As already mentioned, the emission process of most fluorescent molecules is excellently described by the model of an ideal electric dipole emitter. Following Chance et al. [31], a semi-classical modeling of the electromagnetic interaction of an emitting molecule with its environment is based on solving Maxwell’s equations for the electromagnetic field in the given environment with the dipole emitter as the field’s source.

For a planar stratified environment, the geometry of the considered situation is shown in Figure 1. The emitting molecule (electric dipole emitter) is located at position **r**_{0}=(*x*_{0}, *y*_{0}, *z*_{0})=(*ρ*_{0}, *z*_{0}) in a medium with refractive index *n*_{1}. In this medium, Maxwell’s equations lead to the following determining equation for the electric field **E** at position **r**:

Figure 1: General geometry of dipole emission above an interface.

An oscillating electric dipole (red double arrow) is located at position **r**_{0}=(*ρ*_{0}, *z*_{0}) within medium “1” above an interface dividing it from a medium “2” below the interface. The figure shows the vectors that define one of the plane waves that contribute to the dipole’s electromagnetic emission. The wave vector of a directly emitted wave is ${k}_{1}^{+},$ and the wave vector of its reflection from the interface is ${k}_{1}^{-}.$ The unit vectors ${\widehat{e}}_{1\text{p}}^{\pm}$ represent the directions of the electric field vector in the plane of incidence (p-polarization), whereas the unit vector **ê**_{s} points in the direction perpendicular to the plane of incidence (s-polarization). Shown are also the wave vector and the p- and s-polarization vectors of the plane wave transmitted into medium “2”. The projections of all wave vectors into the interface are all equal and denoted by **q**. The projections along the vertical *z*-axis are ±*w*_{1} and *w*_{2}. The angle *ψ* is the angle between the horizontal projections **q** of all wave vectors and the *x*-axis. The “interface” itself can be any stack of planar layers of different materials.

$$rot\text{\hspace{0.17em}}rot\text{\hspace{0.17em}}E\mathrm{(}r\mathrm{)}-{k}_{1}^{2}\text{\hspace{0.05em}}E\mathrm{(}r\mathrm{)}=4\pi {k}_{0}^{2}\text{\hspace{0.05em}}p\text{\hspace{0.17em}}\delta \text{\hspace{0.17em}}\text{\hspace{0.05em}}\mathrm{(}r-{r}_{0}\mathrm{)}\text{.}$$(1)

On the right-hand side, the dipole emitter source is represented as the Dirac delta function. The wave vector amplitudes *k*_{1}=*n*_{1}*k*_{0} refer to medium 1 and the vacuum, respectively. Let us first consider only an infinitely extended homogeneous medium 1. Then, in Fourier space, the solution $\tilde{E}$ of the last equation is found as follows:

$$\tilde{E}\mathrm{(}k\mathrm{)}=\frac{4\pi}{{n}_{1}^{2}}\frac{{k}_{1}^{2}p-k\mathrm{(}k\cdot p\mathrm{)}}{{k}^{2}-{k}_{1}^{2}}\mathrm{exp}\mathrm{(}-ik\cdot {r}_{0}\mathrm{)}\text{\hspace{0.17em}}\text{,}$$(2)

where a tilde denotes the Fourier transform. The field in real space is then found by an inverse Fourier transform. Using Cartesian coordinates, **k**=(*q*_{x}, *q*_{y}, *w*)=(*q*, *w*), we can integrate over *w* applying Cauchy’s residue theorem. Here, only the poles ${w}_{1}=+{\mathrm{(}{k}_{1}^{2}-{q}^{2}\mathrm{)}}^{1/2}$ will be taken into account so that the solution automatically consists of only outgoing plane waves. This integration results in the so-called Weyl representation of the electric field of an oscillating electric dipole:

$$\begin{array}{l}E\mathrm{(}r\mathrm{)}=\frac{i}{2\pi {n}_{1}^{2}}{\displaystyle \int}\frac{{\text{d}}^{2}q}{{w}_{1}}[{k}_{1}^{2}p-{k}_{1}^{\pm}\mathrm{(}{k}_{1}^{\pm}\cdot p\mathrm{)}]\cdot \mathrm{exp}\{i[q\cdot \mathrm{(}\rho -{\rho}_{0}\mathrm{)}\\ \text{}+\text{\hspace{0.17em}}{w}_{1}|z-{z}_{0}|]\}\text{\hspace{0.17em}}\text{,}\end{array}$$(3)

where ${k}_{1}^{\pm}=\mathrm{(}q\mathrm{,}\text{\hspace{0.17em}}\mp {w}_{1}\mathrm{)}$ refers to the wave vectors above (*z*>*z*_{0}) and below (*z*<*z*_{0}) the emitter’s position, respectively. The two-dimensional integration d^{2}**q** extends over the whole **q**-plane; see Figure 1. This Weyl representation of the electric field is a superposition of plane waves and is thus ideally suited to study next the interaction of the dipole field with a planar horizontal interface (*z*=0).

To model such an interaction, we will use Fresnel’s formulas for the reflection and transmission coefficients of a plane electromagnetic wave interacting with an interface [32]. These coefficients are polarization dependent, and we thus separate the plane waves in Eq. (3) into their s- and p-polarization components (electric field vector either perpendicular or within the plane of incidence formed by the wave vector and the normal to the interface, respectively). The corresponding unit vectors of the electric field polarization are given by (see also Figure 1)

${\widehat{e}}_{\text{s}}=\mathrm{(}\frac{-{q}_{y}}{q}\mathrm{,}\text{\hspace{0.17em}}\frac{{q}_{x}}{q}\mathrm{,}\text{\hspace{0.17em}}0\mathrm{)}\text{\hspace{0.17em}and\hspace{0.17em}}{\widehat{e}}_{1\text{p}}^{\pm}=\frac{\pm {w}_{1}}{{k}_{1}q}\mathrm{(}{q}_{x}\mathrm{,}\text{\hspace{0.17em}}{q}_{y}\mathrm{,}\text{\hspace{0.17em}}\frac{{q}^{2}}{\pm {w}_{1}}\mathrm{)}.$

They are both perpendicular to the wave vector **k**_{1}, while **ê**_{s} is parallel to the interface (s-wave); ${\widehat{e}}_{1\text{p}}^{\pm}$ lies within the plane of incidence (p-wave). Thus, we can re-write Eq. (3) as follows:

$$\begin{array}{l}E\mathrm{(}r\mathrm{)}=\frac{i{k}_{0}^{2}}{2\pi}{\displaystyle \iint}\frac{{\text{d}}^{2}q}{{w}_{1}}[{\widehat{e}}_{1\text{p}}^{\pm}\mathrm{(}{\widehat{e}}_{1\text{p}}^{\pm}\cdot p\mathrm{)}+{\widehat{e}}_{\text{s}}\mathrm{(}{\widehat{e}}_{\text{s}}\cdot p\mathrm{)}]\cdot \mathrm{exp}\{i[q\cdot \mathrm{(}\rho -{\rho}_{0}\mathrm{)}\\ \text{}+\text{\hspace{0.17em}}{w}_{1}|z-{z}_{0}|]\}\text{\hspace{0.17em}}\text{.}\end{array}$$(4)

In this way, we have obtained, with (4), an expansion of the dipole’s electric field into plane p- and s-waves. Note that for **q**-vectors where *w*_{1} becomes purely imaginary (*q*>*k*_{1}), the plane wave amplitude decays exponentially with increasing distance from the dipole. Using Fresnel’s well-known relations, Eq. (4) now allows us to directly write down an expression for the reflected field (*z*>0):

$$\begin{array}{l}{E}_{\text{R}}\mathrm{(}r\mathrm{)}=\frac{i{k}_{0}^{2}}{2\pi}{\displaystyle \iint}\frac{{\text{d}}^{2}q}{{w}_{1}}[{\widehat{e}}_{1\text{p}}^{-}{R}_{\text{p}}\mathrm{(}{\widehat{e}}_{1\text{p}}^{+}\cdot p\mathrm{)}+{\widehat{e}}_{\text{s}}{R}_{\text{s}}\mathrm{(}{\widehat{e}}_{\text{s}}\cdot p\mathrm{)}]\cdot \mathrm{exp}\\ \text{}\{i[q\cdot \mathrm{(}\rho -{\rho}_{0}\mathrm{)}+{w}_{1}{z}_{0}+{w}_{1}z]\}\text{\hspace{0.17em}}\text{,}\end{array}$$(5)

with *R*_{p,s} denoting Fresnel’s reflection coefficients for p- and s-waves, respectively. Similarly, the electric field transmitted through the interface at *z*<0 into medium *n*_{2} is given by

$$\begin{array}{l}{E}_{\text{T}}\mathrm{(}r\mathrm{)}=\frac{i{k}_{0}^{2}}{2\pi}{\displaystyle \iint}\frac{{\text{d}}^{2}q}{{w}_{1}}[{\widehat{e}}_{2\text{p}}^{+}{T}_{\text{p}}\mathrm{(}{\widehat{e}}_{1\text{p}}^{+}\cdot p\mathrm{)}+{\widehat{e}}_{\text{s}}{T}_{\text{s}}\mathrm{(}{\widehat{e}}_{\text{s}}\cdot p\mathrm{)}]\cdot \mathrm{exp}\\ \text{}\{i[q\cdot \mathrm{(}\rho -{\rho}_{0}\mathrm{)}+{w}_{1}{z}_{0}-{w}_{2}z]\}\text{\hspace{0.17em}}\text{.}\end{array}$$(6)

Here, the unit vector

${\widehat{e}}_{2\text{p}}^{+}=\frac{{w}_{2}}{{k}_{2}q}\mathrm{(}{q}_{x}\mathrm{,}\text{\hspace{0.17em}}{q}_{y}\mathrm{,}\text{\hspace{0.17em}}\frac{{q}^{2}}{{w}_{2}}\mathrm{)}$

is perpendicular to the wave vector in medium 2 where the wave vector is given by ${k}_{2}^{+}=\{{q}_{x}\mathrm{,}\text{\hspace{0.17em}}{q}_{y}\mathrm{,}\text{\hspace{0.17em}}-\text{}{w}_{2}\}$ (Snell’s law), with ${w}_{2}={\mathrm{(}{k}_{2}^{2}-{q}^{2}\mathrm{)}}^{1/2}$ and *T*_{p,s} being Fresnel’s transmission coefficients. The phase exp(*iw*_{1}*z*_{0}) in **E**_{R} and **E**_{T} accounts for the wave propagation from the dipole’s position to the interface.

The magnitude of both fields depends on the orientation of the dipole vector **p** with respect to the interface, as is captured by the scalar products ${\widehat{e}}_{\text{1p}}^{\pm}\cdot p$ and **ê**_{s}·**p** in the above equations. The reflection and transmission coefficients *T*_{p,s} and *R*_{p,s} are functions of the refractive indices *n*_{1} and *n*_{2} and the angle of incidence (and thus *q*) of the plane waves to the interface. It should be emphasized that expressions (5) and (6) are applicable to a general stratified stack of planar layers between the emitter and a homogeneous medium below – in that case, one has to only replace the Fresnel coefficients for a single interface by those for the stratified stack.

Now, using (6), it is easy to determine the total energy emission of the emitter into the bottom half space. Note that the electric field vector of each plane-wave component traveling along the wave vector **k**_{2}={**q**, −*w*_{2}} is indeed perpendicular to **k**_{2}, since ${k}_{2}\perp {\widehat{e}}_{2\text{p}}^{+}$ and **k**_{2}⊥**ê**_{s}. Furthermore, all plane-wave components with Im(*w*_{2})>0 decay exponentially and do not contribute to the far-field radiation. For wave components where *w*_{2} is real, that is, *q*≤*k*_{2}, we can rewrite **q** and *w*_{2} in terms of the emission angles *ψ* and *θ* and find **q**=*k*_{2}{sin *θ* cos *ψ*, sin *θ* sin *ψ*, 0} and *w*_{2}=*k*_{2} cos *θ*. Here, *θ* is the angle between the downward vertical axis and **k**_{2}, and *ψ* is the angle between the *x*-axis and the projection of **k**_{2} into the (*x*, *y*)-plane. We also find ${\widehat{e}}_{2\text{p}}^{+}=\left\{\mathrm{cos}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\psi \mathrm{,}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\psi \mathrm{,}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta \right\}$ and **ê**_{s}={sin *ψ*, −cos *ψ*, 0}. For an emitter on the optical axis (*ρ*_{0}=0), (6) shows that the electric field amplitude vector connected with emission into the solid angle sin *θdθdψ* is proportional to

$\frac{i{k}_{0}^{3}}{2\pi}\frac{{w}_{2}}{{w}_{1}}[{\widehat{e}}_{2\text{p}}^{+}{T}_{\text{p}}\mathrm{(}{\widehat{e}}_{1\text{p}}^{+}\cdot p\mathrm{)}+{\widehat{e}}_{\text{s}}{T}_{\text{s}}\mathrm{(}{\widehat{e}}_{\text{s}}\cdot p\mathrm{)}]\mathrm{exp}\mathrm{(}i{w}_{1}{z}_{0}\mathrm{)}\text{\hspace{0.17em}}\text{.}$

Knowing this electric field amplitude, we can find the time-averaged energy flux density (Poynting vector) as follows:

$\begin{array}{c}{P}_{-}\mathrm{(}\theta \mathrm{,}\text{\hspace{0.17em}}\psi \mathrm{)}=\frac{c{n}_{2}}{8\pi}|E{|}^{2}\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\propto \frac{c{k}_{0}^{6}{n}_{2}}{32{\pi}^{3}}[|{T}_{p}{|}^{2}\mathrm{(}{\widehat{e}}_{1\text{p}}^{+}\cdot p{\mathrm{)}}^{2}+|{T}_{\text{s}}{|}^{2}\mathrm{(}{\widehat{e}}_{\text{s}}\cdot p{\mathrm{)}}^{2}]\cdot \left|\frac{{w}_{2}}{{w}_{1}}\right|{2}^{}\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{exp}\mathrm{(}-2\text{Im}\mathrm{(}{w}_{1}\mathrm{)}{z}_{0}\mathrm{)}\text{,}\end{array}$

where the proportionality factor is found by comparing this result with the emission of a free dipole ${S}_{0}=c{k}_{0}^{4}{n}_{1}{p}^{2}\mathrm{/}3\mathrm{,}$ which then yields

$\begin{array}{l}{P}_{-}\mathrm{(}\theta \mathrm{,}\text{\hspace{0.17em}}\psi \mathrm{)}=\frac{c{k}_{0}^{4}{n}_{2}}{8\pi}[|{T}_{\text{p}}{|}^{2}\mathrm{(}{\widehat{e}}_{1\text{p}}^{+}\cdot p{\mathrm{)}}^{2}+|{T}_{\text{s}}{|}^{2}\mathrm{(}{\widehat{e}}_{\text{s}}\cdot p{\mathrm{)}}^{2}]\cdot {\left|\frac{{w}_{2}}{{w}_{1}}\right|}^{2}\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{exp}\mathrm{(}-2\text{Im}\mathrm{(}{w}_{1}\mathrm{)}{z}_{0}\mathrm{)}\text{\hspace{0.17em}}\text{.}\end{array}$

A similar equation can be derived, starting from (4) to (5), for the emission into the upper half-space:

$\begin{array}{l}{P}_{+}\mathrm{(}\theta \mathrm{,}\text{\hspace{0.17em}}\psi \mathrm{)}=\frac{c{k}_{0}^{4}{n}_{1}}{8\pi}\mathrm{[}|\mathrm{(}{\widehat{e}}_{1\text{p}}^{-}+{R}_{\text{p}}{\widehat{e}}_{1\text{p}}^{+}{e}^{2i{w}_{1}{z}_{0}}\mathrm{)}\cdot p{|}^{2}+|\mathrm{(}1+{R}_{\text{s}}{e}^{2i{w}_{1}{z}_{0}}\mathrm{)}\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{(}{\widehat{e}}_{\text{s}}\cdot p\mathrm{)}{|}^{2}\mathrm{]}\text{\hspace{0.17em}}\text{,}\end{array}$

where *θ* is now the angle between the propagation direction and the vertical +*z*-axis.

The total emission per unit time from the dipole is obtained by integrating over the respective half spheres.

$S\mathrm{(}\alpha \mathrm{,}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{0}\mathrm{)}={\displaystyle \underset{0}{\overset{\pi /2}{\int}}}\text{d}\theta \mathrm{sin}\text{\hspace{0.17em}}\theta {\displaystyle \underset{0}{\overset{2\pi}{\int}}}\text{d}\psi \text{\hspace{0.17em}}\mathrm{(}{P}_{-}\mathrm{(}\theta \mathrm{,}\text{\hspace{0.17em}}\psi \mathrm{)}+{P}_{+}\mathrm{(}\theta \mathrm{,}\text{\hspace{0.17em}}\psi \mathrm{)}\mathrm{)}\text{\hspace{0.17em}}\text{.}$

To clarify the dependence on the elevation angle *α* we first consider the special case of a vertical dipole, *α*=0, or **p**=*p***ê**_{z}. Using $\mathrm{(}{\widehat{e}}_{1\text{p}}^{+}\cdot {\widehat{e}}_{z}\mathrm{)}=\mathrm{(}{\widehat{e}}_{1\text{p}}^{-}\cdot {\widehat{e}}_{z}\mathrm{)}=q\mathrm{/}{k}_{1}$ we find

${S}_{\perp}=\frac{c{k}_{0}{p}^{2}}{2{n}_{1}^{2}}\text{Re}\left\{{\displaystyle \underset{0}{\overset{\infty}{\int}}}\frac{\text{d}q\text{\hspace{0.05em}}{q}^{3}}{{w}_{1}}\mathrm{(}1+{R}_{\text{p}}{e}^{2i{w}_{1}{z}_{0}}\mathrm{)}\right\}\text{\hspace{0.17em}}\text{.}$

Similarly, for a parallel dipole *α*=*π*/2, or **p**=*p***ê**_{x}, we find

${S}_{\parallel}=\frac{c{k}_{0}^{3}{p}^{2}}{4}\text{Re}\{{\displaystyle \underset{0}{\overset{\infty}{\int}}}\frac{\text{d}q\text{\hspace{0.05em}}q}{{w}_{1}}\left[\frac{{w}_{1}^{2}}{{k}_{1}^{2}}\mathrm{(}1-{R}_{\text{p}}{e}^{2i{w}_{1}{z}_{0}}\mathrm{)}+\mathrm{(}1+{R}_{\text{s}}{e}^{2i{w}_{1}{z}_{0}}\mathrm{)}\right]\}\text{\hspace{0.17em}}\text{,}$

where we have taken into account that $\mathrm{(}{\widehat{e}}_{1\text{p}}^{+}\cdot {\widehat{e}}_{x}\mathrm{)}=-\mathrm{(}{\widehat{e}}_{1\text{p}}^{-}\cdot {\widehat{e}}_{x}\mathrm{)}={w}_{1}\mathrm{/}{k}_{1}\mathrm{cos}\text{\hspace{0.17em}}\psi $ and (**ê**_{s}·**ê**_{x})=sin *ψ*, with *ψ* being the angle between the *x*-axis and the vector **q**, and furthermore, that the average of cos^{2} *ψ* and sin^{2} *ψ* over the full circle is both 1/2. Finally, the emission rate of a dipole oriented at an arbitrary angle *α* with respect to the normal of the surface and at height *z*_{0} can be written as follows:

$$S\mathrm{(}\alpha \mathrm{,}\text{\hspace{0.17em}}{z}_{0}\mathrm{)}={S}_{\perp}\mathrm{(}{z}_{0}\mathrm{)}{\mathrm{cos}}^{2}\alpha +{S}_{\parallel}\mathrm{(}{z}_{0}\mathrm{)}{\mathrm{sin}}^{2}\alpha \mathrm{.}$$(7)

The average lifetime *τ*_{fl} of an ideal dipole will be inversely proportional to the just calculated emission rate. However, the fluorescence quantum yield *η* of real dyes is lower than 1, and the ratio of the lifetime *τ*_{fl} in the presence of an interface to the lifetime *τ*0 within a homogeneous medium is given by

$$\frac{{\tau}_{\text{f}\text{\hspace{0.05em}}\text{l}}\mathrm{(}\alpha \mathrm{,}\text{\hspace{0.17em}}z\mathrm{)}}{{\tau}_{0}}=\frac{{S}_{0}}{\eta S\mathrm{(}\alpha \mathrm{,}\text{\hspace{0.17em}}z\mathrm{)}+\mathrm{(}1-\eta \mathrm{)}{S}_{0}},$$(8)

where *S*_{0} is the radiative emission rate in free space with the refractive index *n*_{1} and far away from any dielectric or metal interfaces. Since a fluorescent molecule emits a spectrum of frequencies, one has to calculate the values of *S* over the full spectral range of the dye and average the result with the normalized emission spectrum as weight function. Often, we are interested in the lifetime of a dye which can freely rotate, with a rotational diffusion time much faster than the typical fluorescence decay time. In that case, one can average the decay rate, Eq. (7), over all orientations:

$$\overline{S\mathrm{(}{z}_{0}\mathrm{)}}={\u3008S\mathrm{(}\alpha \mathrm{,}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{0}\mathrm{)}\u3009}_{\alpha}=\frac{1}{3}[{S}_{\perp}\mathrm{(}{z}_{0}\mathrm{)}+2{S}_{\parallel}\mathrm{(}{z}_{0}\mathrm{)}]\text{\hspace{0.05em}}\text{.}$$(9)

Eq. (8) is the theoretical basis of MIET; it establishes a direct relationship between the measurable fluorescence lifetime and the vertical position of a fluorescing molecule above a surface. Note that there is no free fitting parameter entering this relationship. All involved quantities are well-defined absolute physical or geometric properties.

Nonetheless, the calculations of the Fresnel coefficients for arbitrary multi-layered structures and the integration over the electric field amplitude vectors require some numerical effort. We have developed a freely available software tool that allows anyone to calculate the distance dependence of the fluorescence lifetime for arbitrary planar sample structures; see [33].

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