Decay rates of an electric dipole at 612 nm moving away from a 10 nm silver sphere in a medium of n = 1.5, radial and tangential.

Inputs: Radius 10 nm, host ε = 2.25, λ = 612 nm (printed); Johnson & Christy silver (the paper prints none), which puts the dipole plasmon at the paper's 3.05 eV.
Agreement: Lifetime (the inverse of the total rate) 0.98 at kd = 1 and no oscillation beyond; tangential maximum 1.16 at kd = 0.19 (paper ~1.12 at ~0.2). 7 of 7 checks within tolerance
Try: Switch the axis to wavelength at a fixed 10 nm gap: the l = 1 plasmon appears at ~407 nm (3.05 eV).

Open in calculator Recomputation script

Inside a sphere, and in a shell

A dipole inside a sphere, or in one of its shells, is handled by the same recursion with the jump placed in that shell; the radiative rate then involves the amplitudes transmitted to the host, and the formula gains the host's and the shell's indices as prefactors (Chew; Kim, Leung and George). This is the model of a dye molecule inside a silica bead, of an emitter in the dielectric spacer of a nanomatryoshka, and, with the sphere of the size of a cell, of the autofluorescence seen through a microscope. An emitter in a lossless dielectric sphere shows no quenching at all, only the oscillation of the radiative rate with position and the sharp enhancements at the sphere's whispering-gallery resonances, which for a large sphere of high index can reach factors of thousands over a narrow band: the Purcell effect in its original form.

What the model leaves out

The emitter is a point dipole of fixed frequency and the metal a local, homogeneous medium with a bulk permittivity; within a nanometre or two of the surface the electron gas's nonlocal response, electron spill-out and the discreteness of the lattice change the quenching law, and the calculated rate at contact is a limit rather than a prediction. A dielectric host surrounding the dipole must be lossless for the free-space rate $\Gamma_0$ to be defined. Within those limits the formulas above are exact, and the Verification page lists their checks against independently written Sommerfeld-type integrals, against smuthi and treams, and against the finite-difference and finite-element methods that know nothing about multipoles.

References

R. Ruppin, J. Chem. Phys. 76, 1681 (1982), doi:10.1063/1.443196
H. Chew, J. Chem. Phys. 87, 1355 (1987), doi:10.1063/1.453317
Y. S. Kim, P. T. Leung and T. F. George, Surf. Sci. 195, 1 (1988), doi:10.1016/0039-6028(88)90776-5
A. Moroz, Ann. Phys. 315, 352 (2005), doi:10.1016/j.aop.2004.07.002
R. Carminati, J.-J. Greffet, C. Henkel and J. M. Vigoureux, Opt. Commun. 261, 368 (2006), doi:10.1016/j.optcom.2005.12.009
P. Anger, P. Bharadwaj and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006), doi:10.1103/PhysRevLett.96.113002
L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University Press, 2012), ch. 8, doi:10.1017/CBO9780511794193