Spontaneous decay rate of an excited molecule placed near a circular aperture in a perfectly conducting screen: An analytical approach

We have investigated theoretically the spontaneous decay rate of an excited molecule placed near a circular aperture in a perfectly conducting infinitely thin plane screen. A quasistatic analytical solution for a molecule with an arbitrary position near the aperture is found. In a case with a retardation, an exact analytical solution expressed through spheroidal wave functions is obtained. Analytical results are in good agreement with numerical simulations. The results may be useful in the design and development of optical nanodevices based on the control of elementary quantum systems emission.

Figure 1

(a) A schematic of a scanning optical microscope and (b) a schematic of a plasmon biosensor based on a nanoaperture.

Figure 2

Geometry of the problem: An oscillating dipole with a dipole momentum ${\mathbf{d}}_0$ located near the nanoaperture of the radius $a$ in an infinitely thin PEC plane screen situated at $z=0$.

Figure 3

Schematic of the geometry of the numerical simulation. (a) An electric dipole with the dipole momentum ${\mathbf{d}}_0$ is located in vacuum on the symmetry axis of the hole in a PEC plane screen. The thickness of the screen is $h_p=10\mathrm{nm}$. The aperture radius is $a$. (b) A magnetic dipole with the dipole momentum ${\mathbf{m}}_0={\mathbf{d}}_0$ is located in vacuum on the symmetry axis of a PEC plane disk with a thickness of $h_d=10\mathrm{nm}$. The disk radius is $a$.

Figure 4

The decay rate $\gamma /{\gamma }_0$ of the molecule, located on the symmetry axis of the hole (the aperture radius is $a=50\mathrm{nm}$; the wavelength is ${\lambda }_0=1000\mathrm{nm}$, $z$-oriented dipole). The solid black curve is a quasistatic approximation; the black curve with dots is a direct numerical calculation for the electric dipole; the magenta curve with dots is the numerical calculations using Babinet's principle; the red curve with dots is an analytical calculation; the orange curve is the decay rate near the PEC plane without holes.

Figure 5

The decay rate $\gamma /{\gamma }_0$ of a molecule located on the symmetry axis of the aperture as a function of the distance $z_0$ to the aperture (the aperture radius is $a=50\mathrm{nm}$; the wavelength is ${\lambda }_0=1000\mathrm{nm}$, horizontal dipole momentum). The big black point is a quasistatic approximation (35); the black curve with dots is a direct numerical calculation; the magenta curve with dots is the numerical calculations using Babinet's principle; the orange curve is the decay rate near the PEC plane without holes.

Figure 6

The decay rate $\gamma /{\gamma }_0$ of the molecule with a horizontal dipole momentum located in the center of the hole as a function of the hole radius $a$. The black curve is a quasistatic approximation, the red curve with dots is an analytical calculation, and the black squares are a numerical calculation.

Figure 7

The decay rate $\gamma /{\gamma }_0$ of the molecule with vertical orientation of the electric dipole momentum and located on the axis of the aperture as a function of the coordinate $z_0$ and the hole diameter $D_0$. Babinet's principle has been used [see Eq. (37) and Fig. 3]. The wavelength is ${\lambda }_0=500\mathrm{nm}$. The thickness of the complementary disk is $h_d=10\mathrm{nm}$.

Figure 8

The decay rate $\gamma /{\gamma }_0$ of a molecule with horizontal orientation of the electric dipole momentum and located on the aperture axis as a function of the coordinate $z_0$ and the hole diameter $D_0$. The wavelength is ${\lambda }_0=500\mathrm{nm}$. The thickness of the screen is $h_p=10\mathrm{nm}$.

Figure 9

The decay rate of a molecule with the $z$-oriented dipole momentum as a function of the $x$ and $y$ coordinates on the plane $z=5\mathrm{nm}$ (quasistatic approximation). The hole diameter is 100 nm.

Figure 10

The decay rate of molecule with an $x$-oriented dipole momentum as a function of $x$ and $y$ coordinates on the plane $z=5\mathrm{nm}$ (quasistatic approximation). The diameter of the hole is 100 nm.